Gaussian Processes on Trees 9781316675779

Table of contents :
Contents......Page 6
Preface......Page 7
Acknowledgements......Page 11
1.1 Basic Issues......Page 12
1.2 Extremal Distributions......Page 13
1.3 Level-Crossings and kth Maxima......Page 23
1.4 Bibliographic Notes......Page 24
2.1 Point Processes......Page 26
2.2 Laplace functionals......Page 29
2.3 Poisson Point Processes......Page 30
2.4 Convergence of Point Processes......Page 32
2.5 Point Processes of Extremes......Page 40
2.6 Bibliographic Notes......Page 44
3 Normal Sequences......Page 45
3.1 Normal Comparison......Page 46
3.2 Applications to Extremes......Page 53
3.3 Bibliographic Notes......Page 55
4.1 Setting and Examples......Page 56
4.2 The REM......Page 58
4.3 The GREM, Two Levels......Page 60
4.4 Connection to Branching Brownian Motion......Page 65
4.5 The Galton–Watson Process......Page 66
4.6 The REM on the Galton–Watson Tree......Page 68
4.7 Bibliographic Notes......Page 70
5.1 Definition and Basics......Page 71
5.2 Rough Heuristics......Page 72
5.3 Recursion Relations......Page 74
5.4 The F-KPP Equation......Page 76
5.5 The Travelling Wave......Page 78
5.6 The Derivative Martingale......Page 81
5.7 Bibliographic Notes......Page 86
6.1 Feynman–Kac Representation......Page 87
6.2 The Maximum Principle and its Applications......Page 91
6.3 Estimates on the Linear F-KPP Equation......Page 106
6.4 Brownian Bridges......Page 109
6.5 Hitting Probabilities of Curves......Page 113
6.6 Asymptotics of Solutions of the F-KPP Equation......Page 116
6.7 Convergence Results......Page 123
6.8 Bibliographic Notes......Page 132
7.1 Limit Theorems for Solutions......Page 133
7.2 Existence of a Limiting Process......Page 138
7.3 Interpretation as Cluster Point Process......Page 143
7.4 Bibliographic Notes......Page 155
8.1 The Embedding......Page 156
8.2 Properties of the Embedding......Page 158
8.3 The q-Thinning......Page 160
8.4 Bibliographic Notes......Page 163
9.1 The Construction......Page 164
9.2 Two-Speed BBM......Page 165
9.3 Universality Below the Straight Line......Page 187
9.4 Bibliographic Notes......Page 200
References......Page 202
Index......Page 210

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C A M B R I D G E S T U D I E S I N A DVA N C E D M AT H E M AT I C S 1 6 3 Editorial Board B . B O L L O B Á S , W. F U LTO N , A . K ATO K , F. K I RWA N , P. S A R NA K , B . S I M O N , B . TOTA RO

GAUSSIAN PROCESSES ON TREES Branching Brownian motion (BBM) is a classical object in probability theory with deep connections to partial differential equations. This book highlights the connection to classical extreme value theory and to the theory of mean-field spin glasses in statistical mechanics. Starting with a concise review of classical extreme value statistics and a basic introduction to mean-field spin glasses, the author then focuses on branching Brownian motion. Here, the classical results of Bramson on the asymptotics of solutions of the F-KPP equation are reviewed in detail and applied to the recent construction of the extremal process of BBM. The extension of these results to branching Brownian motion with variable speed are then explained. As a self-contained exposition that is accessible to graduate students with some background in probability theory, this book makes a good introduction for anyone interested in accessing this exciting field of mathematics. Anton Bovier is Professor of Mathematics at the University of Bonn, Germany. He is the author of more than 130 scientific papers and two monographs, Mathematical Statistical Mechanics and Metastability: A Potential-Theoretic Approach, co-written with Frank den Hollander. Bovier is a Fellow of the Institute of Mathematical Statistics and a member of the Clusters of Excellence, Hausdorff Center for Mathematics and ImmunoSensation, both at the University of Bonn.

C A M B R I D G E S T U D I E S I N A DVA N C E D M AT H E M AT I C S Editorial Board: B. Bollobás, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon, B. Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit: www.cambridge.org/mathematics. Already published 124 K. Lux & H. Pahlings Representations of groups 125 K. S. Kedlaya p-adic differential equations 126 R. Beals & R. Wong Special functions 127 E. de Faria & W. de Melo Mathematical aspects of quantum field theory 128 A. Terras Zeta functions of graphs 129 D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, I 130 D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, II 131 D. A. Craven The theory of fusion systems 132 J. Väänänen Models and games 133 G. Malle & D. Testerman Linear algebraic groups and finite groups of Lie type 134 P. Li Geometric analysis 135 F. Maggi Sets of finite perimeter and geometric variational problems 136 M. Brodmann & R. Y. Sharp Local cohomology (2nd Edition) 137 C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, I 138 C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, II 139 B. Helffer Spectral theory and its applications 140 R. Pemantle & M. C. Wilson Analytic combinatorics in several variables 141 B. Branner & N. Fagella Quasiconformal surgery in holomorphic dynamics 142 R. M. Dudley Uniform central limit theorems (2nd Edition) 143 T. Leinster Basic category theory 144 I. Arzhantsev, U. Derenthal, J. Hausen & A. Laface Cox rings 145 M. Viana Lectures on Lyapunov exponents 146 J.-H. Evertse & K. Gy˝ory Unit equations in Diophantine number theory 147 A. Prasad Representation theory 148 S. R. Garcia, J. Mashreghi & W. T. Ross Introduction to model spaces and their operators 149 C. Godsil & K. Meagher Erd˝os–Ko–Rado theorems: Algebraic approaches 150 P. Mattila Fourier analysis and Hausdorff dimension 151 M. Viana & K. Oliveira Foundations of ergodic theory 152 V. I. Paulsen & M. Raghupathi An introduction to the theory of reproducing kernel Hilbert spaces 153 R. Beals & R. Wong Special functions and orthogonal polynomials 154 V. Jurdjevic Optimal control and geometry: Integrable systems 155 G. Pisier Martingales in Banach spaces 156 C. T. C. Wall Differential topology 157 J. C. Robinson, J. L. Rodrigo & W. Sadowski The three-dimensional Navier–Stokes equations 158 D. Huybrechts Lectures on K3 surfaces 159 H. Matsumoto & S. Taniguchi Stochastic analysis 160 A. Borodin & G. Olshanski Representations of the infinite symmetric group 161 P. Webb A course in finite group representation theory 162 C. J. Bishop & Y. Peres Fractals in probability and analysis 163 A. Bovier Gaussian processes on trees

Gaussian Processes on Trees From Spin Glasses to Branching Brownian Motion A N TO N B OV I E R University of Bonn, Germany

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi - 110002, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107160491 10.1017/9781316675779 © Anton Bovier 2017 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2017 Printed in the United Kingdom by Clays, St Ives plc, October 2016 A catalogue record for this publication is available from the British Library ISBN 978-1-107-16049-1 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication, and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

Contents

page vii x

Preface Acknowledgements 1

Extreme Value Theory for iid Sequences 1.1 Basic Issues 1.2 Extremal Distributions 1.3 Level-Crossings and kth Maxima 1.4 Bibliographic Notes

1 1 2 12 13

2

Extremal Processes 2.1 Point Processes 2.2 Laplace functionals 2.3 Poisson Point Processes 2.4 Convergence of Point Processes 2.5 Point Processes of Extremes 2.6 Bibliographic Notes

15 15 18 19 21 29 33

3

Normal Sequences 3.1 Normal Comparison 3.2 Applications to Extremes 3.3 Bibliographic Notes

34 35 42 44

4

Spin Glasses 4.1 Setting and Examples 4.2 The REM 4.3 The GREM, Two Levels 4.4 Connection to Branching Brownian Motion 4.5 The Galton–Watson Process 4.6 The REM on the Galton–Watson Tree 4.7 Bibliographic Notes

45 45 47 49 54 55 57 59

v

vi

Contents 60 60 61 63 65 67 70 75

5

Branching Brownian Motion 5.1 Definition and Basics 5.2 Rough Heuristics 5.3 Recursion Relations 5.4 The F-KPP Equation 5.5 The Travelling Wave 5.6 The Derivative Martingale 5.7 Bibliographic Notes

6

Bramson’s Analysis of the F-KPP Equation 6.1 Feynman–Kac Representation 6.2 The Maximum Principle and its Applications 6.3 Estimates on the Linear F-KPP Equation 6.4 Brownian Bridges 6.5 Hitting Probabilities of Curves 6.6 Asymptotics of Solutions of the F-KPP Equation 6.7 Convergence Results 6.8 Bibliographic Notes

76 76 80 95 98 102 105 112 121

7

The Extremal Process of BBM 7.1 Limit Theorems for Solutions 7.2 Existence of a Limiting Process 7.3 Interpretation as Cluster Point Process 7.4 Bibliographic Notes

122 122 127 132 144

8

Full Extremal Process 8.1 The Embedding 8.2 Properties of the Embedding 8.3 The q-Thinning 8.4 Bibliographic Notes

145 145 147 149 152

9

Variable Speed BBM 9.1 The Construction 9.2 Two-Speed BBM 9.3 Universality Below the Straight Line 9.4 Bibliographic Notes

153 153 154 176 189

References Index

191 199

Preface

The title of this book is owed in large part to my personal motivation to study the material I present here. It is rooted in the problem of so-called mean-field models of spin glasses. I will not go into a discussion of the physical background of these systems (see, e.g., [25]). The key mathematical objects associated with them are random functions (called Hamiltonians on some highdimensional space, e.g. {−1, 1}n . The standard model here is the Sherrington– Kirkpatrick model, introduced in a seminal paper [103] in 1972. Here the Hamiltonian can be seen as a Gaussian process indexed by the hypercube {−1, 1}n whose covariance is a function of the Hamming distance. The attempt to understand the structure of these objects has given rise to the remarkable heuristic theory of replica symmetry breaking developed by Parisi and collaborators (see the book [91]). A rigorous mathematical corroboration of this theory was obtained only rather recently through the work of Talagrand [109, 108, 110, 111], Guerra [63], Aizenman, Simms and Starr [6] and Panchenko [97], to name the most important ones. A second class of models that are significantly more approachable by rigorous mathematics was introduced by Derrida and Gardner [46, 59]. Here the Hamming distance was replaced by the lexicographic ultra-metric on {−1, 1}. The resulting class of models are called the generalised random energy models (GREM). These processes can be realised as branching random walks with Gaussian increments and thus provide the link to the general topic of this book. From branching random walks it is a small step to branching Brownian motion (BBM), a classical object of probability theory, introduced by Moyal [1, 92] in 1962. BBM has been studied over the last 50 years as a subject of interest in its own right, with seminal contributions by McKean [90], Bramson [33, 34], Lalley and Sellke [83], Chauvin and Rouault [39, 40] and others. Recently, the field has experienced a revival with many remarkable contributions and repercussions in other areas. The construction of the extremal provii

viii

Preface

cess in our work with Arguin and Kistler [11], as well as, in parallel, in that of A¨ıd´ekon, Berestycki, Brunet and Shi [5], is one highlight. Many results on BBM were extended to branching random walk, (see, e.g., A¨ıd´ekon [4] and Madaule [87]). Other developments concern the connection to extremes of the free Gaussian random field in d = 2 by Bramson and Zeitouni [36], Ding [48], Bramson, Ding and Zeitouni [37] and Biskup and Louidor [22, 21]. Another topic that turns out to be closely related is the problem of cover times for random walks, in particular in dimension two (see [14] and references therein). This book grew out of lecture notes [26] which I wrote first for a course at the 7th Prague School on Mathematical Statistical Mechanics in 2013 and much expanded for a one-semester graduate course I taught in Bonn. It is mainly motivated by the work we did with Nicola Kistler, Louis-Pierre Arguin and Lisa Hartung. The aim of the book is to give a comprehensive picture of what goes into the analysis of the extremal properties of branching Brownian motions, seen as Gaussian processes indexed by Galton–Watson trees. I attempt to be reasonably self-contained, but the reader is assumed to be familiar with graduate level probability theory and analysis. Chapters 1–3 provide some standard background material on extreme value theory, point processes and Gaussian processes. Chapter 4 gives a brief glimpse of spin glasses, in particular the REM and GREM models of Derrida which provides some important motivation. Chapter 5 introduces branching Brownian motion and its relation to the Fisher–Kolmogorov–Petrovsky–Piscounov (F-KPP) equation and states some by now classical results of Bramson and of Lalley and Sellke. Chapter 6 gives a condensed but detailed review of the analysis of the F-KPP equation that was given in Bramson’s monograph [34]. The remainder of the book is devoted to more recent work. Chapter 7 reviews the derivation and description of the extremal process of BBM contained mostly the paper [11] with Louis-Pierre Arguin and Nicola Kistler. Chapter 8 describes recent work with Lisa Hartung [29] on an extension of that work. Chapter 9 reviews two papers [27, 28] with Lisa Hartung on variable speed branching Brownian motion. The recent activities in and around BBM have triggered a number of lecture notes and reviews, that hopefully are complementary to this one. I mention the review by Gou´er´e [62] that presents and compares the approaches of Arguin et al. and A¨ıd´ekon et al. Ofer Zeitouni has lecture notes on his homepage that deal with branching random walk and the Gaussian free field [117]. Nicola Kistler has written a survey linking REM and GREM to other correlated Gaussian processes [78]. Zhan Shi’s Saint Flour lectures on branching random walks, with an emphasis on spinal decompositions, has recently appeared [104]. The

Preface

ix

lecture notes by Arguin [7] also discuss applications to cover times and the Riemann zeta function. I am deeply grateful to my collaborators on the matters of this book: LouisPierre Arguin, Lisa Hartung, Nicola Kistler and Irina Kurkova. They did the bulk of the work, and without them none of this would have been achieved. ˇ y and Lisa Hartung for pointing out various mistakes I also thank Jiˇr´ı Cern´ ˇ y and in previous versions of the manuscript. I thank Marek Biskup, Jiˇr´ı Cern´ Roman Koteck´y for organising the excellent school in Prague that ultimately triggered the idea to produce this book. Bonn, May 2016

Anton Bovier

Acknowledgements

This work was supported by the German Research Foundation (DFG) through the Collaborative Research Center 1060 The Mathematics of Emergent Effects, by the Cluster of Excellence Hausdorff Center for Mathematics (HCM) , and through the Priority Programme 1590 Probabilistic Structures in Evolution.

x

1 Extreme Value Theory for iid Sequences

The subject of this book is the analysis of extremes in a large class of Gaussian processes on trees. To get this into perspective, we start with the simplest classical situation, the theory of extremes of sequences of independent random variables. This is the subject of this opening chapter. Records and extremes not only fascinate us in all areas of life, they are also of tremendous importance. We are constantly interested in knowing how big, how small, how rainy, how hot, etc. things may possibly be. To answer such questions, an entire branch of statistics, called extreme value statistics, was developed.

1.1 Basic Issues As usual in statistics, one starts with a set of observations, or data, that correspond to partial observations of some sequence of events. Let us assume that these events are modelled as the values of some random variables, Xn , n ∈ N, taking values in the real numbers. Then (Xn , n ∈ N) is a stochastic process (with discrete time) defined on some probability space (Ω, F, P). Our first question is about the distribution of its maximum: given n ∈ N, define the maximum up to time n by n

Mn ≡ max Xi . i=1

(1.1)

We then ask for the distribution of this new random variable, i.e. we ask what is P(Mn ≤ x)? As often, one is interested in this question when n is large, i.e. we are interested in the asymptotics of this probability as n ↑ ∞. Certainly, the sequence of random variables (Mn , n ∈ N) may tend to infinity, and their distributions may have no reasonable limit. Natural questions about Mn are thus: first, can we rescale Mn in some way such that the rescaled 1

2

Extreme Value Theory for iid Sequences

variable converges to a random variable, and second, is there a universal class of distributions that arises as the distribution of the limits? To answer these questions will be our first target. A second major issue is to go beyond just the maximum value. What is the law of the maximum, the second-largest, third-largest, etc.? Is there possibly a universal law that describes these process of extremal marks? This is the second target, and we will see that there is again an answer to the affirmative.

1.2 Extremal Distributions We consider a family of real valued, independent, identically distributed (iid) random variables, Xi , i ∈ N, with common distribution function F(x) ≡ P (Xi ≤ x) .

(1.2)

Recall that, by convention, F is a non-decreasing, right-continuous function F : R → [0, 1]. Note that the distribution function of Mn is simply n    P (Mn ≤ x) = P ∀ni=1 Xi ≤ x = P (Xi ≤ x) = (F(x))n .

(1.3)

i=1

As n tends to infinity, this will converge to a trivial limit, ⎧ ⎪ ⎪ ⎨0, if F(x) < 1, n lim(F(x)) = ⎪ ⎪ ⎩1, if F(x) = 1, n↑∞

(1.4)

which simply says that any value that the variables Xi can exceed with positive probability will eventually be exceeded after sufficiently many independent trials, with probability one. As we have already indicated, to get something more interesting we must rescale. It is natural to try something similar to what is done in the central limit theorem: first subtract an n–dependent constant, then rescale by an n– dependent factor. Thus, the first question is whether one can find two sequences, (bn , n ∈ N) and (an , n ∈ N), and a non-trivial distribution function, G(x), such that lim P (an (Mn − bn )) = G(x). n↑∞

(1.5)

Example: the Gaussian Distribution In probability theory, it is always natural to start playing with the example of a Gaussian distribution. In a book about Gaussian processes, this is even more

3

1.2 Extremal Distributions

natural. So assume that Xi , i ∈ N are iid standard normal random variables, i.e. that F(x) = Φ(x), where x 1 2 e−y /2 dy. (1.6) Φ(x) ≡ √ 2π −∞ We want to compute

   n  −1 . P (an (Mn − bn ) ≤ x) = P Mn ≤ a−1 n x + bn = Φ an x + bn

(1.7)

Setting xn ≡ a−1 n x + bn , this can be written as (1 − (1 − Φ(xn ))n .

(1.8)

For this to converge, we must choose xn such that (1 − Φ(xn )) = n−1 g(x) + o(1/n),

(1.9)

lim (1 − (1 − Φ(xn ))n = e−g(x) .

(1.10)

in which case n↑∞

Thus our task is to find xn such that ∞ 1 2 e−y /2 dy = n−1 g(x). √ 2π xn

(1.11)

At this point it is very convenient to use an approximation for the function 1 − Φ(u) when u is large. Lemma 1.1

For any u > 0, we have the bounds   1 1 2 2 √ e−u /2 1 − 2u−2 ≤ 1 − Φ(u) ≤ √ e−u /2 . u 2π u 2π

(1.12)

For a proof see, e.g., [84]. Note that these bounds are going to be used over and over, and are surely worth memorising. Lemma 1.1 simplifies our problem to that of solving 1 2 (1.13) √ e−xn /2 = n−1 g(x) + o(1/n), xn 2π that is, we can try to solve −1

−2 2

−1

e− 2 (an x+bn ) e−bn /2−an x /2−an bn x n−1 g(x) = √ = √ . 2π(a−1 2π(a−1 n x + bn ) n x + bn ) 1

2

2

(1.14)

Setting x = 0, we find e−bn /2 = n−1 g(0). √ 2πbn 2

(1.15)

4

Extreme Value Theory for iid Sequences √ Let us make the ansatz bn = 2 ln n + cn . Then we get for cn e−

√ 2 ln ncn −c2n /2

=

√

√ 2π( 2 ln n + cn ).

(1.16)

It is convenient to choose g(0) = 1. Then, the leading terms for cn are given by cn = −

ln ln n + ln(4π) . √ 2 2 ln n

(1.17)

The higher–order corrections to cn can be ignored, as they do not affect the validity of (1.9). Finally, inspecting (1.14), we see that we can choose an = √ 2 ln n. Putting all things together we arrive at the following assertion. Theorem 1.2 Let Xi , i ∈ N be iid normal random variables. Let bn ≡

√

2 ln n −

and an =

ln ln n + ln(4π) , √ 2 2 ln n

√

2 ln n.

(1.18)

(1.19)

Then, for any x ∈ R, −x

lim P (an (Mn − bn ) ≤ x) = e−e . n↑∞

(1.20)

Remark It is sometimes convenient to express (1.20) in a slightly different, equivalent form. With the same constants, an , bn , define the functions un (x) ≡ bn + x/an .

(1.21)

Then −x

lim P (Mn ≤ un (x)) = e−e . n↑∞

(1.22)

This is our first result on the convergence of extremes, and the function e , called the Gumbel distribution, is the first extremal distribution that we encounter. The next question to ask is how ‘typical’ the result for the Gaussian distribution is. From the computation we see readily that we made no use of the Gaussian hypothesis to get the general form exp(−g(x)) for any possible limit distribution. The fact that g(x) = exp(−x), however, depended on the particular form of Φ. We will see next that, remarkably, only two other types of functions can occur. −e−x

1.2 Extremal Distributions

5

Some Technical Preparation Our goal is to be as general as possible with regard to the allowed distributions F. Of course we must anticipate that in some cases, no limiting distributions can be constructed (e.g. think of the case of a distribution with support on the two points 0 and 1!). Nonetheless, we are not willing to limit ourselves to random variables with continuous distribution functions, and this will introduce a little bit of complication, that, however, can be seen as a useful exercise. Before we continue, let us explain where we are heading. In the Gaussian case we have seen already that we could make certain choices at various places. In general, we can certainly multiply the constants an by a finite number and add a finite number to the choice of bn . This will clearly result in a different form of the extremal distribution, which, however, we think as morally equivalent. Thus, when classifying extremal distributions, we consider two distributions, G, F, as equivalent, if, for all x ∈ R, F(ax + b) = G(x).

(1.23)

The distributions we are looking for arise as limits of the form G(x) = lim F n (an x + bn ). n↑∞

(1.24)

We want to use the fact that such limits have particular properties, namely that, for some choices of αn , βn , for all n ∈ N and all x ∈ R, Gn (αn x + βn ) = G(x).

(1.25)

This property is called max-stability. Our program is then reduced to classifying all max-stable distributions modulo the equivalence (1.23), and determining their domains of attraction. Note the similarity of the characterisation of the Gaussian distribution as a stable distribution under addition of random variables. Recall the notion of weak convergence of distribution functions: Definition 1.3 A sequence, Fn , of probability distribution functions is said to converge weakly to a probability distribution function F, w

Fn → F,

(1.26)

lim Fn (x) = F(x)

(1.27)

if and only if n↑∞

for all points x ∈ R where F is continuous.

6

Extreme Value Theory for iid Sequences

The next thing we want to do is to define the notion of the (left-continuous) inverse of a non-decreasing, right-continuous function (that may have jumps and flat pieces). Definition 1.4 Let ψ : R → R be a monotone increasing, right-continuous function. Then the inverse function, ψ−1 : R → [−∞, +∞], is defined as ψ−1 (y) ≡ inf{x : ψ(x) ≥ y},

y ∈ R.

(1.28)

We need the following properties of ψ−1 . Lemma 1.5 Let ψ be as in the definition, and a > c and b real constants. Let H(x) ≡ ψ(ax + b) − c. Then (i) (ii) (iii) (iv) (v)

ψ−1 is left-continuous. ψ(ψ−1 (x)) ≥ x. If ψ−1 is continuous at ψ(x) ∈ R, then ψ−1 (ψ(x)) = x.  −1 −1 −1 H (y) = a ψ (y + c) − b . If G is a non-degenerate distribution function, then there exist y1 < y2 , such that G−1 (y1 ) < G−1 (y2 ).

Proof (i) First note that ψ−1 is increasing. Let yn ↑ y. Assume that limn↑∞ ψ−1 (yn ) < ψ−1 (y). This means that, for all yn , inf{x : ψ(x) ≥ yn } < inf{x : ψ(x) ≥ y}. This means that there is a number, x0 < ψ−1 (y), such that, for all n, ψ(x0 ) ≤ yn , but ψ(x0 )) > y. But this means that limn yn ≥ y, which is in contradiction to the hypothesis. Thus ψ−1 is left-continuous. (ii) This is immediate from the definition. (iii) ψ−1 (ψ(x)) = inf{x : ψ(x ) ≥ ψ(x)}, thus obviously ψ−1 (ψ(x)) ≤ x. On the other hand, for any  > 0, ψ−1 (ψ(x) + ) = inf{x : ψ(x ) ≥ ψ(x) + }. But ψ(x ) can only be strictly greater than ψ(x) if x > x, so, for any y > ψ(x), ψ−1 (y ) ≥ x. Thus, if ψ−1 is continuous at ψ(x), this implies that ψ−1 (ψ(x)) = x. (iv) The verification of the formula for the inverse of H is elementary and left as an exercise. (v) If G is not degenerate, then there exist x1 < x2 such that 0 < G(x1 ) ≡ y1 < G(x2 ) ≡ y2 ≤ 1. But then G−1 (y1 ) ≤ x1 , and G−1 (y2 ) = inf{x : G(x) ≥ G(x2 )}. If the latter equals x1 , then for all x ≥ x1 , G(x) ≥ G(x2 ), and since G is right continuous, G(x1 ) = G(x2 ), which is a contradiction. The following corollary is important. Corollary 1.6 If G is a non-degenerate distribution function, and there are constants a > 0, α > 0 and b, β ∈ R, such that, for all x ∈ R, G(ax + b) = G(αx + β),

(1.29)

1.2 Extremal Distributions

7

then a = α and b = β. Proof

Set H(x) ≡ G(ax + b). Then, by (i) of the preceding lemma, H −1 (y) = a−1 (G−1 (y) − b),

(1.30)

H −1 (y) = α−1 (G−1 (y) − β).

(1.31)

but by (1.29) also

On the other hand, by (v) of the same lemma, there are at least two values of y such that G−1 (y) are different, i.e. there are x1 < x2 such that a−1 (xi − b) = α−1 (xi − β), which obviously implies the assertion of the corollary.

(1.32) 

Remark Note that the assumption that G is non-degenerate is necessary. If, e.g., G(x) has a single jump from 0 to 1 at a point a, then it holds that G(5x − 4a) = G(x)! The next theorem is known as Khintchine’s theorem. Theorem 1.7 Let Fn , n ∈ N, be distribution functions, and let G be a nondegenerate distribution function. Let (an > 0, n ∈ N) and (bn ∈ R, n ∈ N) be sequences such that w

Fn (an x + bn ) → G(x).

(1.33)

Then there are sequences, (αn > 0, n ∈ N) and (βn ∈ R, n ∈ N), and a nondegenerate distribution function, G∗ , such that w

Fn (αn x + βn ) → G∗ (x),

(1.34)

if and only if lim a−1 n αn = a, n↑∞

lim(βn − bn )/an = b, n↑∞

(1.35)

and G∗ (x) = G(ax + b).

(1.36)

Remark This theorem says that different choices of the scaling sequences an , bn can lead only to distributions that are related by a transformation (1.36). Proof By changing Fn , we can assume for simplicity that an = 1, bn = 0. Let us first show that if αn → a, βn → b, then Fn (αn x + βn ) → G∗ (x). Let ax + b be a point of continuity of G. Write Fn (αn x + βn ) = Fn (αn x + βn ) − Fn (ax + b) + Fn (ax + b).

(1.37)

8

Extreme Value Theory for iid Sequences

By assumption, the last term converges to G(ax + b). Without loss of generality we may assume that αn x + βn is monotone increasing. We want to show that Fn (αn x + βn ) − Fn (ax + b) ↑ 0, as n ↑ ∞.

(1.38)

Otherwise, there would be a constant, δ > 0, such that along a subsequence,

 (nk , k ∈ N), limk↑∞ Fnk (αnk x + βnk ) − Fnk (ax + b) < −δ. But since αnk x + βnk ↑ 

ax+b, this implies that, for any y < ax+b, limk↑∞ Fnk (y) − Fnk (ax + b) < −δ. Now, if G is continuous at y, this implies that G(y) − G(ax + b) < −δ. But this implies that either F is discontinuous at ax + b, or there exists a neighbourhood of ax + b such that G(x) has no point of continuity within this neighbourhood. But this is impossible since a probability distribution function can only have countably many points of discontinuity. Thus, (1.38) must hold, and hence w

Fn (αn x + βn ) → G(ax + b),

(1.39)

which proves (1.34) and (1.36). Next we want to prove the converse, i.e. we want to show that (1.34) implies (1.35). Note first that (1.34) implies that the sequence αn x + βn is bounded, since otherwise there would be subsequences converging to plus or minus infinity along which Fn (αn x + βn ) would converge to 0 or 1, contradicting the assumption. This implies that the sequence has converging subsequences, αnk , βnk , along which lim Fnk (αnk x + βnk ) → G∗ (x). k↑∞

(1.40)

Then the preceding result shows that ank → a , bnk → b and G∗ (x) = G(a x + b ). Now, if the sequence does not converge, there must be another convergent subsequence ank → a , bnk → b . But then G∗ (x) = lim Fnk (αnk x + βnk ) → G(a x + b ). k↑∞

(1.41)

Thus G(a x + b ) = G(a x + b ), and so, since G is non-degenerate, Corollary 1.6 implies that a = a and b = b , contradicting the assumption that the sequences do not converge. This proves the theorem. 

Max-Stable Distributions We are now prepared to continue the search for extremal distributions. Let us define the notion of max-stable distributions. Definition 1.8 A non-degenerate probability distribution function, G, is called

1.2 Extremal Distributions

9

max-stable if there exist sequences, (an > 0, n ∈ N), (bn ∈ R, n ∈ N), such that, for all x ∈ R, Gn (a−1 n x + bn ) = G(x).

(1.42)

The next proposition gives some important equivalent formulations of maxstability and justifies the term. Proposition 1.9 (i) A probability distribution, G, is max-stable if and only if there exist sequences of probability distributions, (Fn , n ∈ N), and of constants, (an > 0, n ∈ N), (bn ∈ R, n ∈ N), such that, for all k ∈ N, 1/k (x), as n ↑ ∞, lim Fn (a−1 nk x + bnk ) = G n↑∞

(1.43)

for all x ∈ R where G is continuous. (ii) G is max-stable if and only if there exists a probability distribution function, F, and sequences, (an > 0, n ∈ N), (bn ∈ R, n ∈ N), such that lim F n (a−1 n x + bn ) = G(x), n↑∞

(1.44)

for all x ∈ R where G is continuous. Proof We first prove (i). If (1.43) holds, then, by Khintchine’s theorem, there exist constants, αk , βk , such that G1/k (x) = G(αk x + βk ),

(1.45)

for all k ∈ N, and thus G is max-stable. Conversely, if G is max-stable, set Fn = Gn , and let an , bn be the sequence that give (1.42)

1/k nk −1 = G1/k , (1.46) Fn (a−1 nk x + bnk ) = G (ank x + bnk ) which proves the existence of the sequence (Fn , n ∈ N) and of the respective constants. Now let us prove (ii). Assume first that G is max-stable. Then choose F = G. The fact that limn↑∞ F n (a−1 n x + bn ) = G(x) now follows trivially with the sequences (an , n ∈ N), (bn , n ∈ N) from the definition of max-stability. Next assume that (1.44) holds. Then, for any k ∈ N, and x a continuity point of G, lim F nk (a−1 nk x + bnk ) = G(x),

(1.47)

1/k lim F n (a−1 (x), nk x + bnk ) = G

(1.48)

n↑∞

and so n↑∞

so G is max-stable by (i)!



10

Extreme Value Theory for iid Sequences

There is a slight extension of this result. Corollary 1.10 If G is max-stable, then there exist functions, a(s) > 0, b(s) ∈ R, s ∈ R+ , such that G s (a(s)x + b(s)) = G(x). Proof

(1.49)

This follows essentially by interpolation. We have that G[ns] (a[ns] x + b[ns] ) = G(x).

(1.50)

But Gn (a[ns] x + b[ns] = G[ns]/s (a[ns] x + b[ns] )Gn−[ns]/s (a[ns] x + b[ns] ) = G1/s (x)Gn−[ns]/s (a[ns] x + b[ns] ).

(1.51)

As n ↑ ∞, the last factor tends to one (as the exponent remains bounded), and so w

Gn (a[ns] x + b[ns] ) → G1/s (x),

(1.52)

and w

Gn (an x + bn ) → G(x).

(1.53)

Thus, by Khintchine’s theorem, a[ns] /an → a(s),

(bn − b[ns] )/an → b(s),

(1.54)

and G1/s (x) = G(a(s)x + b(s)).

(1.55) 

The Extremal Types Theorem Definition 1.11 Two distribution functions, G, H, are called ‘of the same type’, if and only if there exists a > 0, b ∈ R such that, for all x ∈ R, G(x) = H(ax + b).

(1.56)

We have seen that the only distributions that can occur as extremal distributions are max-stable distributions. The next theorem classifies these distributions. Theorem 1.12 Any max-stable distribution is of the same type as one of the following three distributions:

1.2 Extremal Distributions

11

(i) The Gumbel distribution, −x

G(x) = e−e ,

∀x ∈ R.

(ii) The Fr´echet distribution with parameter α > 0, ⎧ ⎪ ⎪ if x ≤ 0 ⎨0, G(x) = ⎪ ⎪ ⎩e−x−α , if x > 0. (iii) The Weibull distribution with parameter α > 0, ⎧ −(−x)α ⎪ ⎪ , if x < 0 ⎨e G(x) = ⎪ ⎪ ⎩1, if x ≥ 0.

(1.57)

(1.58)

(1.59)

The rather lengthy proof of this theorem can be found in [84]. An immediate corollary is the following extremal types theorem. Theorem 1.13 Let Xi , i ∈ N be a sequence of iid random variables. If there exist sequences an > 0, bn ∈ R and a non-degenerate probability distribution function, G, such that w

P (an (Mn − bn ) ≤ x) → G(x),

(1.60)

then G(x) is of the same type as one of the three extremal-type distributions. Note that it is not true, of course, that, for arbitrary distributions of the variables Xi , it is possible to obtain a non-trivial limit as in (1.60). The following theorem gives necessary and sufficient conditions. We set xF ≡ sup{x : F(x) < 1}. Theorem 1.14 The following conditions are necessary and sufficient for a distribution function, F, to belong to the domain of attraction of the three extremal types: (i) Fr´echet: xF = +∞, lim t↑∞

1 − F(tx) = x−α , 1 − F(t)

∀x > 0, α > 0.

(1.61)

(ii) Weibull: xF =< +∞, lim h↓∞

1 − F(xF − xh) = xα , 1 − F(xF − h)

∀x > 0, α > 0.

(1.62)

(iii) Gumbel: ∃g(t) > 0, lim t↑xF

1 − F(t + xg(t)) = e−x , 1 − F(t)

For proof of this theorem, see [99].

∀x ∈ R.

(1.63)

12

Extreme Value Theory for iid Sequences

1.3 Level-Crossings and kth Maxima In the previous section we have answered the question of the distribution of the maximum of n iid random variables. It is natural to ask for more, i.e. for the joint distribution of the maximum, the second largest, third largest, etc. A natural variable to study is Mkn , the value of the kth largest of the first n variables Xi . It is useful to introduce here the notion of order statistics. Definition 1.15 Let X1 , . . . , Xn be real numbers. We denote by M1n , . . . , Mnn its order statistics, i.e. for some permutation, π, of n numbers, Mkn = Xπ(k) , and ≤ · · · ≤ M2n ≤ M1n ≡ Mn . Mnn ≤ Mn−1 n

(1.64)

We will also introduce the notation S n (u) ≡ #{i ≤ n : Xi > u}

(1.65)

for the number of exceedances of the level u. Obviously we have the relation   (1.66) P Mkn ≤ u = P (S n (u) < k) . The following result states that the number of exceedances of an extremal level un is Poisson distributed. Theorem 1.16 Let Xi be iid random variables with common distribution F. If un is a sequence such that lim n(1 − F(un )) = τ, n↑∞

0 < τ < ∞,

(1.67)

then k−1 s    τ . lim P Mkn ≤ un = lim P (S n (un ) < k) = e−τ n↑∞ n↑∞ s! s=0

(1.68)

Proof The proof of this lemma is quite simple. We just need to consider all possible ways to realise the event {S n (un ) = s}, namely P (S n (un ) = s] =



s       P Xi > un P X j ≤ un

{i1 ,...,i s }⊂{1,...,n} =1

j{1i ,...,i s }

  n = (1 − F(un )) s F(un )n−s s   n! 1 [n(1 − F(un ))] s F n (un ) 1−s/n . = s s! n (n − s)!

(1.69)

13

1.4 Bibliographic Notes But for any s fixed, n(1 − F(un )) → τ, F n (un ) → e−τ , s/n → 0 and Thus τs P (S n (un ) = s) → e−τ . s! Summing over all s < k gives the assertion of the theorem.

n! n s (n−s)!

→ 1.

(1.70) 

Using the same sort of reasoning, one can generalise the result above to obtain the distribution of the numbers of exceedances of several extremal levels. Theorem 1.17 Let u1n > n2n · · · > urn such that lim n(1 − F(u n )) = τ , n↑∞

= 1, . . . , r,

(1.71)

with 0 < τ1 < τ2 < . . . , < τr < ∞.

(1.72)

Then, under the assumptions of Theorem 1.16, with S ni ≡ S n (uin ),   lim P S n1 = k1 , S n2 − S n1 = k2 , . . . , S nr − S nr−1 = kr n↑∞

=

τk11 (τ2 − τ1 )k2 (τr − τr−1 )kr −τr ... e . k1 ! k2 ! kr !

(1.73)

Proof Again, we just have to count the number of arrangements that will place the desired number of variables in the respective intervals. Then   P S n1 = k1 , S n2 − S n1 = k2 , . . . , S nr − S nr−1 = kr  

k1

k2 n = 1 − F(u1n ) F(u1n ) − F(u2n ) k1 , . . . , kr

r kr n−k1 −···−kr r · · · F(ur−1 F (un ). (1.74) n ) − F(un ) Now we write

1

(1.75) F(u −1 n(1 − F(u n )) − n(1 − F(u −1 n ) − F(un ) = n )) , n

and use that n(1 − F(u n )) − n(1 − F(u −1 n )) → τ −τ −1 . Proceeding otherwise as in the proof of Theorem 1.16, we arrive at (1.73) 

1.4 Bibliographic Notes 1. The question of what the distribution of the maximum is was already posed in Nicolas Bernoulli’s Specimina artis conjectandi, ad quaestiones juris applicatae [18]. The Gaussian case was analysed by von Mises [114]. The question

14

Extreme Value Theory for iid Sequences

of the classification of extremal distributions was probably first addressed by Fr´echet [57] and a complete description of the domains of attraction of the extremal types was given by Gnedenko [60]. One of the earliest systematic expositions of extreme value theory is the book by Gumbel [64]. 2. There are a large number of textbooks on the subject of extreme value statistics, my personal favourites being those by Leadbetter, Lindgren and Rootz´en [84] and by Resnick [99]. The material in this chapter is mostly taken from [84].

2 Extremal Processes

In this chapter we develop and complete the description of the collection of ‘extremal values’ of a stochastic process. We do this in the elegant language of point processes. We begin with some background on this subject and the particular class of processes that will turn out to be fundamental, the Poisson point processes.

2.1 Point Processes Point processes are designed to describe the probabilistic structure of point sets in some metric space, for our purposes Rd . For reasons that may not be obvious immediately, a convenient way to represent a collection of points xi in Rd is by associating to them a point measure. Let us first consider a single point x ∈ Rd . On Rd we introduce the Borelsigma algebra, B ≡ B(Rd ), that is, the smallest σ–algebra that contains all open sets in the Euclidean topology of Rd . Given x ∈ Rd , we define the Dirac measure, δ x , such that, for any Borel set A ∈ B, ⎧ ⎪ ⎪ ⎨1, if x ∈ A, (2.1) δ x (A) = ⎪ ⎪ ⎩0, if x  A. Definition 2.1 A point measure on Rd is a measure, μ, on (Rd , B(Rd )), such that there exists a countable collection of points, {xi ∈ Rd , i ∈ I ⊆ N}, such that  μ= δ xi , (2.2) i∈I

and if K ⊂ R is compact, then μ(K) < ∞. d

Note that the points xi need not all be distinct. The set S μ ≡ {x ∈ Rd : μ(x)  15

16

Extremal Processes

0} is called the support of μ. A point measure is called simple if, for all x ∈ Rd , μ(x) ≤ 1. We denote by M p (Rd ) the set of all point measures on Rd . We equip this set with the σ–algebra M p (Rd ), the smallest sigma algebra that contains all subsets of M p (Rd ) of the form {μ ∈ Mm (Rd ) : μ(F) ∈ B}, where F ∈ B(Rd ) and B ∈ B(R+ ). M p (Rd ) is also characterised by saying that it is the smallest σ–algebra that makes the evaluation maps, μ → μ(F), measurable for all Borel sets F ∈ B(Rd ). Definition 2.2 A point process, N, is a random variable on some probability space (Ω, F, P) taking values in M p (Rd ), i.e. a measurable map, N : (Ω, F) → (M p (Rd ), M p (Rd )). This looks very fancy, but in reality things are quite down-to-earth: Proposition 2.3 A map N : Ω → M p (Rd ) is a point process if and only if the map N(·, F) : ω → N(ω, F) is measurable from (Ω, F) → (R+ , B(R+ )), for any Borel set F, i.e. if N(F) is a positive real random variable. Proof Let us first prove necessity, which should be obvious. In fact, since ω → N(ω, ·) is measurable into (M p (Rd ), M p (R p )) and μ → μ(F) is measurable from this space into (R+ , B(R+ )), the composition of these maps is also measurable. Next we prove sufficiency. Define the set G ≡ {A ∈ M p (Rd ) : N −1 A ∈ F}.

(2.3)

This set is a σ–algebra and N is measurable from (Ω, F) → (M p (Rd ), G) by definition. But G contains all sets of the form {μ ∈ M p (Rd ) : μ(F) ∈ B}, since N −1 {μ ∈ M p (Rd ) : μ(F) ∈ B} = {ω ∈ Ω : N(ω, F) ∈ B} ∈ F,

(2.4)

since N(·, F) is measurable. Thus G ⊃ M p (Rd ), and N is measurable a fortiori as a map from the smaller σ–algebra.  We need to find criteria for convergence of point processes. For this we recall some basic notions from standard measure theory. If B is a Borel-sigma algebra of a metric space, E, then T ⊂ B is called a Π–system if T is closed under finite intersections; G ⊂ B is called a λ–system, or a σ–additive class, if (i) E ∈ G, (ii) If A, B ∈ G, and A ⊃ B, then A \ B ∈ G, (iii) If An ∈ G and An ⊂ An+1 , then limn↑∞ An ∈ G. The following useful observation is called Dynkin’s theorem.

17

2.1 Point Processes

Theorem 2.4 If T is a Π–system and G is a λ–system, then G ⊃ T implies that G contains the smallest σ–algebra containing T . The most useful application of Dynkin’s theorem is the observation that, if two probability measures are equal on a Π–system that generates the σ– algebra, then they are equal on the σ–algebra (since the set on which the two measures coincide forms a λ–system containing T ). As a consequence we can further restrict the criteria to be verified for N to be a point process. In particular, we can restrict the class of Fs for which N(·, F) need to be measurable to bounded rectangles. Proposition 2.5 Suppose that T are relatively compact sets in B satisfying (i) T is a Π–system, (ii) The smallest σ–algebra containing T is B, (iii) Either there exists En ∈ T such that En ↑ E, or there exists a partition, {En }, of E with ∪n En = E, with En ⊂ T . Then N is a point process on (Ω, F) in (E, B) if and only if the map N(·, I) : ω → N(ω, I) is measurable for any I ∈ T . It is easy to check that the set of all finite collections of bounded (semi-open) rectangles indeed forms a Π–system for E = Rd that satisfies the hypotheses of Proposition 2.5. Corollary 2.6 Let T satisfy the hypothesis of Proposition 2.5 and set   (2.5) G ≡ {μ : μ(I j ) = n j , 1 ≤ j ≤ k}, k ∈ N, I j ∈ T , n j ≥ 0 . Then the smallest σ–algebra containing G is M p (Rd ) and G is a Π–system. Next we show that the law, PN , of a point process is determined by the law of the collections of random variables N(Fn ), Fn ∈ B(Rd ). Proposition 2.7 Let N be a point process in (Rd , B(Rd ) and suppose that T is as in Proposition 2.5. Define the mass functions   (2.6) PI1 ,...,Ik (n1 , . . . , nk ) ≡ P N(I j ) = n j , ∀1 ≤ j ≤ k for k ∈ N, I j ∈ T , n j ≥ 0. Then PN is uniquely determined by the collection {PI1 ,...,Ik , k ∈ N, I j ∈ T }. We need some further notions.

(2.7)

18

Extremal Processes

Definition 2.8 Two point processes, N1 , N2 are independent, if and only if, for any collection F j ∈ B, G j ∈ B, the vectors (N1 (F j ), 1 ≤ j ≤ k)

and

(N2 (G j ), 1 ≤ j ≤ )

(2.8)

are independent random vectors. Definition 2.9 Given a point process, N, on Rd , we define the set function, λ, on (Rd , B(Rd ) by μ(F)PN (dμ), (2.9) λ(F) ≡ EN(F) = M p (Rd )

for F ∈ B. Then λ is a measure, called the intensity measure of the point process N. For measurable functions f : Rd → R+ , we define N(ω, f ) ≡ f (x)N(ω, dx).

(2.10)

Then N(·, f ) is a random variable. We have that EN( f ) = λ( f ) = f (x)λ(dx).

(2.11)

Rd

Rd

2.2 Laplace functionals If Q is a probability measure on (M p , M p ), the Laplace transform of Q is a map, ψ, from non-negative Borel functions on Rd to R+ , defined as   ψ( f ) ≡ exp − f (x)μ(dx) Q(dμ). (2.12) Mp

Rd

If N is a point process, the Laplace functional of N is

ψN ( f ) ≡ E e−N( f ) = e−N(ω, f ) P(dω)   exp − f (x)μ(dx) PN (dμ). = Mp

(2.13)

Rd

Proposition 2.10 The Laplace functional, ψN , of a point process, N, determines N uniquely.  Proof For k ≥ 1, F1 , . . . , Fk ∈ B, c1 , . . . , ck ≥ 0, let f = ki=1 ci 1Fi (x). Then N(ω, f ) =

k  i=1

ci N(ω, Fi )

(2.14)

2.3 Poisson Point Processes and

⎞⎤ ⎡ ⎛ k ⎟⎟⎟⎥⎥⎥ ⎢⎢⎢ ⎜⎜⎜  ⎢ ⎜ ψN ( f ) = E ⎢⎢⎣exp ⎜⎜⎝− ci N(Fi )⎟⎟⎟⎠⎥⎥⎥⎦ .

19

(2.15)

i=1

This is the Laplace transform of the vector (N(Fi ), 1 ≤ i ≤ k), that uniquely determines its law. Hence the proposition follows from Proposition 2.7.  One can considerably restrict the set of functions that is required for the Laplace functionals to determine the measure, as one can see from the proof above.

2.3 Poisson Point Processes The most important class of point processes for our purposes will be Poisson point processes. Definition 2.11 Let λ be a σ–finite, positive measure on Rd . Then a point process, N, is called a Poisson point process with intensity measure λ (denoted PPP(λ)), if (i) For any F ∈ B(Rd ) and k ∈ N, ⎧ k −λ(F) (λ(F)) ⎪ ⎪ ⎨e k! , P (N(F) = k) = ⎪ ⎪ ⎩0,

if λ(F) < ∞, if λ(F) = ∞.

(2.16)

(ii) If F, G ∈ B are disjoint sets, then N(F) and N(G) are independent random variables. In the next theorem we will assert the existence of a Poisson point process with any prescribed intensity measure. In the proof we will give an explicit construction of such a process. Proposition 2.12 Let λ be a σ–finite measure on Rd . (i) PPP(λ) exists, and its law is uniquely determined by the requirements of the definition. (ii) The Laplace functional of PPP(λ) is given, for f ≥ 0, by   ΨN ( f ) = exp − (1 − e− f (x) )λ(dx) . (2.17) Rd

Proof Since we know that the Laplace functional determines a point process, in order to prove that the conditions of the definition uniquely determine the

20

Extremal Processes

PPP(λ), we show that they determine the form (2.17) of the Laplace functional. Suppose that N is a PPP(λ). Let f = c1F . Then     ΨN ( f ) = E exp (−N( f )) = E exp (−cN(F)) (2.18) ∞ k  (λ(F)) −c = = e(e −1)λ(F) e−ck e−λ(F) k! k=0   − f (x) = exp − (1 − e )λ(dx) ,  which is the desired form. Next, if Fi are disjoint, and f = ki=1 ci 1Fi , it is straightforward to see that ⎡ ⎛ k ⎞⎤ k ⎢⎢⎢ ⎜⎜⎜  ⎟⎟⎟⎥⎥⎥    ΨN ( f ) = E ⎢⎢⎢⎣exp ⎜⎜⎜⎝− ci N(Fi )⎟⎟⎟⎠⎥⎥⎥⎦ = E exp (−ci N(Fi )) , (2.19) i=1

i=1

due to the independence assumption (ii); a simple calculation shows that this again yields the desired form. Finally, for general f , we can choose a sequence, fn , of the form considered, such that fn ↑ f . By monotone convergence, N( fn ) ↑ N( f ). On the other hand, since e−N(g) ≤ 1, we get from dominated convergence that



ΨN ( fn ) = E e−N( fn ) → E e−N( f ) = ΨN ( f ). (2.20) But, since 1 − e− fn (x) ↑ 1 − e− f (x) , monotone convergence gives once more     (1 − e− fn (x) )λ(dx) ↑ exp (1 − e− f (x) )λ(dx) . (2.21) ΨN ( fn ) = exp On the other hand, given the form of the Laplace functional, it is trivial to verify that the conditions of the definition hold, by choosing suitable functions f. Finally we turn to the construction of PPP(λ). Let us first consider the case when λ is a finite measure, i.e. λ(Rd ) < ∞. Then construct (i) a Poisson random variable, τ, of parameter λ(Rd ), (ii) a family, Xi , i ∈ N, of independent, Rd –valued random variables with common distribution λ/λ(Rd ). This family is independent of τ. Then set N∗ ≡

τ 

δ Xi .

(2.22)

i=1

The easiest way to verify that N ∗ is a PPP(λ) is to compute its Laplace functional. This is left as an easy exercise.

2.4 Convergence of Point Processes

21

To deal with the case when λ(Rd ) is infinite, notice first that, if PPP(λ1 ) and PPP(λ2 ) are independent Poisson point processes, then PPP(λ1 ) + PPP(λ2 ) = PPP(λ1 + λ2 ), in distribution. Thus, decompose λ into a countable sum of finite measures, λk , that are just the restriction of λ to a finite set Fk , where the Fk form a partition of Rd . Then N ∗ is just the sum of independent PPP(λk ) Nk∗ . 

2.4 Convergence of Point Processes Before we turn to applications to extremal processes, we still have to discuss the notion of convergence of point processes. As point processes are probability distributions on the space of point measures, we will naturally think of weak convergence. This means that we will say that a sequence of point processes, Nn , converges weakly to a point process, N, if, for all continuous functions, f , on the space of point measures, lim E[ f (Nn )] = E[ f (N)]. n↑∞

(2.23)

However, to understand what this means, we must discuss what continuous functions on the space of point measures are, i.e. we must introduce a topology on the set of point measures. The appropriate topology for our purposes will be that of vague convergence.

Vague convergence We consider the space R equipped with its natural Euclidean metric. Clearly Rd is a complete, separable metric space. We will denote by C0 (Rd ) the set of continuous, real valued functions on Rd that have compact support; C0+ (Rn ) denotes the subset of non-negative functions. We consider M+ (Rd ), the set of all σ–finite, positive measures on (Rd , B(Rd )). We denote by M+ (Rd ) the smallest σ–algebra of subsets of M+ (Rd ) that makes the maps m → m( f ) measurable for all f ∈ C0+ (Rd ). d

Definition 2.13 A sequence of measures, μn ∈ M+ (Rd ), converges vaguely to a measure μ ∈ M+ (Rd ) if, for all f ∈ C0+ (Rd ), μn ( f ) → μ( f ).

(2.24)

Vague convergence defines a topology on the space of measures. Typical open neighbourhoods are of the form B f1 ,..., fk , (μ) ≡ {ν ∈ M+ (Rd ) : ∀ki=1 |ν( fi ) − μ( fi )| < },

(2.25)

22

Extremal Processes

i.e. to test the contiguity of two measures, we test it on their expectations on finite collections of continuous, positive functions with compact support. Given this topology, one can of course define the corresponding Borel-sigma algebra, B(M+ (Rd )), which (fortunately) turns out to coincide with the sigma algebra M+ (Rd ) introduced before. The following properties of vague convergence are useful. Proposition 2.14 Let (μn , n ∈ N) be a sequence in M+ (Rd ). Then the following statements are equivalent: v

(i) (μn , n ∈ N) converges vaguely to μ, μn → μ. (ii) limn↑∞ μn (B) = μ(B), for all relatively compact sets B for which μ(∂B) = 0. (iii) lim supn↑∞ μn (K) ≤ μ(K) and lim inf n↑∞ μn (G) ≥ μ(G), for all compact sets K, and all open, relatively compact sets G. In the case of point measures, we would of course like to see that the points where the sequence of vaguely convergent measures are located converge. The following proposition tells us that this is true. v

Proposition 2.15 Let μn , n ∈ N and μ be in M p (Rd ), and μn → μ. Let K be a compact set with μ(∂K) = 0. Then there exists a labelling of the points of μn , for all n ≥ n(K) large enough, such that μn (· ∩ K) =

p 

δ xin ,

μ(· ∩ K) =

i=1

p 

δ xi ,

(2.26)

i=1

and (x1n , . . . , xnp ) → (x1 , . . . , x p ), as n ↑ ∞. Another useful and unsurprising fact is stated in the following proposition: Proposition 2.16 The set M p (Rd ) is vaguely closed in M+ (Rd ). Thus, in particular, the limit of a sequence of point measures will, if it exists as a σ–finite measure, again be a point measure. Finally, we need a criterion that describes relatively compact sets. Proposition 2.17 (Proposition 3.16 in [99]) A subset M of M+ (E) or M p (E) is relatively compact, if and only if one of the following holds: (i) For all f ∈ C0+ (E), sup μ( f ) < ∞.

μ∈M

(2.27)

(ii) For all relatively compact B ∈ B(E), sup μ(B) < ∞.

μ∈M

(2.28)

23

2.4 Convergence of Point Processes For the proof, see [99].

Proposition 2.18 The topology of vague convergence can be metrised and turns M+ into a complete, separable metric space. Although we will not use the corresponding metric directly, it may be nice to see how this can be constructed. We therefore give a proof of the proposition that constructs such a metric. Proof The idea is to first find a countable collection of functions, hi ∈ C0+ (Rd ), v

such that μn → μ if and only if, for all i ∈ N, μn (hi ) → μ(hi ). The construction below is from [75]. Take a family Gi , i ∈ N, that forms a base of relatively compact sets, and assume it to be closed under finite unions and finite intersections. One can find (by Uryson’s lemma [99, Lemma 3.11]) families of functions fi,n , gi,n ∈ C0+ , such that fi,n ↑ 1Gi ,

gi,n ↓ 1Gi .

(2.29)

Take the countable set of functions gi,n , fi,n as the collection hi . Now μ ∈ M+ is determined by its values on the h j . First of all, μ(Gi ) is determined by these values, since μ( fi,n ) ↑ μ(Gi )

and μ(gi,n ) ↓ μ(Gi ).

(2.30)

But the family Gi is a Π–system that generates the σ–algebra B(Rd ), and so the values μ(Gi ) determine μ. v Now, μn → μ if and only if, for all hi , μn (hi ) → ci = μ(hi ). From here the idea is simple: define d(μ, ν) ≡

∞ 

  2−i 1 − e−|μ(hi )−ν(hi )| .

(2.31)

i=1

Indeed, if D(μn , μ) ↓ 0, then, for each , |μn (h )−μ(h )| ↓ 0, and conversely.  It is not very difficult to verify that this metric is complete and separable.

Weak Convergence Having established the space of σ–finite measures as a complete, separable metric space, we can think of weak convergence of probability measures on this space just as if we were working on a Euclidean space. One very useful fact about weak convergence is Skorokhod’s theorem, which relates weak convergence to almost sure convergence.

24

Extremal Processes

Theorem 2.19 Let Xn , n = 0, 1, . . . be a sequence of random variables on a complete separable metric space. Then Xn converges weakly to a random variable X0 if and only if there exists a family of random variables, Xn∗ , defined on the probability space ([0, 1], B([0, 1]), m), where m is the Lebesgue measure, such that D

(i) for each n, Xn = Xn∗ , and (ii) limn↑∞ Xn∗ = X0∗ , almost surely. For a proof, see [20]. While weak convergence usually means that the actual realisation of the sequence of random variables does not converge at all and oscillates widely, Skorokhod’s theorem says that it is possible to find an ‘equally likely’ sequence of random variables, Xn∗ , that does converge, with probability one. Such a construction is easy in the case when the random variables take values in R. In that case, we associate with the random variable Xn (whose distribution function is Fn , which for simplicity we may assume strictly increasing) the random variable Xn∗ (t) ≡ Fn−1 (t). It is easy to see that 1 ∗ 1Fn−1 (t)≤x dt = Fn (x) = P(Xn ≤ x). (2.32) m(Xn ≤ x) = 0

On the other hand, if P(Xn ≤ x) → P(X0 ≤ x) for all points of continuity of F0 , that means that, for Lebesgue almost all t, Fn−1 (t) → F0−1 (t), i.e. Xn∗ → X0∗ , m–almost surely. Skorokhod’s theorem is very useful for extracting important consequences from weak convergence. In particular, it allows one to prove the convergence of certain functionals of sequences of weakly convergent random variables, which otherwise would not be obvious. When proving weak convergence, one usually does this in two steps: (i) Prove tightness of the sequence of laws. (ii) Prove convergence of finite-dimensional distributions. We need to recall the definition of conditional compactness and tightness. Definition 2.20 Let S be a topological space. A subset, J ⊂ S , is called conditionally compact if its closure is compact. J is called conditionally sequentially compact if its closure is sequentially compact. If S is a metrisable space, then any conditionally compact set is conditionally sequentially compact. Remark The terms conditionally compact and relatively compact are used interchangeably by different authors, with the same meaning.

2.4 Convergence of Point Processes

25

The usefulness of this notion for us lies in the following observation. Assume that we are given a sequence of probability measures, μn , on some space, S . If the set {μn , n ∈ N} is conditionally sequentially compact in the weak topology, then there exist limit points, μ ∈ M1 (S ), and subsequences, nk , such that μnk → μ, in the weak topology. Definition 2.21 A subset, H ⊂ M1 (S ), is called tight if and only if there exists, for any  > 0, a compact set, K ⊂ S , such that, for all μ ∈ H, μ(K ) > 1 − .

(2.33)

Theorem 2.22 (Prohorov) If S is a Polish space, then a subset H ⊂ M1 (S ) is conditionally compact if and only if it is tight. Moreover, since the spaces M1 (S ) are metrisable, conditionally compact may be replaced by sequentially conditionally compact in statements. A useful tightness criterion is the following (see [99, Lemma 3.20]). Lemma 2.23 A sequence of point processes (ξn , n ∈ N) is tight if and only if, for any f ∈ C0+ (E), the sequence (ξn ( f ), n ∈ N) is tight as a sequence of real random variables. + (ξ Proof  all f ∈ C0 (E), the sequence n ( f ), n ∈ N) is tight.  Assume that, for + Let γi ∈ C0 (E), i ∈ N be a sequence of functions such that gi ↑ 1, as i ↑ ∞. Then, for any  > 0, there exists ci < ∞ such that

P (ξn (gi ) > ci ) ≤ 2−i .

"

(2.34)

Let M ≡ i≥1 {μ ∈ M+ (E) : μ(gi ) ≤ ci }. We claim that this implies that, for any f ∈ C0+ (E), we have that supμ∈M μ( f ) < ∞. To see this, since gi ↑ 1, for any f there will be K0 < ∞ and i0 < ∞, such that f ≤ Kgi0 , and hence sup μ( f ) ≤ sup K0 μ(gi0 ) ≤ K0 ci0 .

μ∈M

μ∈M

This implies that M is relatively compact in M+ (E). Finally,    P (ξn (gi ) > ci ) ≤ 2. P ξn  M ≤ P (ξ  M) ≤

(2.35)

(2.36)

i≥1

Hence, for any  > 0 there is a compact set such that, for all n, ξn is in this set with probability at least 1 − . This implies that ξn is tight. The converse statement is trivial.  Corollary 2.24 Let E = R. Assume that for all a the sequence (ξn (1 x>a ), n ∈ N) is tight. Then (ξn , n ∈ N) is tight. Proof

Just choose gi in the proof above as gi (x) = 1 x>ai , for ai ↑ ∞.



26

Extremal Processes

Clearly, Laplace functionals are a good tool to verify weak convergence. Proposition 2.25 Let (Nn , n ∈ N) be a sequence of point processes. Then this sequence converges weakly to N if and only if all Laplace functionals converge, more precisely if and only if, for all f ∈ C0+ (E), lim ΨNn ( f ) = ΨN ( f ). n↑∞

(2.37)

Proof We just sketch the argument. Equation (2.37) asserts that the Laplace transforms of the positive random variables Nn ( f ) converge to the Laplace transform of random variable N( f ), which implies convergence in distribution and hence tightness. On the other hand, ΨN ( f ) determines the law of N, and hence there can be only one limit point of the sequences Nn , which implies weak convergence. The converse assertion is trivial.  Of course we do not really need to check convergence for all f ∈ C0+ . For instance, in the case E = R we may choose the class of functions of the form  f (x) = k =1 c 1 x>u , k ∈ N, ci > 0, u ∈ R. Clearly, the Laplace functionals evaluated on these functions determine the Laplace transforms of the vectors (N((u1 , ∞)), . . . , Nk ((uk , ∞))), and hence the probabilities (assume the ui are an increasing sequence) P (N((u1 , ∞)) = m1 , . . . , Nk ((uk , ∞) = mk )

= P N((u1 , u2 ]) = m1 − m2 , N((u2 , u3 ]) = m2 − m3 , . . . ,  . . . , N((uk−1 , uk ]) = mk−1 − mk , N((uk , ∞)) = mk ,

(2.38)

and hence the mass functions, that we know to determine the law of N by Proposition 2.7. Another useful criterion for weak convergence of point processes is provided by Kallenberg’s theorem [75]. Theorem 2.26 Assume that ξ is a simple point process on a metric space E, and T is a Π–system of relatively compact open sets, and that, for I ∈ T , P (ξ(∂I) = 0) = 1.

(2.39)

If ξn , n ∈ N are point processes on E, and for all I ∈ T , lim P (ξn (I) = 0) = P (ξ(I) = 0) ,

(2.40)

lim E[ξn (I)] = E[ξ(I)] < ∞,

(2.41)

n↑∞

and n↑∞

then w

ξn → ξ.

(2.42)

2.4 Convergence of Point Processes

27

Remark The Π–system, T , can be chosen, in the case E = Rd , as finite unions of semi-open rectangles. Proof The key observation needed to prove the theorem is that simple point processes are uniquely determined by their avoidance function. This seems rather intuitive, in particular in the case E = R: if we know the probability that in an interval there is no point, we know the distribution of the gaps between points, and thus the distribution of the points. Let us note that we can write a point measure, μ, as  μ= cy δy , (2.43) y∈S

where S is the support of the point measure and cy are integers. We can associate to μ the simple point measure  T ∗ μ = μ∗ = δy . (2.44) y∈S ∗

Then it is true that the map T is measurable and that, if ξ1 and ξ2 are point measures such that, for all I ∈ T , P (ξ1 (I) = 0) = P (ξ2 (I) = 0) ,

(2.45)

then D

To see this, let

ξ1∗ = ξ2∗ .

(2.46)

  C ≡ {μ ∈ M p (E) : μ(I) = 0}, I ∈ T .

(2.47)

The set C is easily seen to be a Π–system. Thus, since by assumption the laws, Pi , of the point processes ξi coincide on this Π–system, they coincide on the σ–algebra generated by it. We must now check that T ∗ is measurable as a map from (M p , σ(C)) to (M p , M p ), which will hold, if, for each I, the map T 1∗ : μ → μ∗ (I) is measurable from (M p , σ(C)) to {0, 1, 2, . . . }. Now introduce a family of finite coverings of (the relatively compact set) I, An, j , with An, j ’s whose diameter is less than 1/n. We will choose the family such that, for each j, An+1, j ⊂ An,i , for some i. Then T 1∗ μ = μ∗ (I) = lim n↑∞

kn 

μ(An, j ) ∧ 1,

(2.48)

j=1

since eventually no An, j will contain more than one point of μ. Now set T 2∗ μ = (μ(An, j ) ∧ 1). Clearly, (T 2∗ )−1 {0} = {μ : μ(An, j ) = 0} ⊂ σ(C),

(2.49)

28

Extremal Processes

and so T 2∗ is measurable as desired, and so is T 1∗ , being a monotone limit of finite sums of measurable maps. But now     

P ξ1∗ ∈ B = P (T ∗ ξ1 ∈ B) = P ξ1 ∈ (T ∗ )−1 (B) = P1 (T ∗ )−1 (B) . (2.50) But since (T ∗ )−1 (B) ∈ σ(C), by hypothesis,     (2.51) P1 (T ∗ )−1 (B) = P2 (T ∗ )−1 (B) ,   which is also equal to P ξ1∗ ∈ B . This proves (2.45). Now, as we have already mentioned, (2.41) implies tightness of the sequence ξn . Thus, for any subsequence n , there exists a sub-subsequence, n , such that ξn converges weakly to a limit, η. By Proposition 2.16 this is a point process. Let us assume for a moment that (a) η is simple and (b) for any relatively compact A, P (ξ(∂A) = 0)

⇒

P (η(∂A) = 0) .

(2.52)

Then, the map μ → μ(I) is almost surely continuous with respect to η, and w therefore, if ξn → η, then P (ξn (I) = 0) → P (η(I) = 0) .

(2.53)

P (ξn (I) = 0) → P (ξ(I) = 0) ,

(2.54)

But we assumed that

so that, by the foregoing observation and the fact that both η and ξ are simple, ξ = η. It remains to check simplicity of η and (2.52). To verify the latter, we will show that, for any compact set, K, P (η(K) = 0) ≥ P (ξ(K) = 0) .

(2.55)

We use that, for any such K, there exist sequences of functions, f j ∈ C0+ (Rd ), and compact sets, K j , such that

1K ≤ f j ≤ 1K j ,

(2.56)

    P (η(K) = 0) ≥ P η( f j ) = 0 = P η( f j ) ≤ 0 .

(2.57)

and 1K j ↓ 1K . Thus,

But ξn ( f j ) converges to η( f j ), and so       P η( f j ) ≤ 0 ≥ lim sup P ξn ( f j ) ≤ 0 ≥ P ξn (K j ) ≤ 0 . n

(2.58)

2.5 Point Processes of Extremes

29

Finally, we can approximate K j by elements I j ∈ T , such that K j ⊂ I j ↓ K, so that       P ξn (K j ) ≤ 0 ≥ lim sup P ξn (I j ) ≤ 0 = P ξ(K j ) ≤ 0 , (2.59) n

so that (2.55) follows. Finally, to show simplicity, we take I ∈ T and show that η has multiple points in I with zero probability. Now P (η(I) > η∗ (I)) = P (η(I) − η∗ (I) < 1/2) ≤ 2 (Eη(I) − Eη∗ (I))) .

(2.60)

The latter, however, is zero, due to the assumption of convergence of the intensity measures.  Remark The main requirement in the theorem is the convergence of the socalled avoidance function, P (ξn (I) = 0), Eq. (2.41). The convergence of the mean (the intensity measure) provides tightness. It may be replaced by any other tightness criterion (see [43]). Note that, by Chebyshev’s inequality, (2.41) implies tightness via Corollary 2.24. We will have to deal with situations where (2.41) fails, e.g. in branching Brownian motion.

2.5 Point Processes of Extremes We are now ready to describe the structure of extremes of random sequences in terms of point processes. There are several aspect of these processes that we may want to capture: (i) The distribution of the largest values of the process: if un (x) is the scaling w function such that P (Mn ≤ un (x)) → G(x), it is natural to look at the point process n  δu−1 . (2.61) Nn ≡ n (Xi ) i=1

As n tends to infinity, most of the points un (Xi ) will disappear to minus infinity, but we may hope that, as a point process, this object will converge. (ii) The ‘spatial’ structure of the large values: we may fix an extreme level, un , and ask for the distribution of the values i for which Xi exceeds this level. Again, only a finite number of exceedances will be expected. To represent the exceedances as a point process, it will be convenient to embed 1 ≤ i ≤ n in the unit interval (0, 1], via the map i → i/n. This leads us to consider

30

Extremal Processes the point process of exceedances on (0, 1], Nn ≡

n 

δi/n 1Xi >un .

(2.62)

i=1

(iii) We may consider the two aspects together and consider the point process on R × (0, 1], n  Nn ≡ δ(u−1 . (2.63) n (Xi ),i/n) i=1

In this chapter we consider only the case of iid sequences, although this is far too restrictive. We turn to the non-iid case later when we study Gaussian processes.

The Point Process of Exceedances We begin with the simplest object, the process Nn of exceedances of an extremal level un . Theorem 2.27 Let (Xi )i∈N be iid random variables with marginal distribution function F. Let τ > 0 and assume that there is un ≡ un (τ) such that limn↑∞ n(1− F(un (τ))) = t. Then the point process #n ≡ N

∞ 

δi/n 1Xi >un (τ)

(2.64)

i=−∞

# on R with intensity measure converges weakly to the Poisson point process N τdx. Proof

We use Kallenberg’s theorem. First note that, trivially, E [Nn ((c, d])] =

n 

P (Xi > un (τ)) 1i/n∈(c,d]

(2.65)

i=1

= n(d − c)(1 − F(un (τ))) → τ(d − c), as n ↑ ∞, so that the intensity measure converges to the desired one. Next we need to show that P (Nn (I) = 0) → e−τ|I| , for I any finite union of disjoint intervals. But

 P (Nn (I) = 0) = P ∀i/n∈I Xi ≤ un → e−τ|I| ,

(2.66)

(2.67)

from the basic result on convergence of the law of the maximum. This proves the theorem. 

2.5 Point Processes of Extremes

31

The Point Process of Extreme Values Next we consider the point process of the largest values, i.e. that of the values of the largest maxima. We use the occasion to show how to employ Laplace functionals. Theorem 2.28 Let (Xi , i ∈ N) be a sequence of iid random variables with marginal distribution function F. Assume that, for all τ ∈ (0, ∞), lim n ↑ ∞n(1 − F(un (τ))) = τ,

(2.68)

uniformly on compact intervals in τ. Then the point process En ≡

n 

δu−1 n (Xi )

(2.69)

i=1

converges weakly to the Poisson point process on (0, ∞) with intensity measure the Lebesgue measure. Proof For φ : R → R+ a continuous, non-negative function with compact support in (0, ∞), we write Ψn (φ) for the Laplace functional of En , i.e. ⎡ ⎛ n ⎞⎤ $  % ⎟⎟⎥⎥ ⎢⎢⎢ ⎜⎜⎜   −1 ⎢ ⎜ φ un (Xi ) ⎟⎟⎟⎠⎥⎥⎥⎦ . (2.70) Ψn (φ) ≡ E exp − φ(x)En (dx) = E ⎢⎣exp ⎜⎝− i=1

The computations in the iid case are very simple. By independence, ⎞⎤ ⎡ ⎛ n n ⎟⎟⎥⎥     ⎢⎢⎢ ⎜⎜⎜   −1 ⎢ ⎜ φ un (Xi ) ⎟⎟⎟⎠⎥⎥⎥⎦ = E exp −φ u−1 E ⎢⎣exp ⎜⎝− n (Xi ) i=1

i=1

 n    . = E exp −φ u−1 n (X1 )

(2.71)

We know that the probability for un (Xi ) to be in the support of φ is of order 1/n. Thus, if we write



−1 −1 (2.72) E e−φ(un (Xi )) = 1 + E e−φ(un (X1 )) − 1 , the second term will be of order 1/n. Therefore,

n

   −1 −1 E e−φ(un (X1 )) ∼ exp nE e−φ(un (X1 )) − 1 . Finally,

  −1 e−φ(τ) − 1 dτ. lim nE e−φ(un (Xi )) − 1 = n↑∞

(2.73)

(2.74)

To show the latter, note first that, assuming that φ is differentiable and using

32

Extremal Processes

integration by parts,

−φ(u−1 n (X1 ))

nE e



∞

−1 =



0

=−

   −φ(τ) nP u−1 −1 n (X1 ) ∈ dτ e ∞

0

(2.75)

   d  −φ(τ) − 1 nP u−1 e n (X1 ) > τ dτ. dτ

Since φ has compact support and the integrand converges uniformly on compact sets, we get that ∞

 d  −φ(τ) −φ(u−1 (X1 )) n −1 =− e − 1 τdτ, (2.76) lim nE e n↑∞ 0 dτ and one further integration by parts yields (2.74). A standard approximation argument shows that the same holds for all φ ∈ C0+ . From (2.74) we get that the Laplace functional converges to that of the PPP(dτ), which proves the theorem. 

Complete Poisson Convergence We now come to the final goal of this section, the characterisation of the spacevalue process of extremes as a two–dimensional point process. We consider again un (τ) such that n(1 − F(un (τ))) → τ. Then we define Nn ≡

∞ 

δ(i/n,u−1 , n (Xi ))

(2.77)

i=1

as a point process on R2 (or, more precisely, on R+ × R+ ). Theorem 2.29 Let un (τ) be as above. Then the point process Nn converges to the Poisson point process N on R2+ with intensity measure given by the Lebesgue measure. Proof The easiest way in the iid case to prove this theorem is to use Laplace functionals just as in the proof of Theorem 2.28. For φ ∈ C0+ (R2+ ), clearly ΨNn (φ) =

∞  



   1 + E exp −φ u−1 n (Xi ), i/n − 1

(2.78)

i=1

⎞ ⎛ ∞   

⎟⎟ ⎜⎜⎜ −1  −1 ⎜ nE exp −φ un (Xi ), i/n − 1 ⎟⎟⎟⎠ . ∼ exp ⎜⎝n i=1

Note that, since φ has compact support, there are only finitely many non-zero

2.6 Bibliographic Notes terms in the sum. We write   

∞  (X ), i/n − 1 = e−φ(τ,i/n) − 1 dτ lim nE exp −φ u−1 i n n↑∞ 0 ∞     d  −φ(τ,i/n) −1 − − 1 nP un (Xi ) > τ − τ dτ. e 0 dτ

33

(2.79)

The first term is what we want, while, again due to the fact that φ has compact support (and for the moment has bounded derivatives), say I × J, the second term is in absolute value smaller than     (2.80) 1i/n∈J K nP u−1 n (X1 ) ≤ τ − τ dτ. I

Since the term in the brackets tends to zero, inserting this into the sum over i still gives a vanishing contribution, while ∞ ∞ ∞ ∞    (2.81) e−φ(τ,i/n) − 1 dτ = e−φ(τ,z) − 1 dτdz. lim n−1 n↑∞

i=1

0

0

0

From here the claimed result is obvious.



2.6 Bibliographic Notes 1. The exposition in this chapter follows partly Resnick’s book [99] and that of Leadbetter et al. [84]. The most comprehensive treatise of point processes is the two–volume monograph by Daley and Verre-Jones [42, 43]. A further nice reference is Kingman [77].

3 Normal Sequences

The assumption that the underlying random processes are iid is, of course, rather restrictive and unrealistic in real applications. A lot of work has been done to extend extreme value theory to other processes. It turns out that the results of the iid model are fairly robust, and survive essentially unchanged under relatively weak mixing assumptions. Since this book is about Gaussian processes, we present some of the main classical results for this case in the present chapter. More precisely, we consider the case of stationary Gaussian sequences. We will introduce a very powerful tool, Gaussian comparison, that makes the study of extremes in the Gaussian case more amenable. There is of course much more (see, e.g., the monograph by Adler and Taylor [3]). Let us recall the definition of a Gaussian process indexed by an arbitrary set. Definition 3.1 A collection of n ∈ N real random variables X1 , . . . , Xn is called jointly Gaussian with mean m ∈ Rn and covariance matrix C if there exists m ∈ Rn and a positive definite n × n–matrix C such that the law of (X1 , . . . , Xn ) is absolutely continuous with respect to the Lebesgue measure on Rn with density   1 1 −1 fC (x1 , . . . , xn ) ≡ ((x − m), C (x − m)) . (3.1) exp − √ 2 (2π)n/2 det C A simple computation shows that the elements of the matrix C are the covariances of the random variables Xi , that is, Ci j = E[Xi X j ] − E[Xi ]E[X j ]

(3.2)

and of course mi = E[Xi ]. Definition 3.2 Let I be a set and let X = (Xt , t ∈ I) be a stochastic process with index-set I and state space R. Then X is called a Gaussian process if all finite-dimensional marginals are jointly Gaussian. 34

3.1 Normal Comparison

35

One can easily see that this definition implies that there exists a map C : I × I → R with the property that its restrictions to J × J for any finite subset of I is a positive definite matrix, which then is the covariance matrix of the J–marginal of X. process X is called normal if, for all t ∈ I, E [Xt ] = 0 and

A Gaussian E Xt2 = 1. A stationary normal Gaussian sequence, (Xi , i ∈ Z), is then a Gaussian process indexed by Z that satisfies (3.3) E [Xi ] = 0, E Xi2 = 1, E[Xi X j ] = ri− j , i, j ∈ Z, where rk = r−k for all k ∈ Z and the infinite-dimensional matrix with entries ci j = ri− j is positive definite. The target of this chapter is to show that, under the so-called Berman condition [17], lim rn ln n = 0, n↑∞

(3.4)

the extremes of stationary normal sequences behave like those of the corresponding iid normal sequence. We shall see later that the logarithmic decay of correlations is indeed a boundary for irrelevance of correlation.

3.1 Normal Comparison In the context of Gaussian random variables, a recurrent idea is to compare one Gaussian process to another, simpler one. The simplest processes for comparison are, of course, iid variables, but the concept goes much further. Let us consider a family of Gaussian random variables, ξ1 , . . . , ξn , normalised to have mean zero and variance one (we refer to such Gaussian random variables as centred normal random variables), and let Λ1 denote their covariance matrix. Similarly, let η1 , . . . , ηn be centred normal random variables with covariance matrix Λ0 . Generally speaking, one is interested in comparing functions of these two processes that are of the form E[F(X1 , . . . , Xn )],

(3.5)

where F : Rn → R. For instance, we could take F(X1 , . . . , Xn ) = 1X1 ≤x1 ,...,Xn ≤xn ,

(3.6)

to get the probability that the maximum does not exceed xn . An extraordinarily

36

Normal Sequences

efficient tool to compare such processes turns out to be interpolation. Given the families ξ and η, we define X1h , . . . , Xnh , for h ∈ [0, 1], by √ √ (3.7) X h = hξ + 1 − hη. Then X h is normal and has covariance Λh = hΛ1 + (1 − h)Λ0 .

(3.8)

The following Gaussian comparison lemma is a fundamental tool in the study of Gaussian processes. Lemma 3.3 Let η, ξ, X h be as above. Let F : Rn → R be twice differentiable and of moderate growth. Set f (h) ≡ EF(X1h , . . . , Xnh ). Then $ 2 % 1 1 1 ∂ F f (1) − f (0) = (Λi j − Λ0i j )E (X1h , . . . , Xnh ) dh. (3.9) 2 0 i j ∂xi ∂x j Proof

Trivially,

f (1) − f (0) = 0

and

1

df (h)dh, dh

$ n  ∂F % 1   −1/2 df −1/2 (h) = E h ξi − (1 − h) ηi , dh 2 i=1 ∂xi

(3.10)

(3.11)

∂F where of course ∂x is evaluated at X h . To continue, we use a remarkable fori mula for Gaussian processes, known as the Gaussian integration by parts formula.

Lemma 3.4 Let (Xi )ni=1 be a multivariate Gaussian process, and let g : Rn → R be a differentiable function of moderate growth. Then n

$ ∂g(X) %    E g(X)Xi = E Xi X j E . (3.12) ∂x j j=1 Proof We start with the case n = 1. Write X1 = X, for simplicity. Let σ2 = E[X 2 ]. Then ∞ x2 1 E[Xg(X)] = √ g(x)x e− 2σ2 dx (3.13) 2 2πσ −∞ ∞ 1 d & 2 − x22 ' = √ g(x) −σ e 2σ dx dx 2πσ2 −∞ ∞ x2 dg(x) 1 dx = E X 2 E[g (x)], e− 2σ2 = σ2 √ 2 dx 2πσ −∞

37

3.1 Normal Comparison 2 − x2 2σ

where we have used integration by parts and the assumption that g(x)e is integrable. This is (3.12) for n = 1. For the proof of the general case, let X be i] a centred Gaussian random variable. Set Xi ≡ Xi − X E[XX . Then E[X 2 ] E[Xi X] = E[Xi X] − E[Xi X] = 0,

(3.14)

and so X is independent of the vector (Xi , i = 1, . . . , n). Now compute $



% E[Xi X] E[Xn X]  +X , . . . , Xn + X (3.15) E Xg(X1 , . . . , Xn ) = E Xg E[X 2 ] E[X 2 ] $ $   ( %% E[Xi X] E[Xn X] ((  = E E Xg X1 + X , . . . , Xn + X (F , E[X 2 ] E[X 2 ] 



X1

where F is the σ–algebra generated by the random variables (Xi , i = 1, . . . , n). Now we can use (3.12) with n = 1 for the conditional expectation. This gives $   % E[Xi X] E[Xn X] (((   E Xg X1 + X + X , . . . , X (F n E[X 2 ] E[X 2 ]  % $ ∂  E[Xi X] E[Xn X] (((   g X1 + X = E X2 E + X , . . . , X (F n ∂X E[X 2 ] E[X 2 ]  $  % n  E[Xn X] E[Xi X] E[Xn X] (((  ∂ 2   g X1 + X =E X E , . . . , Xn + X (F ∂xi E[X 2 ] E[X 2 ] E[X 2 ] i=1 $ n (( %  ∂ = E[Xn X]E g (X1 , . . . , Xn ) ((F . (3.16) ∂x i i=1 Inserting this into (3.15) with X = Xi yields the assertion of the lemma.



Applying Lemma (3.4) in (3.11) yields $ 2

% ∂ F 1 df E E ξi ξ j − ηi η j (h) = dh 2 i j ∂x j ∂xi  $ ∂2 F  % 1  1 0 h h = Λ ji − Λ ji E X , . . . , Xn , 2 i j ∂x j ∂xi 1 which is the desired formula.

(3.17)



The general comparison lemma can be put to various good uses. The first is a monotonicity result that is sometimes known as Kahane’s theorem [74]. Theorem 3.5

Let ξ and η be two independent n–dimensional Gaussian vec-

38

Normal Sequences

tors. Let D1 and D2 be subsets of {1, . . . , n} × {1, . . . , n}. Assume that E[ξi ξ j ] ≥ E[ηi η j ],

if

(i, j) ∈ D1 ,

E[ξi ξ j ] ≤ E[ηi η j ], if (i, j) ∈ D2 ,

E[ξi ξ j ] = E ηi η j , if (i, j)  D1 ∪ D2 .

(3.18)

Let F be a function on Rn of moderate growth, such that its second derivatives satisfy ∂2 F(x) ≥ 0, ∂xi ∂x j

if

(i, j) ∈ D1 ,

∂2 F(x) ≤ 0, ∂xi ∂x j

if

(i, j) ∈ D2 .

(3.19)

Then E[F(ξ)] ≤ E[F(η)].

(3.20)

Proof The proof of the theorem can be trivially read off the preceding lemma by inserting the hypotheses into the right-hand side of (3.9).  One often wants to obtain similar results in situations when the function F is not differentiable. This can be done easily, provided the function f can be approximated by smooth functions. Theorem 3.6 Let ξ and η be as in Theorem 3.5. Let F : Rn → R be a function of moderate growth and assume that there exists a sequence of functions Fk : Rn → R, k ∈ N of moderate growth that are twice differentiable and that converge to F in L1 with respect to the Gaussian measures induced by ξ, η. Assume further that, for all k ∈ N, ∂2 Fk (x) ≥ 0, ∂xi ∂x j

if

(i, j) ∈ D1 ,

∂2 Fk (x) ≤ 0, ∂xi ∂x j

if

(i, j) ∈ D2 .

(3.21)

Then E[F(ξ)] ≤ E[F(η)]. Proof

(3.22)

Theorem 3.5 implies that, for all k ∈ N, E[Fk (ξ)] ≤ E[Fk (η)].

(3.23)

By the assumption of convergence in L2 , we may take the limit as k ↑ ∞ to get (3.22). 

39

3.1 Normal Comparison We consider two examples. The first is known as Slepian’s lemma [106].

Lemma 3.7 Let ξ and η be two independent n–dimensional Gaussian vectors. Assume that E[ξi ξ j ] ≥ E[ηi η j ],

for all i  j

E[ξi ξi ] = E[ηi ηi ],

for all i.

Then

) * ) * n n E max(ξi ) ≤ E max(ηi ) . i=1

Proof

i=1

Let −1

Fβ (x1 , . . . , xn ) ≡ β

⎛ n ⎞ ⎜⎜⎜ βx ⎟⎟⎟ i ln ⎜⎜⎝ e ⎟⎟⎠ .

(3.24)

(3.25)

(3.26)

i=1

Clearly, by Jensen’s inequality, |Fβ (x1 , . . . , xn )| ≤

n 

|xi |,

(3.27)

i=1

and, for Lebesgue-almost all x ∈ Rn , n

lim Fβ (x1 , . . . , xn ) = max xi . β↑∞

i=1

(3.28)

Therefore, by Lebesgue’s dominated convergence theorem, Fβ converges in L1 with respect to any Gaussian measure to max. A simple computation shows that, for i  j, ∂2 F eβ(xi +x j ) = −β  2 , < 0, ∂xi ∂x j n eβxk

(3.29)

k=1

and so Theorem 3.5 implies that, for all β > 0, E[Fβ (ξ)] ≤ E[Fβ (η)].

(3.30)

lim E[Fβ (ξ1 , . . . , ξn )] ≤ lim E[Fβ (η1 , . . . , ηn )],

(3.31)

Thus β↑∞

and hence (3.22) holds.

β↑∞



As a second application, we want to study P (X1 ≤ u1 , . . . , Xn ≤ un ). This corresponds to choosing F(X1 , . . . , Xn ) = 1X1 ≤u1 ,...,Xn ≤un .

40

Normal Sequences

Lemma 3.8 Let ξ, η be as above. Set ρi j ≡ max(Λ0i j , Λ1i j ), and denote x+ ≡ max(x, 0). Then P (ξ1 ≤ u1 , . . . , ξn ≤ un ) − P (η1 ≤ u1 , . . . , ηn ≤ un ) ⎞ ⎛ 1 0 ⎜⎜⎜ u2i + u2j ⎟⎟⎟ 1  (Λi j − Λi j )+ ⎟⎟⎠ . ⎜ ≤ exp ⎜⎝− + 2π 1≤i< j≤n 2(1 + ρ ) 2 i j 1−ρ

(3.32)

ij

Proof

We define Fσ (x1 , . . . , xn ) =

n 

Φσ (ui − xi ),

(3.33)

 z2 exp − 2 . 2σ −∞

(3.34)

i=1

where 1 Φσ (x) = (2πσ2 )1/2



x



Clearly, Fσ is bounded uniformly in σ, smooth for all σ > 0, and converges pointwise to F at all continuity points of F. Now, for any σ > 0, (3.9) holds with F replaced by Fσ . We have that, for i  j, ⎡ ⎤ % $ 2  ⎢⎢⎢ ⎥⎥⎥ ∂ Fσ h h  h  h h ⎢ (X1 , . . . , Xn ) = E ⎢⎣⎢Φσ (Xi − ui )Φσ (X j − u j ) Φσ (Xk − uk )⎥⎥⎥⎦ E ∂xi ∂x j ki∨ j

 h  h (3.35) ≤ E Φσ (Xi − ui )Φσ (X j − u j ) . The last expression is explicitly given by

E Φσ (Xih − ui )Φσ (X hj − u j ) (3.36)  2 2 (xi − ui ) + (x j − u j ) , = dxi dx j φh (xi , x j )(2πσ2 )−1 exp − 2σ2 where φh denotes the density of the bivariate normal distribution with covariance Λhij , i.e. ⎛ 2 ⎞ ⎜⎜⎜ x + y2 − 2Λhij xy ⎟⎟⎟ ⎟⎟⎠ . exp ⎜⎜⎝− φh (x, y) = + 2(1 − (Λhij )2 ) 2π 1 − (Λhij )2 1

(3.37)

It is well known and easy to show that   (xi − ui )2 + (x j − u j )2 = φh (ui , u j ). lim dxi dx j φh (xi , x j )(2πσ2 )−1 exp − σ↓0 2σ2 (3.38)

41

3.1 Normal Comparison Finally, some algebra shows that u2i + u2j − 2Λhij ui u j 2(1 − (Λhij )2 )

= ≥

(u2i + u2j )(1 − Λhij ) + Λhij (ui − u j )2 2(1 − (Λhij )2 ) (u2i + u2j ) 2(1 + |Λhij |)

≥

(u2i + u2j ) 2(1 + ρi j )

,

where ρi j = max(Λ0i j , Λ1i j ). Inserting this into (3.35) gives ⎛ ⎞ $ 2 % ⎜⎜⎜ (u2i + u2j ) ⎟⎟⎟ ∂ Fσ h 1 h ⎜ ⎟⎟ , (X , . . . , Xn ) ≤ exp ⎜⎝− lim E + σ↓0 ∂xi ∂x j 1 2(1 + ρi j ) ⎠ 2π 1 − ρ2i j

(3.39)

(3.40) 

from which (3.32) follows immediately.

Remark The proof above shows that we would have obtained the correct result by pretending that the indicator function is differentiable with derivative d 1 x≤u = δ(x − u), dx where δ(x) is Dirac’s delta-function.

(3.41)

Remark It is often convenient to replace the assertion of Lemma 3.8 by |P (ξ1 ≤ u1 , . . . , ξn ≤ un ) − P (η1 ≤ u1 , . . . , ηn ≤ un )| ⎛ ⎞ 1 0 ⎜⎜⎜ u2i + u2j ⎟⎟⎟ 1  |Λi j − Λi j | ⎜ ⎟⎟ . ≤ exp ⎜⎝− + 2π 1≤i< j≤n 1 − ρ2 2(1 + ρi j ) ⎠ ij

(3.42)

A simple but useful corollary is the specialisation of this lemma to the case when ηi are independent random variables. Corollary 3.9 Let ξi , i = 1, . . . , n be centred normal variables with covariance matrix Λ, and let ηi , i = 1, . . . , n be iid centred normal variables. Then P (ξ1 ≤ u1 , . . . , ξn ≤ un ) − P (η1 ≤ u1 , . . . , ηn ≤ un ) ⎞ ⎛ u2i + u2j ⎟⎟⎟ ⎜⎜⎜ (Λi j )+ 1  ⎟⎟ . ⎜ ≤ exp ⎜⎝− + 2π 1≤i< j≤n 1 − Λ2 2(1 + |Λi j |) ⎠ ij In particular, if |Λi j | < δ ≤ 1, |P (ξ1 ≤ u, . . . , ξn ≤ u) − [Φ(u)]n | ≤

Proof

(3.43)

  |Λi j | 1  u2 . exp − √ 2π 1≤i< j≤n 1 − δ2 1 + |Λi j | (3.44)

The proof of the corollary is straightforward and left to the reader.



42

Normal Sequences

Another simple corollary is a version of Slepian’s lemma: Corollary 3.10 Let ξ, η be as above. Assume that Λ0ii = Λ1ii for all 1 ≤ i ≤ n, and Λ0i j ≤ Λ1i j for all 1 ≤ i  j ≤ n. Then ' & ' & n n (3.45) P max ξi ≤ u − P max ηi ≤ u ≤ 0. i=1

i=1

Proof The proof of the corollary is again obvious from (3.43), since under  the assumption of the corollary, (Λ0i j − Λ1i j )+ = 0.

3.2 Applications to Extremes The comparison results of Section 3.1 can readily be used to give criteria under which the extremes of correlated Gaussian sequences are distributed as in the independent case. Lemma 3.11 Let ξi , i ∈ Z, be a stationary normal sequence with covariance rn . Assume that supn≥1 rn ≤ δ < 1. Let un be a sequence such that   n  u2n lim n = 0. (3.46) |ri | exp − n↑∞ 1 + |ri | i=1 Then n (1 − Φ(un )) → τ ⇔ P (Mn ≤ un ) → e−τ .

(3.47)

Proof Using Corollary 3.9, we see that (3.46) implies that |P (Mn ≤ un ) − Φ(un )n | ↓ 0.

(3.48)

n (1 − Φ(un )) → τ ⇔ Φ(un )n → e−τ ,

(3.49)

Since

the assertion of the lemma follows.



Since the condition n (1 − Φ(un )) → τ determines un (if 0 < τ < ∞), one can easily derive a criterion for (3.46) to hold. Lemma 3.12 Assume that the Berman condition, rn ln n ↓ 0, holds, and that un is such that n (1 − Φ(un )) → τ, 0 < τ < ∞. Then (3.46) holds. Proof We know that, if n(1 − Φ(un )) ∼ τ,   1 2 exp − un ∼ Kun n−1 , 2

(3.50)

3.2 Applications to Extremes and un ∼

√

43

2 ln n. Thus     u2 u2 |ri | = n|ri | exp −u2n + n . n|ri | exp − n 1 + |ri | 1 + |ri |

(3.51)

Let α > 0 and i ≥ nα . Then n|ri |e−un ∼ 2n−1 |ri |ln n

(3.52)

u2n |ri | ≤ 2|ri | ln n. 1 + |ri |

(3.53)

2

and

But then |ri | ln n = |ri | ln i

ln n ln n ≤ |ri | ln i = α−1 |ri | ln i, ln i ln nα

which tends to zero as i ↑ ∞, due to the Berman condition. Thus      u2 ≤ 2α−1 sup |ri |ln i exp 2α−1 |ri |ln i ↓ 0 n|ri | exp − n 1 + |ri | i≥nα i≥nα

(3.54)

(3.55)

as n ↑ ∞. On the other hand, since there exists δ > 0, such that 1 − ri ≥ δ,    u2n ≤ n1+α n−2/(2−δ) (2 ln n)2 , n|ri | exp − (3.56) 1 + |r | i α i≤n which tends to zero as well, provided we choose α such that 1 + α < 2/(2 − δ), i.e. α < δ/(2 + δ). This proves the lemma.  The following theorem summarises the results on the stationary Gaussian case. Theorem 3.13 Let Xi , i ∈ N, be a stationary centred normal sequence with covariance rn , such that rn ln n → 0. Then (i) for 0 ≤ τ ≤ ∞,

& ' n n (1 − Φ(un )) → τ ⇔ P max Xi ≤ un → e−τ , i=1

(3.57)

(ii) un (x) can be chosen as in the iid normal case, and (iii) the extremal process is Poisson; more precisely, n 

δu−1 → PPP(e−x dx), N (Xi )

(3.58)

i=1

where PPP(μ) denotes the Poisson point process on R with intensity measure μ.

44

Normal Sequences

Proof Combining Lemmas 3.11 and 3.12, one immediately proves items (i) and (ii). For the proof of the convergence of the extremal process, we show the convergence of the Laplace functional. Let φ : R → R+ have compact support (say in [a, b]), and assume without loss of generality that φ has bounded derivatives. Then, by Lemma 3.3, n n



−1 −1 E e− i=1 φ(un (Xi )) − E e−φ(un (Xi ))

(3.59)

i=1

=

 1 h  −1 h − ni=1 φ(u−1 n (Xi )) 2 ln n. Λi j E φ (u−1 n (Xi ))φ (un (X j )) e 2 i j

By assumption on the function φ, the right-hand side of (3.59) is bounded in absolute value by    C Λi j P Xih > uN (a), X hj > un (a) 2 ln n, (3.60) i j

for some constant C. Using the formula for the joint density of the variables Xih and X hj together with the bound (3.39), we see that ∞ ∞ 2 2   1 − x +y dx dy e 2(1+Λi j ) P Xih > uN (a), X hj > un (a) ≤ , 2π (1 − Λi j ) un (a) un (a) ≤

u2 (a) (1 + Λi j ) − n e (1+Λi j ) . , 2πun (a)2 (1 − Λi j )

(3.61)

Using that u2n (a) ∼ ln n and inserting this bound into (3.60), we see that (3.59) tends to zero as n ↑ ∞, whenever the assumption of Lemma 3.11 is satisfied. This proves the theorem. 

3.3 Bibliographic Notes 1. Gaussian processes are a vast field and there is an enormous literature on them. We mention the monographs by Adler [2] and Piterbarg [98]. Many relevant results can also be found in the monograph by Ledoux and Talagrand [85]. 2. The method of Gaussian comparison goes back to Slepian [106] and was further developed by Fernique [53, 55, 54], Gordan [61] and Kahane [74]. 3. The Berman condition was established by Berman in [17].

4 Spin Glasses

The motivation for the results I present in this book comes from spin glass theory. I will not go into this at any depth and I will only sketch a few key concepts.

4.1 Setting and Examples Spin glasses are spin systems with competing random interactions. To remain close to the main frame of this book, let us remain in the realm of mean-field spin glasses of the type proposed by Sherrington and Kirkpatrick in [103]. Here we have a state space, SN ≡ {−1, 1}N . On this space we define a Gaussian process HN : SN → R, characterised by its mean E[HN (σ)] = 0, for all σ ∈ SN , and covariance E[HN (σ)HN (σ )] ≡ Ng(σ, σ ),

σ, σ ∈ SN ,

(4.1)

where g : SN × SN → R is a positive definite quadratic form. The random functions HN are called Hamiltonians. In physics, they represent the energy of a configuration of spins, σ ∈ SN . In so-called mean-field spin glasses, g is assumed to depend only on some distance between σ and σ . There are two popular distances: (i) The Hamming distance, 

dham (σ, σ ) ≡

N 

1σi σi

i=1

⎛ ⎞ N  ⎟ 1 ⎜⎜⎜⎜ ⎟ = ⎜⎝N − σi σi ⎟⎟⎟⎠ . 2 i=1

(4.2)

In this case one chooses

 g(σ, σ ) = p 1 − 2dham (σ, σ )/N , 45

(4.3)

46

Spin Glasses with p a polynomial with non-negative coefficients. The case p(x) = x2 corresponds to the original Sherrington–Kirkpatrick model [103]. One denotes the quantity 1 − 2dham (σ, σ )/N ≡ RN (σ, σ )

(4.4)

as the overlap between σ and σ . (ii) The lexicographic distance dlex (σ, σ ) = N + 1 − min(i : σi  σi ).

(4.5)

Note that dlex is an ultrametric. By analogy with the overlap RN we define the ultrametric overlap qN (σ, σ ) ≡ 1 − N −1 dlex (σ, σ ).

(4.6)

g(σ, σ ) = A(qN (σ, σ )),

(4.7)

In this case,

where A : [0, 1] → [0, 1] can here be chosen as any non-decreasing function such that A(0) = 0 and A(1) = 1. The models obtained in this way were introduced by Gardner and Derrida [58, 59] and are called generalised random energy models (GREM). The GREMs have a more explicit representation in terms of iid normal random variables. Let us fix a number, k ∈ N, of levels. Then introduce k natural numbers, 0 < 1 < 2 < · · · < k = N, and k real numbers, a1 , . . . , ak , such that - . , be iid centred normal rana21 + a22 + · · · + a2k = 1. Finally, let Xσm 1≤m≤k σ∈{−1,1} 1 +···+ m dom variables with variance N. The Hamiltonian of a k–level GREM is then given as a random function HN : {−1, 1}N → R, defined as HN (σ) ≡

k 

am Xσm1 ···σm ,

(4.8)

m=1

where we write σ ≡ σ1 σ2 · · · σk and σk ∈ {−1, 1} m . It is straightforward to see that HN (σ) is Gaussian with variance N, and that E[HN (σ)HN (τ)] = N

k 

a2m 1σi =τi ,∀i≤ 1 +···+ m .

(4.9)

m=1

Thus, if we define the function Σ2 (z) = N

k  m=1

a2m 1 1 +···+ m ≤z ,

z ∈ [0, N],

(4.10)

4.2 The REM

47

Eq. (4.9) can be written in the form E[HN (σ)HN (τ)] = Σ2 (min(i : σi  τi ) − 1).

(4.11)

This is the form of the correlation given in (4.3) with A(z) = N −1 Σ2 (Nz). Clearly the representation (4.8) is very useful for doing explicit computations. The main objects one studies in statistical mechanics are the Gibbs measures associated with these Hamiltonians. They are probability measures on SN that assign to σ ∈ SN the probability μβ,N (σ) ≡

e−βHN (σ) , Zβ,N

(4.12)

where the normalising factor Zβ,N is called the partition function. The parameter β is called the inverse temperature. Note that these measures are random variables on the underlying probability space (Ω, F, P) on which the Gaussian processes HN are defined. The objective of statistical mechanics is to understand the geometry of these measures for very large N. This problem is usually interesting in the case when β is large, where the Gibbs measures will feel the geometry of the random process HN . In particular, one should expect that, for large enough β (and finite N), μβ,N will assign mass only to configurations of almost minimal energy. One way to acquire information on the Gibbs measures is thus to first analyse the structure of the minima of HN .

4.2 The REM In order to illustrate what can go on in spin glasses, Derrida [44, 45] had the bright idea to introduce a simple toy model, the random energy model (REM). In the REM, the values HN (σ) of the Hamiltonian are simply independent Gaussian random variables with mean zero and variance N. One might think that this would be too simple but, remarkably, quite a lot can be learned from this example. Understanding the structure of the ground states in this model turns into the classical problem of extreme value theory for iid random variables, which we have discussed in the first two chapters. We will set HN (σ) ≡ −Xσ .

Rough Estimates, the Second Moment Method Although we already know everything about the extremes of this process, we begin by taking a fresh look, introducing methods that are also useful in more

48

Spin Glasses

complicated situations. To get an initial feeling, we ask for the right order to the maximum. More precisely, we ask for the right choice of functions uN : R → R, such that the maximum is of that order uN (x) with positive, x–dependent probability. To do so, we introduce the counting function  1Xσ >uN (x) , (4.13) KN (x) ≡ σ∈SN

which is the number of σ’s such that Xσ exceeds uN (x). Then   P max Xσ > uN (x) = P (KN (x) ≥ 1) ≤ E [KN (x)] σ∈SN

= 2N P(Xσ > uN (x)).

(4.14)

This may be called a first moment estimate. So a good choice of uN (x) would be such that 2N P(Xσ > uN (x)) = O(1). For later use, we even choose 2N P(Xσ > uN (x)) ∼ e−x .

(4.15)

From the computations in the proof of Theorem 1.2 we get (replacing n by 2N ) that √ 1 ln(N ln 2) + ln(4π) x + √ . (4.16) uN (x) = N 2 ln 2 − √ 2 2 ln 2 2 ln 2 The fact that E[KN (x)] = O(1) precludes the possibility of too many exceedances of the level uN (x), by Chebyshev’s inequality. But it may still be true that the probability that KN (x) > uN (x) tends to zero. To show that this is not the case, one may want to control, e.g. the variance of KN (x). Namely, using the Cauchy–Schwarz inequality, we have that  

 (E [KN (x)])2 = E(KN (x)1KN (x)≥1 ) 2 ≤ E KN (x)2 P (KN (x) ≥ 1) , (4.17) so that P (KN (x) ≥ 1) ≥

(E [KN (x)])2

 , E KN (x)2

(4.18)

which is a special version of the Paley–Szygmund inequality. It is easy to compute

E KN (x)2 = 2N P (Xσ > uN (x)) + 2N (2N − 1)P (Xσ > uN (x))2 = (E [KN (x)])2 (1 − 2−N ) + E [KN (x)] .

(4.19)

Hence P (KN (x) ≥ 1) ≥

1−

2−N

1 1 ∼ , + 1/E [KN (x)] 1 + e x

(4.20)

4.3 The GREM, Two Levels

49

for N large. This is called the second moment estimate. Together with the Chebyshev upper bound (4.14), this already gives good control of the probability that the maximum exceeds uN (x), at least for large x:   e−x ≤ P max Xσ > uN (x) ≤ e−x . (4.21) σ∈SN 1 + e−x In particular,





lim e lim P max Xσ > uN (x) = 1, x

x↑∞

N↑∞

σ∈SN

(4.22)

which is a bound on the upper tail of the distribution of the maximum. First and second moment estimates are usually easy to use and give the desired control on maxima, if they work. Unfortunately, they do not always work as nicely as here.

Maximum and the Extremal Process In this simple case, we can of course do better. In fact, the result on extremes of iid random variables from Chapters 1 and 2 imply that   −x (4.23) P max Xσ ≤ uN (x) → e−e σ∈SN

as N ↑ ∞. Moreover, with EN ≡

 σ∈SN

δu−1 , N (Xσ )

(4.24)

we have the following corollary. Corollary 4.1 In the REM, the sequence of point processes EN converge in law to the Poisson point process with intensity measure e−z dz on R.

4.3 The GREM, Two Levels To understand what happens if correlation is introduced into the game, we consider the simplest version of the generalised random energy model with just two hierarchies. That is, for σ1 ∈ SN/2 and σ2 ∈ SN/2 , we define the Hamiltonian HN (σ1 σ2 ) ≡ a1 Xσ1 1 + a2 Xσ2 1 σ2 , where all X are iid centred Gaussian with variance N, and a21 + a22 = 1.

(4.25)

50

Spin Glasses Note that this corresponds to the Gaussian process with covariance E[HN (σ)HN (σ )] = NA(qN (σ, σ )),

where

⎧ ⎪ 0, ⎪ ⎪ ⎪ ⎪ ⎨ 2 A(x) = ⎪ a1 , ⎪ ⎪ ⎪ ⎪ ⎩1,

(4.26)

if x < 1/2, if 1/2 ≤ x < 1,

(4.27)

if x = 1.

Second Moment Method We may be tempted to retry the second moment method that worked so nicely in the REM. Of course we get again E[KN (x)] = 2N P(HN (σ) > uN (x)),

(4.28)

as in the REM. A first guess would be to choose uN (x) as in the REM. But the mean of KN (x) does not see any correlation, which should make us suspicious. We know how to check whether this is significant: compute the second moment. This is easily done: 

E KN (x)2 = E 1HN (σ1 σ2 )>uN (x) 1HN (τ1 τ2 )>uN (x) (4.29) σ1 ,σ2 ,τ1 ,τ2

=



E



1HN (σ1 σ2 )>uN (x)

σ1 ,σ2

+

 

E



σ1 σ2 τ2

+

 

E

1HN (σ1 σ2 )>uN (x) 1HN (σ1 τ2 )>uN (x)







1HN (σ1 σ2 )>uN (x) E 1HN (σ1 τ2 )>uN (x) .

σ1 τ1 σ2 ,τ2

The first line equals 2N P(HN (σ) > uN (x)) and the last line is equal to (2N − 1)2 P(HN (σ) > uN (x))2 , so we already know that these are ok, i.e. of order one. But the middle terms are different. In fact, a straightforward Gaussian computation shows that ⎛ ⎞

⎜⎜ uN (x)2 ⎟⎟⎟ ⎟⎠ . (4.30) E 1HN (σ1 σ2 )>uN (x) 1HN (σ1 τ2 )>uN (x) ∼ exp ⎜⎜⎝− N(1 + a21 ) Thus, with uN (x) as in the REM, the middle term gives a contribution of order 23N/2 2−2N/(1+a1 ) . 2

(4.31)

This does not explode only if a21 ≤ 13 . What is going on here? To see this, look

51

4.3 The GREM, Two Levels in detail into the computations:

⎛ ⎞ ⎜⎜⎜ x2 (u − x)2 ⎟⎟⎟ ⎟⎠ dx, P HN (σ σ ) ∼ u ∧ HN (σ τ ) ∼ u ∼ exp ⎝⎜− 2 − 2a1 N (1 − a21 )N (4.32) which is obtained by first integrating over the value x of Xσ1 1 and then asking that both Xσ2 1 σ2 and Xσ2 1 τ2 fill the gap to u. Now this integral, for large u, gets its main contribution from values of x that maximise the exponent, which is 2ua2 easily found to be reached at xc = 1+a12 . For u = uN (x), this indeed yields 1 (4.30). However, in this case, √ √ √ 2a21 2 ln 2 2 2a1 = a N ln 2. (4.33) xc ∼ N 2 2 1 1 + a1 1 + a1 

1

2

1 2





√ Now a1 N ln 2 is the maximum that any of the 2N/2 variables a1 Xσ1 1 can reach, √ and the value xc is larger than that as soon as a1 > 2 − 1. This indicates that this moment computation is nonsensical above that value of a1 . On the other hand, when we compute P(HN (σ1 σ2 ) ∼ uN ) in the same way, we find that√ the corresponding of the first variable is xc = uN a21 = √ √ critical value 1 a1 2a1 N ln 2 = a1 2 maxσ1 (a1 Xσ1 ). Thus, here a problem occurs only for √ a1 > 1/ 2. In the latter case we must expect a change in the value of the maximum, but for smaller values of a1 we should just be more clever.

Truncated Second Moments We see that the problem comes with the excessively large values of the first component contributing to the second moments. A natural idea is to cut these off. Thus we introduce  /N (x) ≡ K 1HN (σ1 σ2 )>uN (x) 1a1 X1 ≤a1 N √ln 2 . (4.34) σ1

σ1 ,σ2

A simple computation shows that

/N (x) = E [KN (x)] (1 + o(1)), E K

(4.35)

/N (x)2 , the as long as a21 < 1/2. On the other hand, when we compute E K previously annoying term becomes 3N/2

2

E

)

1 HN

(σ1 σ2 )>u

N (x)

1 HN

(σ1 τ2 )>u

1

√ N (x) X 1 ≤N ln 2 1 σ

*

−N

∼2

√ (1− 2a1 )2 1−a2 1

,

(4.36)

52

Spin Glasses

/N (x), it follows that which is insignificant for a21 < 1/2. Since KN (x) ≥ K 

2 /N (x)] E[K /N (x) ≥ 1 ≥

= O(1). (4.37) E [KN (x)] ≥ P (KN (x) ≥ 1) ≥ P K /N (x)2 E K 



So the result can be used to show that, as long as a21 < 1/2, the maximum is of the same order as in the REM. If a21 = 1/2, the bound on (4.36) is order one. Thus we still get the same behaviour for the maximum. If a1 is even larger, then the √ behaviour of the maximum changes. Namely, one cannot reach the value 2 ln 2N any more, and the maximum will be achieved by adding the maxima of the first hierarchy to the maximum in one of the branches of the √ second hierarchy. This yields (a1 + a2 ) 2N to leading order.

Finer Computations As in the REM, we do of course want to compute things more precisely. This means we want to compute   (i) P max(σ1 ,σ2 ) HN (σ1 σ2 ) ≤ uN (x) , or, better, (ii) the limit of the Laplace functionals of the extremal process, ⎡ ⎛ ⎞⎤ ⎢⎢⎢ ⎜⎜⎜   ⎟⎟⎟⎥⎥⎥ 1 2 ⎟⎟⎟⎥⎥⎥⎥ . E ⎢⎢⎢⎢⎣exp ⎜⎜⎜⎜⎝− φ u−1 (4.38) N (HN (σ σ )) ⎟ ⎠⎦ σ1 ,σ2 ∈SN/2

Technically, there is rather little difference between the two cases. Let us for simplicity compute the Laplace functional for the case φ(x) = λ1(uN (x),∞) . Then ⎡ ⎛ ⎞⎤ ⎢⎢⎢ ⎜⎜⎜ ⎟⎟⎟⎥⎥⎥  E ⎢⎢⎢⎢⎣exp ⎜⎜⎜⎜⎝−λ 1HN (σ1 σ2 )>uN (x) ⎟⎟⎟⎟⎠⎥⎥⎥⎥⎦ (4.39) σ1 ,σ2 ∈SN/2 ⎡ ⎛ ⎞⎤ ⎢⎢⎢ ⎜⎜⎜ ⎟⎟⎟⎥⎥⎥   ⎢ ⎜ = E ⎢⎢⎢⎣exp ⎜⎜⎜⎝−λ 1HN (σ1 σ2 )>uN (x) ⎟⎟⎟⎟⎠⎥⎥⎥⎥⎦ σ1 ∈SN/2

$



t2 1 e− 2N = √ 2πN $ t2 1 = √ e− 2N 2πN

σ2 ∈SN/2



 N/2 E exp −λ1a2 X>uN (x)−a1 t 2 dt 

%2N/2 2N/2

1 + (e−λ − 1)P(a2 X > uN (x) − a1 t)

%2N/2 dt

.

Here X is a centred Gaussian random variable with variance N. √ We are interested in the situation when we can still choose uN (x) ∼ N 2 ln 2

4.3 The GREM, Two Levels

53

as above. The probability appearing in the last line of (4.39) can be well estimated by the standard Gaussian tail estimates, provided t  un (x)/a1 .

(4.40)

Namely, for t satisfying (4.40), we use ' & 2 1 t) exp − (uN (x)−a    2a22 N N . P (a2 X > uN (x) − a1 t) = √ √ 1 + O (uN (x)−a 2 1 t) 2π(uN (x) − a1 t)/a2 N (4.41) Define √ uN (x) a2 N ln 2 − . (4.42) tc = a1 a1 Then, for t ≤ tc ,

 √  √ P (a2 X > uN (x) − a1 t) ≤ P X > N ln 2 ≤ 2−N/2 / 2π ln 2N.

On the other hand,

(4.43)



∞ 2N/2 t2  0 ≤ √ e− 2N 1 + (e−λ − 1)P(a2 X > uN (x) − a1 t) dt 2πN tc & 2' t &√ '2 &√ ' √ exp − 2Nc 2−a2 2−a −N (2 2)−1 a 2 −1/2 1 . (4.44) ≤ √ √ ≤ 2 2 a1 N 2πtc / N

1

√

2 ≥ 1, with equality only in the case a1 = a2 . Thus, Since a21 + a22 = 1, 2−a a1 in both cases, this integral is much smaller than 2−N/2 . For t ≤ tc , we can approximate  2N/2 1 + (e−λ − 1)P(a2 X > uN (x) − a1 t) ≈ 1+2N/2 (e−λ −1)P(a2 X > uN (x)−a1 t). (4.45) Using this bound together with (4.41) and changing variables, we obtain that tc 2N/2 t2  1 e− 2N 1 + (e−λ − 1)P(a2 X > uN (x) − a1 t) dt √ 2πN −∞ s2  tc −uN (x)a1 e− 2a2 N  −N/2 −x −λ e e −1 ds. (4.46) ∼1+2 √ −∞ 2πNa2

The integral is essentially equal to one, if tc − uN (x)a1 > 0, which is the case when a21 < 1/2. In the case a21 = 1/2, the integral equals 1/2. Inserting this into (4.39), we get, in both cases, ⎡ ⎛ ⎞⎤ ⎢⎢⎢ ⎜⎜⎜ ⎟⎟⎟⎥⎥⎥      1HN (σ1 σ2 )>uN (x) ⎟⎟⎟⎟⎠⎥⎥⎥⎥⎦ = exp e−x e−λ − 1 K , (4.47) lim E ⎢⎢⎢⎢⎣exp ⎜⎜⎜⎜⎝−λ N↑∞ σ1 ,σ2 ∈SN/2

54

Spin Glasses

which is the Laplace functional for a Poisson process with intensity Ke−x dx, K being 1 or 1/2, depending on whether a21 < 1/2 or a21 = 1/2, respectively. In [30] the full picture is explored when the number of levels is arbitrary (but finite). The general result is that the extremal process remains Poisson with intensity measure e−x dx whenever the function A is strictly below the straight line A(x) = x, and it is Poisson with intensity Ke−x dx when A(x) touches the straight line finitely many times. The value of the constant K < 1 can be expressed in terms of the probability that a Brownian bridge (see Section 5.2) stays below 0 in the points where A(x) = x. If a21 > 1/2, the entire picture changes. In that case, the maximal values of the process are achieved by adding up the maxima of the two hierarchies, and this leads to a lower value even on the leading scale N. The extremal process also changes, but it is simply a concatenation of Poisson processes. This has all been fully explored in [30].

4.4 Connection to Branching Brownian Motion We have seen that the general case corresponds to a Gaussian process indexed by SN with covariance E[X s Xσ ] = NA(qN (σ, σ )), for a non-decreasing function A. Now we are invited to think of SN as the leaves of a tree with binary branching and N levels. Then it makes sense to extend the Gaussian process from the leaves of the tree to the entire tree. If we think of the edges of the tree to be of length one, this process should be indexed by t ∈ [0, N], with covariance E [Xσ (t)Xσ (s)] ≡ NA((t ∧ s)/N ∧ qN (σ, σ )).

(4.48)

These models were introduced by Gardner and Derrida [58, 59] and further investigated in [31]. It turns out that the leading order behaviour of the maximum in these models can be computed by approximating the function A by step functions. The point here is that if A is a step function, one can show easily, using the methods explained above, that 0 1 √ dA(x) dx, (4.49) lim N −1 max X s = 2 ln 2 σ∈S N↑∞ dx N 0 where A denotes the concave envelope of A (i.e. the smallest concave function larger than or equal to A). It is straightforward to show that this formula remains true for arbitrary non-decreasing A. The proof, given in [31], uses the Gaussian comparison method explained in Chapter 3. A similar feature is not true for the sub-leading corrections. They can be

4.5 The Galton–Watson Process

55

computed for all step functions, but the expression has no obvious limit when these converge to some continuous function. If the concave hull of A is the linear function x, we have already seen in Section 4.3.3 in the case when A has two steps, that the intensity of the Poisson process of extremes changes when A touches the straight line. In general, a rather complicated-looking formula is obtained if A touches the straight line at several points [31]. These observations make the case A(x) = x an obviously interesting target. Another interesting observation is that the processes introduced above can all be constructed with the help of Brownian motions. In the case A(x) = x, one sees easily that this can be realised as follows. Start at time 0 with a Brownian motion that splits into two independent Brownian motions at time t = 1. Each of these splits again into two independent copies at time t = 2, and so on. The case of A non-decreasing just corresponds to a time change of this process. The case studied above, in which A is a step function, corresponds to singular time changes where for some time the Brownian motions stop moving and just split and then recover their appropriate size instantaneously.

4.5 The Galton–Watson Process So far we have looked at Gaussian processes on {−1, 1}N which could be seen as the leaves of a binary tree of depth N. We could also decide to take some different sort of tree. A relevant example is the continuous time Galton–Watson tree (see, e.g. [13]). The Galton–Watson process [116] is the most canonical continuous-time branching process. Start with a single particle at time zero. After an exponential time of parameter one, this particle splits into k particles according to some probability distribution p on N. Then each of the new particles splits at independent exponential times independently according to the same branching rule, and so on. For the purposes of this book, we will always assume that p0 = 0, i.e. no deaths occur. At time t, there are n(t) ’individuals’ (ik (t), 1 ≤ k ≤ n(t)). The point is that the collection of individuals is endowed with a genealogical structure. Namely, for each pair of individuals at time t, ik (t), i (t), there is a unique time 0 ≤ s ≤ t when they shared a common ancestor for the last time. We call this time d(ik (t), i (t)). It is sometimes useful to provide a labelling of the Galton–Watson process in terms of multi-indices. For convenience we think of multi-indices as infinite

56

Spin Glasses

sequences of non-negative integers. Let us set I ≡ ZN +,

(4.50)

and let F ⊂ I denote the subset of multi-indices that contain only finitely many entries that are different from zero. Ignoring leading zeros, we see that F=

∞ 1

Zk+ ,

(4.51)

k=0

where Z0+ is the one-element set containing the multi-index made of zeros only. A continuous-time Galton–Watson process will be encoded by the set of branching times, {t1 < t2 < · · · < tw(t) < . . . } (where w(t) denotes the number of branching times up to time t) and by a consistently assigned set of multi-indices for all times t ≥ 0. To do so, we construct for a given tree the sets of multi-indices τ(t) at time t as follows: (i) Set τ(0) = {(0, 0, . . . )}. (ii) For all j ≥ 0, for all t ∈ [t j , t j+1 ), set τ(t) = τ(t j ). (iii) If i ∈ τ(t j ), then i + (0, . . . , 0, k, 0, . . . ) ∈ τ(t j+1 ), 2345 w(t j )×0

for all 0 ≤ k ≤ i (t j+1 ) − 1, where i (t j ) = #{offspring of the particle corresponding to i at time t j }. (4.52) Note that here we use the convention that, if a given branch of the tree does not branch at time ti , it is considered to have one offspring (see Figure 4.1). We can relate the assignment of labels in a backwards-consistent fashion as follows. For i ≡ (i1 , i2 , i3 , . . . ) ∈ ZN + , we define the function i(r), r ∈ R+ , through ⎧ ⎪ ⎪ ⎨i , if t ≤ r, i (r) ≡ ⎪ (4.53) ⎪ ⎩0, if t > r. Clearly, if i(t) ∈ τ(t) and r ≤ t, then i(r) ∈ τ(r). This allows us to define the boundary of the tree at infinity as follows: ∂T ≡ {i ∈ I : ∀t < ∞, i(t) ∈ τ(t)} .

(4.54)

Note that ∂T is an ultrametric space equipped with the ultrametric m(i, j) ≡ e−d(i,j) , where d(i, j) = sup{t ≥ 0 : i(t) = j(t)} is the time of the most recent common ancestor of i and j. Note that, from knowledge of the set of multi-indices in ∂T and the set of

4.6 The REM on the Galton–Watson Tree

57

branching times, the entire tree can be reconstructed. Similarly, knowing τ(t) allows to reconstruct the tree up to time t. At any given time t ∈ R+ , we can and will label the n(t) multi-indices in τ(t) by ik (t), k = 1, . . . , n(t). Remark The labelling of the GW-tree is a slight variant of the familiar Ulam– Neveu–Harris labelling (see, e.g. [65]). In our labelling the added zeros keep track of the order in which branching occurred in continuous time. The construction above is also nicely explained for discrete time GW processes in [115].

Figure 4.1 Construction of T#: the green nodes are the artificially added one-child branching.

4.6 The REM on the Galton–Watson Tree A Gaussian process can then be constructed in which the covariance is a function of this distance, just like on the binary tree. The simplest Gaussian process on such trees is of course the REM, i.e. at time t there are n(t) iid Gaussian random variables xk (t), k = 1, . . . , n(t), of variance t. We will always be interested  in the case when k kpk > 1, i.e. when the branching process is supercritical. In that case it is known ([13]; see Lemmas 5.3 and 5.4 in the next chapter) that limt↑∞ t−1 ln n(t) = c > 1, and limt↑∞ n(t)/En(t) = M, where M is a random

58

Spin Glasses

variable with mean 1. We will discuss this further in the next chapter. What does this mean for the REM? If we condition on n(t), standard estimates will apply and we get that   −x P max xk (t) ≤ un (x)|n(t) = n ∼ e−e , (4.55) k≤n(t)

provided n depends on t in such a way that

 P xk (t) > un,t (x) ∼ 1/n.

(4.56)

Using the computations in Chapter 1 and taking into account that the variance of xk (t) is equal to t, this implies that √ x ln ln n + ln(4π) + √ . (4.57) un,t (x) = t 2t−1 ln n − √ −1 −1 2 2t ln n 2t ln n Passing to the limit, since the right-hand side of (4.55) does not depend on n, one easily sees that   −x P max xk (t) ≤ un(t),t (x) → e−e . (4.58) k≤n(t)

Taking into account the convergence properties of n(t), we see that, asymptotically as t ↑ ∞, √ x ln rt + ln(4π) ln(M) un(t),t (x) ∼ t 2c − + √ + √ . (4.59) √ 2c 2c 2c From this we can see that, instead of the random rescaling, we can also use a deterministic rescaling, √ x ln(ct) ut (x) ≡ t 2c − √ + √ . (4.60) 2c 2c In this case we get that  %  $  M √ P max xk (t) ≤ ut (x) → E exp − e− 2cx , (4.61) k≤n(t) 4π where the average is over the random variable M. Here we see for the first time the appearance of a random shift of a Gumbel distribution. Note that I have moved some other constants around to arrive at a form for the right-hand side that is conventional in the context of branching Brownian motion. In the same way, we see that, for the REM on the Galton–Watson tree, the extremal process acquires a random element if we choose the same deterministic rescaling. Namely, we have that % $  n(t)  M − √2cx δu−1 → E dx . (4.62) PPP e M t (xk (t)) 4π k=1

4.7 Bibliographic Notes

59

In other words, the limiting process is a Poisson process with random intensity, also known as a Cox process.

4.7 Bibliographic Notes 1. The literature on spin glasses in physics is enormous. The most famous work is the text by M´ezard, Parisi and Virasoro [91]. As an easy introduction, one should consult the book by Newman and Stein [95]. There are far fewer books from a mathematical point of view on the subject, e.g. [25], [108] or [97]. The REM and GREM models discussed here are presented in some detail in [25]. 2. The random energy model (REM) was introduced by Derrida in 1980 [45]. In [44] he related it to the p ↑ ∞–limit of the p–spin Sherrington–Kirkpatrick models. 3. The generalised random energy model (GREM) was introduced by Derrida in 1985 [46] and analysed in two joint papers with Gardner shortly after that [58, 59]. Ruelle [102] effectively described the limiting model in terms of Poisson cascade processes. Neveu, in an unpublished note [94], related this process to a continuous state branching process. A detailed analysis and the connection to the Bolthausen-Sznitman coalescent [24] can be found in a paper by Bertoin and LeGall [19]. 4. A complete analysis of the thermodynamics of the GREM is given by Bovier and Kurkova [30]. Part of the results had already been obtained earlier by Capocaccia et al. [38]. Models in which the function A is not a stepfunction had already been considered in Gardner and Derrida [59]. Rigorous results on the free energy and the overlap distribution were given in [31]. 5. Classical references for branching processes are the textbooks by Athreya and Ney [13] and by Harris [68]. The Galton–Watson process was introduced by Galton and Watson in 1875 [116] to analyse the probabilities of extinction of family names of noble British families. A rigorous probabilistic formulation including the Neveu–Ulam–Harris labelling [68, 113] was given by Neveu in [93].

5 Branching Brownian Motion

In this chapter we start to look at branching Brownian motion (BBM) as a continuous-time version of the GREM. We collect some basic facts that will be needed later.

5.1 Definition and Basics The simplest way to describe branching Brownian motion is as follows. At time zero, a single particle x1 (0), starting at the origin, begins to perform Brownian motion in R. After an exponential time, τ, of parameter one, the particle splits into two identical, independent copies of itself that start Brownian motion at x1 (τ). This process is repeated ad infinitum, producing a collection of n(t) particles xk (t), 1 ≤ k ≤ n(t). This construction can be easily extended to the case of more general offspring distributions where particles split into k ≥ 1 particles with probabilities    pk . We will always assume that k≥1 pk = 1, k≥1 kpk = 2 and k≥2 k(k − 1)pk < ∞. Note that, while of course nothing happens if the particle splits into one particle, we keep this option to be able to maintain the mean number of offspring equal to 2. Another way of constructing BBM is to first built a Galton–Watson tree with the same branching mechanism as above. Branching Brownian motion (at time t) can then be interpreted as a Gaussian process, xk (t), indexed by the leaves of a GW-process, such that E[xk (t)] = 0 and, given a realisation of the Galton– Watson process, E [xk (t)x (t)] = d(ik (t), i (t)).

(5.1)

This observation makes BBM perfectly analogous to the GREM [31] with co60

5.2 Rough Heuristics

61

variance function A(x) = x, which we have seen to be the critical covariance.

5.2 Rough Heuristics Already in the GREM we have seen that when A touches the straight line x, there appears a constant in the intensity measure that reflects the fact that the process is not allowed to get close to the maximal values possible at this point. The basic intuition of what happens in BBM is that √ the ancestral paths of BBM have to stay below the straight line with slope 2 all the time. This will force the maximum down slightly. The rough heuristics of the correct rescaling can be obtained from second moment calculations. More precisely, we should consider the events    √  (5.2) Er,t ≡ ∃k≤n(t) : {xk (t) − m(t) > x} ∧ xk (s) ≤ 2s , ∀ s∈(r,t−r) , for r fixed and for a function m(t) that gives these events a non-trivial probability. Here we write, for given t, and s ≤ t, xk (s) for the ancestor at time s of the particle xk (t). One of the observations of Bramson [33] is that the event in (5.2) has the same probability as the event . (5.3) ∃k≤n(t) : xk (t) − m(t) > x , when first t ↑ ∞ and then r ↑ ∞. This is not very easy to show, but also not too hard. Let us first recall that a Brownian bridge from 0 to 0 in time t can be represented as s (5.4) z(s) = B(s) − B(t). t More generally, we will denote by ztx,y a Brownian bridge from x to y in time zero. Clearly, ztx,y (s) = x + (y − x) st + zt0,0 (s).

(5.5)

It is clear that the ancestral path from xk (t) to zero is just a Brownian bridge from xk (t) to 0. That is, we can write xk (t − s) =

t−s t xk (t)

+ ztk (s),

(5.6)

where ztk (s), s ∈ [0, t], is a Brownian bridge from 0 to 0, independent of xk (t). The events Er,t in (5.2) can then be expressed as    √ ∃k≤n(t) : {xk (t) − m(t) > x} ∧ ztk (t − s) ≤ 2s − st xk (t), ∀ s∈(r,t−r) , (5.7)

62

Branching Brownian Motion

where of course the Brownian bridges are not independent. The point, however, is that, as far as the maximum is concerned, they might as well be independent. We need to bound the probability of a Brownian bridge to stay below a straight line. The following lemma is taken from Bramson [34], but is of course classic. Lemma 5.1 Then

Let y1 , y2 > 0. Let z be a Brownian bridge from 0 to 0 in time t.  P z(s) ≤ st y1 +

t−s t y2 , ∀0≤s≤t



= 1 − exp(−2y1 y2 /t).

(5.8)

The proof and further discussions on properties of Brownian bridges will be given in the next chapter. The first moment estimate (the calculations are a bit intricate and constitute a good part of Bramson’s first paper [33]) shows that

P Er,t



e−

(m(t)+x)2 2t

m(t)2

√ e− 2t Cx2 ∼e √ ∼ C  et 3/2 x2 e− 2x . √ t 2πm(t)/ t t t

(5.9)

The t–dependent term is equal to one if m(t) =

√

2t −

3 √ ln t. 2 2

(5.10)

To justify that this upper bound is not bad, one has to perform a second moment computation in which one retains the condition that the particle is not above √ the straight line 2s. This is not so easy here and actually yields a lower bound there the x2 is not present. Indeed, it also turns out that the upper bound is not optimal, and that the x2 should be replaced by x. We will show that in the next chapter. One can actually localise the ancestral paths of extremal particles much more precisely, as was shown in [8]. Theorem 5.2 ([8]) Define, for γ > 0, ⎧ γ ⎪ ⎪ ⎨s , ft,γ (s) ≡ ⎪ ⎪ ⎩(t − s)γ ,

0 ≤ s ≤ t/2, t/2 ≤ s ≤ t.

(5.11)

Let D = [dl , du ] ∈ R be a bounded interval. Then, for any 0 < α < 12 < β, and any  > 0, there exists r0 ≡ r0 (D, , α, β) such that, for all r > r0 and t > 3r,  (5.12) P ∀k≤n(t) s.t.xk (t) − m(t) ∈ D, it holds that  s s ∀ s∈[r,t−r] , xk (t) − ft,β (s) ≤ xk (s) ≤ xk (t) − f s,α (s) ≥ 1 − . t t

5.3 Recursion Relations

63

Basically this theorem says that the paths of maximal particles lie below the straight line to their endpoint along the arc ft,1/2 (s). This is just due to a property of Brownian bridges: in principle, a Brownian bridge wants to oscillate (a √ little bit) more than t at its middle. But if it is not allowed to cross zero, then it will also not get close to zero, because each time it does so, it is unlikely to not do the crossing. This phenomenon is known as entropic repulsion in statistical mechanics. It will be useful to give a heuristic explanation for Theorem 5.2 from a more BBM point of view. Let us consider the position of the ancestor at time t/2 of √ a particle that reaches a √level 2t − m(t) at time t. We know that the maximal particle at time t/2 is at √2t/2−m(t/2)+y, where y is bounded. For √ an offspring of this particle to reach 2t − m(t), it would need to grow by 2t/2 − m(t) + m(t/2) − y. But since m(t) = c ln t, −m(t) +√m(t/2) = −c ln 2, so the offspring of this particle would need to reach a level 2t/2, which has a probability that tends to zero like some power as t ↑ ∞. Therefore, the only way that particles can reach the maximal level is to take advantage of the fact that as one goes below the maximum at t/2, the density of particles increases (exponentially), so that the unlikeliness of catching up until time t is upset by the increasing number of candidates. Thus, from the single fact that the level of the maximum is linear minus logarithm in t, it follows that the particles at time t/2√ whose offspring makes it to the maximum must lie in some region below 2t/2 − √ m(t/2). It turns out that this level is of order t. The same holds of course for other times than t/2. Note √ of particles that end up much lower √ also that, for the ancestral paths than 2t, say around at with a < 2, there is no such effect present. The ancestral paths of these particles are just Brownian bridges from zero to their end position which are localised around the straight line by a positive upper and a negative lower envelope fγ,t , with any γ > 12 .

5.3 Recursion Relations The presence of the tree stucture in BBM of course suggests the derivation of recursion relations. To get into the mood, let us prove the following lemma: Lemma 5.3

Let n(t) denote the number of particles of BBM at time t. Then E[n(t)] = et .

(5.13)

Proof For notational simplicity we consider only the case of pure binary branching. The idea is to use the recursive structure of the process. Let τ

64

Branching Brownian Motion

be the first branching time. Clearly, if t < τ, then n(t) = 1; otherwise, it is n (t − τ) + n (t − τ), where n and n are the numbers of particles in the two independent offspring processes. This reasoning leads to t E[n(t)] = P(τ > t) + P(τ ∈ ds)2E[n(t − s)] (5.14) 0 t = e−t + 2 e−s E[n(t − s)]ds. 0

Differentiating this equation yields d E[n(t)] = −e−t + 2e−t E[n(0)] + 2 dt



t

d E[n(t − s)]ds (5.15) dt 0 t d = −e−t + 2e−t E[n(0)] − 2 e−s E[n(t − s)]ds ds 0 t = −e−t + 2E[n(t)] − 2 e−s E[n(t − s)]ds = E[n(t)], e−s

0

where we have used integration by parts in the second equality and the fact that n(0) = 1. The assertion follows by solving this differential equation with E[n(0)] = 1.  We can also show the following classical result (see the standard textbook by Athreya and Ney [13] for this and many further results on branching processes): Lemma 5.4

If n(t) is the number of particles of BBM at time t, then M(t) ≡ e−t n(t)

(5.16)

is a martingale. Moreover, M(t) converges, a.s. and in L1 , to a non-trivial random variable M of mean 1. In the case of binary branching, M is exponentially distributed. Proof Again we write things only for the binary case. The verification that M(t) is a martingale is elementary. Since M(t) is positive with mean 1, it is bounded in L1 and hence, by Doob’s martingale convergence theorem, M(t) converges, a.s., to a random variable M. To show that the martingale is uniformly integrable and hence converges in L1 , we show that

φ(t) ≡ E M(t)2 = 2 − e−t . (5.17) This can be done by noting that E[M(t)2 ] satisfies the recursion t  2 φ(t) = e−3t + 2 e−3s φ(t − s)ds + 1 − e−3t . 3 0

(5.18)

5.4 The F-KPP Equation

65

Differentiating yields the differential equation φ (t) = 2 − φ(t),

(5.19)

of which (5.17) is the unique solution with φ(0) = 1. As explained in the lecture notes by Lalley [82], once convergence of M(t) to a limit M is proven, one

can derive a differential equation for its Laplace transform, ψ(u) = E e−uM , u ∈ R+ , namely ∞  pk ψ(u)k . (5.20) uψ (u) = −ψ(u) + k=1

In the case p2 = 1, the functions γ/(γ+θ) are the only solutions of this ode with ψ(0) = 1. These are the Laplace transforms of exponential random variables with expectation 1/α. Since we know EM = 1, we are done. In the non-binary case, there is no explicit form of the solutions of (5.20). 

5.4 The F-KPP Equation The following lemma relates functionals of BBM to solutions of a non-linear PDE. This relation is fundamental to what follows. Lemma 5.5 ([90]) Let f : R → [0, 1] and {xk (t) : k ≤ n(t)} a branching Brownian motion starting at 0. Let ⎤ ⎡ n(t) ⎥⎥⎥ ⎢⎢⎢ ⎢ f (x − xk (t))⎥⎥⎥⎦ . (5.21) v(t, x) ≡ E ⎢⎢⎣ k=1

Then, u(t, x) ≡ 1 − v(t, x) is the solution of the F-KPP equation (5.22), 1 2 ∂ u + F(u), 2 x with initial conditions u(0, x) = 1 − f (x), where ∂t u =

F(u) = (1 − u) −

∞ 

pk (1 − u)k .

(5.22)

(5.23)

k=1

Remark The reader may wonder why we introduce the function v. It is easy to check that v(t, x) itself solves Equation (5.22) with F(u) replaced by −F(1 − v). The form of F given in the lemma is the one customarily known as the F-KPP equation and will be used throughout this text. Proof The derivation of the F-KPP equation is quite similar to the arguments used in the previous section. Again we restrict ourselves to the case of binary

66

Branching Brownian Motion

branching. Let f : R → [0, 1]. Define v(t, x) by (5.21). Then, distinguishing the cases when the first branching occurs before or after t, we get − z2 t z2 e− 2s e 2t −s e v(t − s, x − z)2 dzds. (5.24) f (x − z)dz + v(t, x) = e−t √ √ 0 2πs 2πt Differentiating with respect to t and using integration by parts as before, together with the fact that the heat kernel satisfies the heat equation, we find that 1 (5.25) ∂t v = ∂2x v + v2 − v. 2 Obviously, v(0, x) = f (x). Following the remark above, u = 1 − v then solves (5.22). There is a smart way to see that (5.25) and (5.24) are the same thing. Namely, 2

−z

2t H(t, x) ≡ e−t e√2πt is the Green kernel for the linear operator

1 ∂t − ∂2x + 1. 2 In other words, the solution of the inhomogeneous linear equation 1 ∂t v − ∂2x v + v = r 2 with initial conditions v0 is given by t H(s, y)r(t − s, x − y)dsdy. v(t, x) = H(t, x − y)v0 (y)dy +

(5.26)

(5.27)

(5.28)

0

Inserting r(t, x) = v2 (t, x), we obtain what is known as the mild formulation of Eq. (5.25), and this is precisely (5.24). Solutions of (5.24) are called mild solutions of the PDE (5.25).  The first example is obtained by choosing f (x) = 1 x≥0 . Then ⎡ n(t) ⎤   ⎢⎢⎢ ⎥⎥ v(t, x) = E ⎢⎢⎣⎢ 1(x−xk (t)≥0) ⎥⎥⎥⎦⎥ = P max xk (t) ≤ x . k≤n(t)

(5.29)

k=1

A second example is obtained by choosing f (x) = exp(−φ(x)) for φ a nonnegative continuous function with compact support. Then ⎤ ⎡ n(t) $ % ⎥⎥ ⎢⎢⎢ −φ(x−xk (t)) ⎥ ⎥ ⎢ ⎥ ⎢ e (5.30) v(t, x) = E ⎢⎣ ⎥⎦ = E exp(− φ(x − z)Et (dz)) , k=1

n(t)

where Et ≡ k=1 δ xk (t) is the point process associated with BBM at time t. We see that in this case u is the Laplace functional of the point process Et .

5.5 The Travelling Wave

67

Definition 5.6 We call Eq. (5.22) with a general non-linear term the F-KPP equation. We say that F satisfies the standard conditions if F ∈ C 1 ([0, 1]), F(0) = F(1) = 0, 

F (0) = 1,



F(u) > 0,

F (u) ≤ 1,

∀u ∈ (0, 1),

∀u ∈ [0, 1],

(5.31)

and 1 − F  (u) = O(uρ ), as u → 0.

(5.32)

5.5 The Travelling Wave Bramson [34] studied convergence of solutions to a large class of F-KPP equations for a large class of functions F that include those arising from BBM. In particular, he established the conditions on the initial data, under which solutions converge to travelling waves. Part of this is based on earlier results by Kolmogorov et al. [80] and Uchiyama [112]. For a purely probabilistic analysis of the travelling wave solutions, see also Harris [66]. We will always be interested in initial conditions that lie between zero and one. It is easy to see that then the solutions remain between zero and one forever. A travelling wave moving with speed λ would be a solution, u, of the F-KPP equation such that d u(t, x + λt) = 0. (5.33) dt A simple computation shows that this implies that u(t, x + λt) = wλ (x), where 1 2 ∂ wλ (x) + λ∂ x wλ (x) + F(wλ (x)) = 0. (5.34) 2 x If we want a solution that decays to zero at +∞, for large positive x, w must be close to the solution of the linear equation 1 2 (5.35) ∂ wλ (x) + λ∂ x wλ (x) + wλ (x) = 0. 2 x √ But this equation can be solved explicitly. If λ√ 2, then there are two linearly independent solutions, e−b± x , where b± = λ± λ2 − 2. Clearly, these values are √ real only solutions. √we have non-oscillatory √ √ if λ ≥ 2, so only in this case can − 2x − 2x and x e . Kolmogorov et al. If λ = 2, then there are two solutions, e [80] and Uchiyama [112] showed by a phase-space analysis that in both cases the heavier-tailed solution describes the correct asymptotics of the travelling wave solution, and that it is unique, up to translations.

68

Branching Brownian Motion

Lemma 5.7 ([80, 112]) Let F satisfy the standard conditions. Then, for λ ≥ √ 2, Eq. (5.34) has a unique solution satisfying 0 < wλ (x) < 1, wλ (x) → 0, as x → +∞, and wλ (x) → 1, as x → −∞, up to translation, i.e. if w, w are two solutions, then there exists a ∈ R s.t. wλ (x) = wλ (x + a). Proof Note first that if wλ ∈ [0, 1] then, unless wλ is constant, it must hold that, for all x ∈ R, wλ (x) ∈ (0, 1). For if for some x ∈ R, wλ (x) = 0, then it must also hold that ∂ x wλ (x) = 0. But then, since F(wλ (x)) = 0, the initial value problem with these initial data at x has the unique solution wλ ≡ 0. The same holds if wλ (x) = 1. Next we look at the equation in phase space. It reads q = p

(5.36)



p = −2F(q) − 2λp. Clearly, this has the two fixpoints (0, 0) and (1, 0). The Hessian matrices at these fixpoints are     0 1 0 1 and . (5.37) −2F  (0) −2λ −2F  (1) −2λ Since√F  (0) = 1, the eigenvalues of the Hessian at the fixpoint (0, , 0) are b± = 2 −λ ± λ − 2, whereas those at the fixpoint (1, 0) are a± = −λ ± λ2 − 2F  (1). Since of necessity F  (1) ≤ 0, the eigenvalues a± are both real, but unless F  (1) = 0, one is positive and the other negative. Hence (1, 0) is a saddle √ point. The eigenvalues b± have negative real part, but are real only if λ ≥ 2. In any case, the fixpoint is stable, but in the case when the eigenvalues are non-real, the solutions of the linearised equations have oscillatory behaviour and cannot stay positive. In the other cases, there exists an integral curve from (1, 0) that approaches (0, 0) along the direction of the smaller of the two eigenvalues, i.e. a map γ : [0, 1] → R2 , such that γ (τ) = V(γ(τ)),

(5.38)

where V(x) is in the direction of√W(q, p) ≡ (p, −2 f (q)−2λp) but has |V(x)| ≡ 1. In the degenerate case λ = 2, one can show that the solution analogously has the behaviour of the heavier-tailed solution of the linear equation. From the existence of such an integral curve it follows that, for any function τ : [0, 1] → R such that τ (t) = |W(γ(τ(t)))|,

(5.39)

with the property that limt↓−∞ τ(t) = 0 and limt↓+∞ τ(t) = 1, we have that w(t) ≡ γ(τ(t)) 

(5.40)

is a solution of w (t) = W(w(t)) and satisfies the right conditions at ±∞. Clearly

5.5 The Travelling Wave

69

the same is true for w(t) ˜ ≡ w(t + a), for any a ∈ R, so solutions are unique only up to a translation.  √ We will be mainly interested in the case λ = 2. The following theorem slightly specialises Bramson’s Theorems A and B from [34] for this case. Theorem 5.8 Let u be solution of the F-KPP equation (5.22) satisfying the standard conditions with 0 ≤ u(0, x) ≤ 1. Then there is a function m(t) such that u(t, x + m(t)) → ω(x), a.s.,

(5.41)

uniformly in x, t ↑ ∞, where ω is one of the solutions of (5.34) from Lemma 5.7, if and only if 6 t(1+h) √ u(0, y)dy ≤ − 2, and (i) for some h > 0, lim supt→∞ 1t ln t 6 x+N (ii) for some ν > 0, M > 0, N > 0, x u(0, y)dy > ν for all x ≤ −M. √ Moreover, if lim x→∞ ebx u(0, x) = 0 for some b > 2, then one may choose √ 3 m(t) = 2t − √ ln t. (5.42) 2 2 Theorem 5.8 is one of the core results of Bramson’s work on Brownian motion and most of the material in his monograph [34] is essentially needed for its proof. We will recapitulate his analysis in the next chapter. We see that Condition (i) is satisfied for Heaviside initial conditions. Hence, this result implies that   (5.43) P max xk (t) − m(t) ≤ x → ω(x) ≡ 1 − w √2 (x), k≤n(t)

where w √2 solves Eq. (5.34). It should be noted that it follows already from the results of Kolmogorov et al. √ [80] that (5.43) must hold for some function m(t) that satisfies that m(t)/t → 2 (see the next chapter). But only Bramson’s precise evaluation of m(t) shows that the shift is changed from the iid case where it would have to be m (t) =

√

2t −

1 √ ln t, 2 2

which is larger than m(t). One can also check that the law of the maximum for the REM on the GW tree, (4.61), does not solve (5.34). In the Section 5.6 we will see that there is nonetheless a close relation to with the Gumbel distribution.

70

Branching Brownian Motion

Remark Bramson also shows that convergence to travelling waves takes place if the forward tail of the initial conditions decays more slowly, i.e. if t(1+h) 1 u(0, y)dy = −b, (5.44) lim sup ln t t→∞ t √ with b < 2. These cases will not be relevant for us.

5.6 The Derivative Martingale While Bramson analyses the asymptotic behaviour of the functions ω(x), he does not give an explicit form for these solutions, and in particular he does not provide a link to the Gumbel distribution from classical extreme value theory. This was achieved a few years later by Lalley and Sellke (see also the paper by Harris [66]). They wrote a short and very insightful paper [83] that provides, after Bramson’s work, one of the most significant insights into BBM that we have. Its main achievement was to give a probabilistic interpretation for the limiting distribution of the maximum of BBM. But this is not all. For 1 − f satisfying the hypothesis of Theorem 5.8, let u be given by (5.21). Now define ⎡ n(t) ⎤ ⎢⎢⎢ √ ⎥⎥⎥ √ ⎢ vˆ (t, x) ≡ E ⎢⎢⎣ f ( 2t + x − xi (t))⎥⎥⎥⎦ = v(t, 2t + x). (5.45) i=1

One checks that vˆ solves the equation √ 1 2 ∂ x vˆ + 2∂ x vˆ − F(1 − vˆ ). 2

(5.46)

√ 1 2 ∂ x ω + 2∂ x ω − F(1 − ω) = 0. 2

(5.47)

∂t vˆ = Now let ω solve

Then vˆ (t, x) ≡ ω(x) is a stationary solution of (5.46) with initial conditions vˆ (0, x) = ω(x). Therefore we have the stochastic representation ⎤ ⎡ n(t) ⎥⎥⎥ ⎢⎢⎢ √ ⎢ ω( 2t + x − xi (t))⎥⎥⎥⎦ . (5.48) ω(x) = E ⎢⎢⎣ i=1

Now probability enters the game. √ 7 Lemma 5.9 The function W(t, x) ≡ n(t) i=1 ω( 2t + x − xi (t)) is a martingale with respect to the natural filtration Ft of BBM. Moreover, W(t, x) converges almost surely and in L1 to a non-trivial limit, W ∗ (x), and E[W ∗ (x)] = ω(x).

5.6 The Derivative Martingale

71

Proof The proof is straightforward. It helps to introduce the notation xky (t) for BBM started in y. Clearly W(t, x) is integrable. By the Markovian nature of BBM, W(t + s, x) =

ni (s) n(t)  

ω

√  2(t + s) − xi−x (t) − xi,0 j (s) ,

(5.49)

i=1 j=1

where the xi, j are independent BBMs. Now ⎤ ⎡n (s) n(t) i √

(( ⎥⎥⎥ ((  ⎢⎢⎢⎢ −x 0 E W(t + s, x)(Ft = E ⎢⎢⎣ ω 2(t + s) − xi (t) − xi, j (s) (Ft ⎥⎥⎥⎦ i=1

=

j=1

n(t)  i=1

=

⎤ ⎡n (s) √ &√ ' ( ⎥⎥ i ⎢⎢⎢ xi−x (t)− 2t ⎥ ( ⎢ E ⎢⎢⎣ ω 2(s) − xi, j (s) (Ft ⎥⎥⎥⎦ j=1

n(t) 

ω

√  2t − xi−x (t) = W(t, x).

(5.50)

i=1

W(t, x) is bounded and therefore converges almost surely and in L1 to a limit, W ∗ (x), whose expectation is the ω(x).  There are more martingales. First, by a trivial computation,

Y(t) ≡

n(t) 

√

e

2xi (t)−2t

(5.51)

i=1

is a martingale. Since it is positive and E[Y(t)] = 1, it converges almost surely to a finite non-negative limit. But this means that the exponents in the sum must all go to minus infinity, as otherwise no convergence is possible. This means that min

i≤n(t)

 √ 2t − xi (t) ↑ +∞, a.s.

(5.52)

(this argument is convincing but a bit fishy; see the remark below). One of Bramson’s results (we will see the proof of this in Chapter 7; see Corollary 6.48) is that 1 − ω(x) ∼ Cx e−

√ 2x

,

as

x ↑ ∞.

(5.53)

72

Branching Brownian Motion

Hence

⎛ n(t) ⎞ ⎜⎜⎜ ⎟⎟⎟ √ ⎜ W(t, x) = exp ⎜⎜⎝ ln ω( 2t + x − xi (t))⎟⎟⎟⎠ i=1 ⎞ ⎛ n(t) √ √ ⎟⎟⎟ ⎜⎜⎜  √ − 2( 2t+x−x (t)) ⎟⎟⎟ i ( 2t + x − xi (t)) e ∼ exp ⎜⎜⎝⎜−C ⎠ i=1 ' & √ √ = exp −Cx e− 2x Y(t) − Ce− 2x Z(t) ,

(5.54)

where n(t)    √ √ √ 2t − xi (t) e− 2( 2t−xi (t)) .

Z(t) ≡

(5.55)

i=1

Z(t) is also a martingale, called the derivative martingale, with E[Zt ] = 0. The fact that Z(t) is a martingale can be verified by explicit computation, but it will actually not be very important for us. In any case, Z(t) is + not even bounded

2t in L1 (in fact, an easy calculation shows that E[|Zt |] = π and therefore it is a priori not clear that Z(t) converges). By the observation (5.52), Z(t) is much bigger than Y(t), which implies that unless Y(t) → 0, a.s., it must be true that lim inf Z(t) = +∞, which is impossible since this would imply that W(t, x) → 0, which we know to be false. Hence Y(t) → 0, and thus Z(t) → Z, a.s., where Z is finite and positive. It follows that & √ ' lim W(t, x) = exp −CZe− 2x . (5.56) t↑∞

Remark While the argument of Lalley and Sellke for (5.52) may not convince everybody, 1 the following gives an alternative proof for the fact that Yt → 0. By a simple Gaussian computation, 

P ∃k≤n(t) : But this implies that

√



√

e 2K 2t − xk (t) < K ≤ √ . 4πt

(5.57)



√  Z(t) e 2K P 0, a.s. Then, for large enough t, √

e 2K P (Z(t) < 2Ka) ≤ √ . 4πt 1

Thanks to Marek Biskup for voicing some doubt about this claim.

(5.59)

73

5.6 The Derivative Martingale But this implies that, for all K ∈ R+ , by the Borel–Cantelli lemma, ' & P lim inf Z(2n ) < 2aK = 0 n

and hence

(5.60)





P lim sup Z(t) < 2aK = 0.

(5.61)

t↑∞

But this implies that lim inf W(t, x) = 0, a.s., t↑∞

(5.62)

and since we know that W(t, x) converges almost surely, it must hold that the limit is zero. But this is in contradiction to the fact that the limit is a nonnegative random variable with positive expectation and that convergence holds in L1 . Hence it must be the case that Yt → 0, almost surely. Remark One may interpret Y(t) as a partition function. Namely, if we set n(t) 

eβxi (t) ,

(5.63)

Z √2 (t)

. E Z √2 (t)

(5.64)

Zβ (t) ≡

i=1

then Y(t) =

√ 2 has the natural interpretation of the critical inverse temperature for this model, which can be interpreted as the value where the ’law of large numbers’ starts to fail in a strong sense, namely that Zβ (t)/E[Zβ√(t)] no longer converges to a non-trivial limit. In the REM, the critical value is 2 ln 2, and it was shown in [32] that, in this case, at the critical value, this ratio converges to 1/2. For BBM, one can show (see Chapter 9) that Zβ (t) E[Zβ (t)]

(5.65)

is a uniformly integrable martingale for all values β < 1 that converges, a.s. and in L1 , to a positive random variable of mean 1. The reason for the name derivative martingale is that Zt looks √ like the derivative of the martingale Yt with respect to β at the value β = 2. This is indeed a strange animal: its limit is almost surely positive, but the limit of its expectation is zero.

74

Branching Brownian Motion Finally, we return to the law of the maximum. We have that, for any s ≥ 0,   (( lim P max xk (t) − m(t) ≤ x(F s t↑∞ k≤n(t)   (( = lim P max xk (t + s) − m(t + s) ≤ x(F s t↑∞

k≤n(t+s)

 n(s)   (( P max xi,k(t) − m(t + s) − xi (s) ≤ x(F s = lim t↑∞

k≤n(t)

i=1

= lim

n(s) 

t↑∞

u(t, x + m(t + s) − xi (s)).

(5.66)

i=1

where 1 − u is the solution of the F-KPP equation with Heaviside initial conditions. Next we use that √ (5.67) m(t + s) − m(t) − 2s → 0, as t ↑ ∞, for fixed s. This shows that  lim P max xk (t) − m(t) ≤ t↑∞

k≤n(t)

 n(s)  (( √ ( x F s = lim u(t, x + m(t) − 2s − xi (s)) t↑∞

=

n(s) 

i=1

ω(x −

√

2s − xi (s)) = W(s, x).

i=1

As now s ↑ ∞, W(s, x) → e−CZe

√ − 2x

, a.s.,

(5.68)

which proves the main theorem of Lalley and Sellke [83]: Theorem 5.10 ([83]) For BBM,   ) √ * − 2x lim P max xk (t) − m(t) ≤ x = E e−CZe . t↑∞

Moreover

k≤n(t)





lim lim P max xk (t) − m(t) ≤ x|F s = e−CZe s↑∞ t↑∞

k≤n(t)

√ − 2x

, a.s.

(5.69)

(5.70)

Remark Of course, the argument above shows that any solution of (5.34) satisfying the conditions of Lemma 5.7 has a representation of the form 1 − ) √ * −CZe− 2x E e , with only different constants C.

5.7 Bibliographic Notes

75

Remark Lalley and Sellke conjectured in [83] and Arguin et al proved in [10] that √ 1 T − 2x lim 1maxk≤n(t) xk (t)−m(t)≤x dt → e−CZe , a.s. (5.71) T ↑∞ T 0

5.7 Bibliographic Notes 1. Branching Brownian motion can be traced back to a 1962 paper by Moyal [92] and, more specifically, to Adke and Moyal [1]. 2. For the use of moment methods, see in particular the very complete paper by Harris and Roberts [67]. The method is used by Roberts [101] to analyse the maximum of BBM. Truncated second moment estimates are also used in spin glass theory [25, 108]. 3. The fundamental link between BBM and the F-KPP equation [80, 56] is generally attributed to McKean [90], but appears already in Skorohod [105] and Ikeda, Nagasawa and Watanabe [69, 70, 71]. It was used to identify the logarithmic correction to the maximum of BBM by Bramson in [33]. 4. Kyprianou [81] gave a proof of the convergence of the derivative martingale via purely probabilistic techniques.

6 Bramson’s Analysis of the F-KPP Equation

In this chapter we recapitulate the essential features of Bramson’s proof of his main theorems on travelling wave solutions. The material is taken from [34] with occasionally a few details added in proofs and some omissions.

6.1 Feynman–Kac Representation Bramson’s analysis of the asymptotics of solutions of the F-KPP equation in [34] relies on a Feynman–Kac representation. The derivation of this can be seen as a combination of the mild formulation of the PDE together with the standard Feynman–Kac representation of the heat kernel. Namely, if u(t, x) is the solution of the linear equation ∂t u =

1 2 ∂ u + k(t, x)u 2 x

(6.1)

with initial conditions u(0, x), then the standard Feynman–Kac formula yields the representation ([72], see also[76, 107])     t u(t, x) = E x exp k(t − s, Bs )ds u(0, Bt ) , (6.2) 0

where the expectation is with respect to ordinary Brownian motion started at x. To get back to the full equation, we use this with k(t, x) = F(u(t, x))/u(t, x),

(6.3)

where u itself is the solution of the F-KPP equation. It may seem that this is just a rewriting of the original equation as an integral equation. Still, the ensuing representation is very useful as it allows us to process a priori information on the solution into finer estimates. Note that, under the standard conditions on 76

6.1 Feynman–Kac Representation

77

F stated in (5.31) and (5.32), 0 ≤ k(t, x) ≤ 1

(6.4)

and k(t, x) ∼ Cu(t, x)ρ ,

when u(t, x) ↓ 0.

(6.5)

Bramson’s key idea was to use the Feynman–Kac representation not all the way down to the initial conditions, but back to some arbitrary time 0 ≤ r < t. 1 Theorem 6.1 Let u be the solution of the F-KPP equation with initial conditions u(0, x). Let k(t, x) ≡ F(u(t, x))/u(t, x). Then, for any 0 ≤ r ≤ t, u satisfies the equation     t−r u(t, x) = E x exp k(t − s, Bs )ds u(r, Bt−r ) , (6.6) 0

where (Bs , s ∈ R+ ) is Brownian motion starting in x. This can conveniently be rewritten in terms of Brownian bridges. Recall the  t definition, in Chapter 5, of a Brownian bridge z x,y (s), 0 ≤ s ≤ t from x to y in time t. Then (6.6) can be written as $  t−r % ∞ 2 e−(x−y) /(2(t−r)) t−r E exp u(t, x) = dy u(r, y) √ k(t − s, z x,y (s))ds . (6.7) 2π(t − r) −∞ 0 Note that, under the standard conditions, k(t, x) lies between 0 and 1 and tends to 1, as x ↑ ∞. Bramson’s basic idea to exploit this formula is to prove a priori estimates on the function k(t, x). Note that k(t, x) > 0, and by condition (5.32), k(x, t) ∼ 1, when u(t, x) ∼ 0 (think about the simplest case when k(t, x) = 1 − u(t, x)). We will see that k(s, x) ∼ 1 if x is above a certain curve, M. On the other hand, Bramson has shown that the probability that the Brownian bridge descends below a curve M is negligibly small. The following theorem, which is a slight rephrasing of Proposition 8.3 in [34], is the main outcome of this strategy and the main technical result in Bramson’s monograph. Theorem 6.2 Let u be a solution of the F-KPP equation (5.22) with initial conditions satisfying ∞ √ ye 2y u(0, y)dy < ∞. (6.8) 0 1

One should appreciate the beauty of this construction: start with a probabilistic model (BBM), derive a PDE whose solutions represent quantities of interest, and then use a different probabilistic representation for the solution (in terms of Brownian motion) to analyse these solutions.

78

Bramson’s Analysis of the F-KPP Equation

Define √ ⎛ ∞ 3 ln t) ⎞ (z+ √ √ √ √ (y−z)2 ⎜ ⎟⎟⎟ 2 2 e− 2z 2y − 2(t−r) ⎜ −2y ⎜ t−r ⎜⎝1 − e ⎟⎠ dy. ψ(r, t, z+ 2t) ≡ √ u(r, y+ 2r)e e 2π(t − r) 0 (6.9) 3 √ Then, for r large enough, t ≥ 8r, and z ≥ 8r − 2 2 ln t,

γ−1 (r)ψ(r, t, z +

√ √ √ 2t) ≤ u(t, z + 2t) ≤ γ(r)ψ(r, t, z + 2t),

(6.10)

where γ(r) ↓ 1, as r → ∞. We see that ψ basically controls u, but of course ψ still involves u. The fact that u is bounded by u may seem strange, but we shall see that this is very useful. Proof The better part of this chapter will be devoted to proving the following two facts: 1. For r large enough, t ≥ 8r and x ≥ m(t) + 8r, (6.11) u(t, x) ≥ ψ1 (r, t, x) (x−y)2 ∞   x e− 2(t−r) P zt−r ≡ C1 (r)et−r u(r, y) √ x,y (s) > Mr,t (t − s), ∀ s∈[0,t−r] dy 2π(t − r) −∞ and u(t, x) ≤ ψ2 (r, t, x) (6.12) (x−y)2 ∞   e− 2(t−r)  P zt−r ≡ C2 (r)et−r u(r, y) √ x,y (s) > Mr,t (t − s), ∀ s∈[0,t−r] dy, 2π(t − r) −∞ x

where the functions Mr,t (t − s), Mr,t (t − s) satisfy x

Mr,t (t − s) ≤ nr,t (t − s) ≤ Mr,t (t − s).

(6.13)

√ (s − r) (m(t) − 2r). t−r

(6.14)

Here nr (s) ≡

√

2r +

Moreover, C1 (r) ↑ 1, C2 (r) ↓ 1, as r ↑ ∞. 2. The bounds ψ1 (r, t, x) and ψ2 (r, t, x) satisfy 1≤ where γ(r) ↓ 1, as r ↑ ∞.

ψ2 (r, t, x) ≤ γ(r), ψ1 (r, t, x)

(6.15)

79

6.1 Feynman–Kac Representation

Assuming these facts, we can conclude the proof of the theorem rather quickly. Define

(x−y)2

  e− 2(t−r) P zt−r u(r, y) √ x,y (s) > nr,t (t − s), ∀ s∈[0,t−r] dy. 2π(t − r) −∞ (6.16) By (6.13), we have ψ1 ≤ ψ ≤ ψ2 . Therefore, for r, t and x large enough, / ψ(r, t, x) ≡ et−r

∞

u(t, x) ψ2 (r, t, x) ψ2 (r, t, x) ≤ ≤ ≤ γ(r) ψ(r, t, x) ψ(r, t, x) ψ1 (r, t, x)

(6.17)

u(t, x) 1 ≥ . ψ(r, t, x) γ(r)

(6.18)

and √ Combining (6.17) and (6.18) we get, for x = z + 2t with z ≥ 8r − that √ √ √ γ−1 (r)ψ(r, t, z + 2t) ≤ u(t, z + 2t) ≤ γ(r)ψ(r, t, z + 2t).

3 √ 2 2

ln t,

(6.19)

The nice thing is that the probability involving the Brownian bridge in the definition of ψ can be explicitly computed; see Lemma 5.1. √ √ Since our bridge goes from z + 2t to y and the straight line is between 2r and m(t), to length t − r, √ and it stays above this line, we have to adjust this √ y1 = 2t + x − m(t) = x + 2 √3 2 ln t > 0 (for t > 1) and y2 = y − 2r. Obviously   √ √ P zt−r 2r, and for y > 2r Lemma 5.1 x,y (s) > nr,t (t − s), ∀ s∈[0,t−r] = 0 for y ≤ gives that ⎛ & 3 ' √ ⎞ & ' ⎜⎜ z+ √ ln t (y− 2r) ⎟⎟⎟ t−r√ P z (s) > nr,t (t − s), ∀ s∈[0,t−r] = 1 − exp ⎜⎜⎜⎝− 2 2 ⎟⎟⎠ . (6.20) z+ 2t,y

t−r

√ = y − 2r in the Changing variables to y √ √ integral appearing in the definition / of ψ, we get that ψ(t, z + 2t) = ψ(t, z + 2t), as given in (6.9). This, together with (6.19), concludes the proof of the proposition.  Remark In a later paper [35], Bramson gives an even√more explicit representation for the asymptotics of the solutions with speed 2, namely, ∞ 2 e−(z+m(t)−y) /2t −1 t yu(0, y) dy, (6.21) u(t, z + m(t)) = 2C(t, z)zt e √ 0 2πt with limz↑∞ limt↑∞ C(t, z) = 1. We will not make use of this and therefore do not give the proof. In the remainder of this chapter we will recall Bramson’s analysis of the F-KPP equation that leads to the proof of the facts we have used in the proof

80

Bramson’s Analysis of the F-KPP Equation

above. In Chapter 7 we use this proposition crucially to obtain the extremal process of BBM. Let us make some comments about what needs to be shown. In the bounds (6.11) and (6.12), the exponential factor involving k(s, x) in the Feynman–Kac representation (6.7) is replaced by 1. For an upper bound, this is easy, since k ≤ 1; it is, however, important that we are allowed to smuggle in the condition that the Brownian bridge stays above the curve Mr,t , which lies somewhat, but not too much, below the straight line nr,t . For the lower bound, we can of course introduce a condition for the Brownian bridge to stay above the curve M x r,t , which lies above the straight line. We then have to show that, in this case, k(t − s, zt−r x,y (s)) is very close to 1. For this we seem to need to know how the solution u(t, x) behaves, but an upper bound suffices. A trivial upper bound is always given by the solution of the linear F-KPP equation for which the Feynman–Kac representation is explicit, and allows us to get very precise bounds. Finally, one needs to show that the two probabilities concerning the Brownian bridge are almost the same. This holds because of what is known as entropic repulsion: a Brownian bridge of length t wants to fluctuate on the √ oder of t. If one imposes a condition not to make such fluctuations in the negative direction, the bridge will prefer to stay positive and in fact behaves as if one would force it to stay above a curve that goes only up to some tδ , with δ < 1/2. The remainder of this chapter explains how these simple ideas are implemented.

6.2 The Maximum Principle and its Applications Let us first remark that, under the standard conditions on F, the existence and uniqueness of solutions of the F-KPP equation follow easily via the standard tools of Picard iteration and the Gronwall lemma. Theorem 6.3 If F satisfies the standard conditions, then the F-KPP equation with initial data 0 ≤ u0 (x) ≤ 1 has a unique solution for all t ∈ R+ and 0 ≤ u(t, x) ≤ 1, for all t ∈ R+ and x ∈ R. Remark Existence and uniqueness hold for general bounded initial conditions, if F is assumed to be Lipschitz, but we are only interested in solutions that stay in [0, 1]. Proof The best way to prove existence is to set up a Picard iteration for the mild form of the equation, i.e. to define u1 (t, x) as the unique solution of the heat equation ∂t u1 (t, x) =

1 2 ∂ u(t, x), 2 xx

t ∈ R+ , x ∈ R,

u1 (0, x) = u0 (x).

(6.22)

6.2 The Maximum Principle and its Applications This can be written as



u (t, x) = 1



t

∞

ds −∞

0

g(t − s, x − y)u0 (y),

81

(6.23)

where x2 1 g(t, x) = √ e− 2t 2πt

is the heat kernel. Then define recursively ∞ t dy g(t, x − y)u0 (y) + ds un (t, x) = −∞ n−1

≡ T (u

0

∞ −∞

(6.24)

dy g(t − s, x − y)F(un−1 (s, y))

)(t, x).

(6.25)

If this sequence converges, the limit will be a solution of t ∞ ∞ dy g(t, x −y)u0 (y)+ ds dy g(t − s, x −y)F(u(s, y)), (6.26) u(t, x) = −∞

0

−∞

which is a mild formulation of the F-KPP equation. To prove convergence, one shows that the map T is a contraction in the sup-norm (on [0, t0 ] × R), for t0 sufficiently small. Since F is Lipschitz continuous on [0, 1], we have, for two functions u, v both satisfying the same initial conditions, |T (u)(t.x) − T (v)(t, x)| t ∞ ds dyg(t − s, x − y)C|u(s, y) − v(s, y)| ≤ 0 −∞ t ∞ ≤ C sup |u(s, y) − v(s, y)| ds dyg(t − s, x − y) y∈R,s∈[0,t]

= Ct

sup

0

−∞

|u(s, y) − v(s, y)|.

(6.27)

y∈R,s∈[0,t]

Hence, sup x∈R,t∈[0,t0 ]

|T (u)(t.x) − T (v)(t, x)| ≤ Ct0

sup

|u(s, y) − v(s, y)|,

(6.28)

y∈R,s∈[0,t0 ]

by which T is a contraction if t0 < 1/C. This proves that a solution exists up to time 1/C. From here we may start with u(t0 , x) as a new initial contain to construct the solution up to time 2/C, and so on. This shows global existence. To prove uniqueness, we could use the Gronwall lemma, but it follows immediately from the maximum principle, which we state next.  One of the fundamental tools in Bramson’s analysis is the following maximum principle for solutions of the F-KPP equation. We take the theorem from Bramson, who attributes it to Aronson and Weinberger [12].

82

Bramson’s Analysis of the F-KPP Equation

Proposition 6.4 Let F satisfy the standard conditions (5.31). Assume that u1 (t, x) and u2 (t, x) satisfy the inequalities 0 ≤ u1 (0, x) ≤ u2 (0, x) ≤ 1,

∀x ∈ (a, b),

(6.29)

and 1 1 ∂t u2 (t, x) − ∂2x u2 (t, x) − F(u2 (t, x)) ≥ ∂t u1 (t, x) − ∂2x u1 (t, x) − F(u1 (t, x)), 2 2 (6.30) for all (t, x) ∈ (0, T ) × (a, b). If a > −∞, assume further that 0 ≤ u1 (t, a) ≤ u2 (t, a) ≤ 1,

∀t ∈ [0, T ],

(6.31)

∀t ∈ [0, T ].

(6.32)

and, if b < ∞, assume that 0 ≤ u1 (t, b) ≤ u2 (t, b) ≤ 1,

Then u2 ≥ u1 , and if the inequality (6.4) is strict in an open interval of (a, b), then u2 > u1 on (0, T ] × (a, b). Proof The proof consists of reducing the argument to the usual maximum principle for parabolic linear equations. Condition (6.30) can be written as 1 ∂t (u2 −u1 )− ∂2x (u2 −u1 ) ≥ F(u2 )−F(u1 ) = F  (u1 +θ(u2 −u1 ))(u2 −u1 ), (6.33) 2 for some θ ∈ [0, 1], by the mean value theorem. Now set α ≡ max F  (u),

(6.34)

  v(t, x) ≡ e−2αt u2 (t, x) − u1 (t, x) .

(6.35)

u∈[0,1]

and set

Then, by (6.33), v satisfies the differential inequality   1 ∂t v(t, x) − ∂2x v(t, x) ≥ F  (u1 + θ(u2 − u1 )) − 2α v(t, x), 2

(6.36)

where by construction the coefficient to v on the right-hand side is non-positive. If for some (t0 , x0 ), v(t0 , x0 ) < 0, then there must be some point (t1 , x1 ), with t1 ≤ t0 , such that v takes its minimum in [0, t0 ]×(a, b) at this point. Note that x1 cannot be in the boundary of (a, b), due to the boundary conditions. But then at (t1 , x1 ), ∂t ≤ 0 and ∂2xx v ≥ 0, so that the left-hand side of (6.36) is non-positive, whereas the left-hand side is strictly positive, which leads to a contradiction. Hence ν ≥ 0, and so u2 ≥ u1 . 

6.2 The Maximum Principle and its Applications

83

Clearly, the preceding theorem implies that two solutions of the F-KPP equation with ordered initial (and boundary) conditions remain ordered for all times. Since, moreover, the constant functions 0 and 1 are solutions of the F-KPP equation, any solution with initial conditions within [0, 1] will remain in this interval. This also follows, of course, from the McKean representation (5.21). The following corollary compares solutions of the linear and non-linear equations. Corollary 6.5 (Corollary 1 of [34]) Let u1 be a solution of the F-KPP equation with F satisfying (5.31) and let u2 satisfy the linear equation ∂t u2 (t, x) =

1 2 2 ∂ u (t, x) + u2 (t, x). 2 x

(6.37)

If u1 (0, x) = u2 (0, x) for all x, then u1 (t, x) ≤ u2 (t, x)

(6.38)

for all t > 0, x ∈ R. Therefore, if u1 (0, x) = 1 x≤0 , the re-centring m(t) satisfies √ ˜ + C. (6.39) m(t) < 2t − 2−3/2 ln t + C = m(t) Proof Since F(u) ≤ u, the two functions satisfy the hypothesis of Proposition 6.4. Thus, (6.38) holds. On the other hand, one can solve u2 explicitly, using the heat kernel, as +∞ et 2 2 e−(x−y) /(2t) u2 (0, y)dy. (6.40) u (t, x) = √ 2πt −∞ For Heaviside initial conditions this becomes 0 et 2 u2 (t, x) = √ e−(x−y) /(2t) dy = et P (N(0, t) > x) . 2πt −∞

(6.41)

Thus, we know that u2 (t, z + m(t)) ˜ → c e−cz .

(6.42)

By monotonicity, we see that m(t) must be less than m(t), ˜ up to constants. Of course, we already know this probabilistically.  An extension of the maximum principle due to McKean also controls the sign changes of solutions. Proposition 6.6 (Proposition 3.2 in [34]) Let ui , i = 1, 2, be solutions of the F-KPP equation with F satisfying (5.31). Assume that the initial conditions are such that, if x1 < x2 , then u2 (0, x1 ) > u1 (0, x1 ) implies that u2 (0, x2 ) ≥ u1 (0, x2 ). Then this property is preserved for all times, i.e. for all t > 0, if x1 < x2 , then u2 (t, x1 ) > (≥)u1 (t, x1 ) implies that u2 (t, x2 ) > (≥)u1 (t, x2 ).

84

Bramson’s Analysis of the F-KPP Equation

Proof We set v = u2 − u1 . Then ∂t v =

1 2 ∂ v + F  (u1 + θ(u2 − u1 ))v. 2 x

(6.43)

We now use the Feynman–Kac representation in the form (6.6) with k(t, x) ≡ F  (u1 (t, x) + θ(t, x)(u2 (t, x) − u1 (t, x))), the coefficient of ν on the right of (6.43). Set   r k(t − s, Bs )ds v(t − r, Br ). M(r) ≡ exp

(6.44)

(6.45)

0

Then ν(t, x) = E x [M(r)], for any t ≥ r ≥ 0; in fact, M(r) is a martingale. Next we replace the constant time r by the stopping time τ ≡ inf{0 ≤ s ≤ t : M(s) = 0} ∧ t.

(6.46)

Now choose an x1 such that v(t, x1 ) > 0. Then 0 < v(t, x1 ) = E x1 [M(τ)],

(6.47)

so that P(M(τ) > 0) > 0. But this means that, with positive probability τ = t, which can only happen if there is a continuous path, γ, starting at x1 , such that, for all 0 ≤ s ≤ t, v(t − s, γ(s)) > 0. In particular, v(0, γ(t)) > 0. Now define another stopping time, σ, by σ ≡ inf{0 ≤ s ≤ t : Bs = γ(s)} ∧ t,

(6.48)

where B0 = x2 . Now M(σ) ≥ 0. If σ < t, the inequality is strict, and if σ = t, it must be the case that B(t) ≥ γ(t), since otherwise the BM would have had to cross γ. But then ν(0, B(t) > 0 by assumption on the initial conditions. On the other hand, we have again v(t, x2 ) = E x2 [M(σ)].

(6.49)

But Brownian motion starting at x2 hits the path γ with positive probability before time t, so the left-hand side of (6.49) is strictly positive, hence v(t, x2 ) > 0.  The first main goal is to prove an a priori convergence result for travelling wave solutions that was already known to Kolmogorov, Petrovsky and Piscounov [80]. We will use the notation me (t) ≡ sup{x ∈ R : u(t, x) ≥ }.

(6.50)

6.2 The Maximum Principle and its Applications

85

Theorem 6.7 ([80]) Let u be a solution of the F-KPP equation with F satisfying (5.31) and Heaviside initial conditions. Then u(t, x + m1/2 (t)) → w

√

2

(x),

(6.51)

√ 2

uniformly in x, as t ↑ ∞, where w is the solution of the stationary F-KPP equation 1  √  w + 2w + F(w) = 0, (6.52) 2 with w(0) = 1/2. Moreover, m1/2 (t)/t → Proof

√

2.

(6.53)

We need two results that are consequences of Proposition 6.6.

Corollary 6.8 (Corollary 1 in [34]) For u as in Theorem 6.7, for any 0 <  < 1, u(t, m (t) + x) ↑ w(x),

for x > 0,

(6.54)

u(t, m (t) + x) ↓ w(x),

for x ≤ 0,

(6.55)

for some functions w. Proof

We fix t0 , a ∈ R+ and set u1 (t, x) ≡ u(t, x + m (t0 )),

(6.56)

u (t, x) ≡ u(t + a, x + m (t0 + a)).

(6.57)

2

Clearly these functions satisfy the hypothesis of Proposition 6.6. Moreover, since by definition u(t, m (t)) = , for all t ≥ 0, it follows that u2 (t0 , 0) = u1 (t0 , 0).

(6.58)

Hence, by Proposition 6.6, for x ≥ 0, u2 (t0 , x) ≥ u1 (t0 , x), i.e. u(t0 + a, x + m (t0 + a)) ≥ u(t0 , x + m (t0 )).

(6.59)

Likewise, for x < 0, if for some x < 0 it were true that u2 (t0 , x) > u1 (t0 , x), then the same would need to hold at 0, which is not the case. Hence, u2 (t0 , x) ≤ u1 (t0 , x), i.e. u(t0 + a, x + m (t0 + a)) ≤ u(t0 , x + m (t0 )).

(6.60)

Hence u(t, m (t)+ x is monotone increasing and bounded from above for x > 0, and decreasing and bounded from below for x ≤ 0. This implies that it must converge to some function w, as claimed. 

86

Bramson’s Analysis of the F-KPP Equation

We need one more corollary that involves the notion of being more stretched. For g, h monotone decreasing functions, g is more stretched than h if, for any c ∈ R and x1 < x2 , g(x1 ) > h(x1 + c) implies g(x2 ) > h(x2 + c), and the same with > replaced by ≥. Corollary 6.9 (Corollary 2 in [34]) If ui are solutions of the F-KPP equation as in Theorem 6.7, then if the initial conditions of u2 are more stretched than those of u1 then for any time, u2 is more√ stretched than u1 . In particular, if u1 has Heaviside initial conditions, then w 2 is more stretched than u1 (t, ·) for all times. 

Proof This is straightforward from Proposition 6.6.

So now we know that u(t, m1/2 (t) + x) converges to some function, w(x). moreover, so is the limit. Moreover, u(t, √ since u is monotone, √ √ x) is less stretched 2 2 than w . Thus w(x) ≤ w (x), for x > 0, √and w(x) ≥ w 2 (x), for x ≤ 0. It is not hard to show that the limit is indeed w 2 ; if we set v(t, x) ≡ u(t, x + m(t)), then 1 2 ∂ v + m (t)∂ x v + F(v). 2 x (6.61) Integrating the left-hand side over x twice starting from zero, and then integrating the result over t over an interval of length 1, we get x y dy dz (u(t + 1, z + m(t + 1)) − u(t, z + m(t)) ∂t v(t, x) = ∂t u(t, m(t) + x) + m (t)∂ x u(t, m(t) + x) =

0

0 2

≤ x /2 sup (u(t + 1, z + m(t + 1)) − u(t, z + m(t)) ,

(6.62)

0≤z≤x

which tends to zero by Corollary 6.8, as t ↑ ∞. Applying the same procedure on the right-hand side yields  t+1 x y  1 ds dy dz ∂2x v + m (t)∂ x v + F(v) (6.63) 2 t 0 0 t+1  1 x 1 ds v(s, x) − v(s, 0) − ∂ x v(s, 0) = 2 2 2 t x x y  +m (s) dy(v(s, y) − v(s, 0)) + dy dzF(v(s, z)) . 0

0

0

Using the convergence of v to w, as t ↑ ∞, and setting a ≡ limt↑∞ ∂ x v(t, x), the first line converges to t+1  1 x 1 a 1 1 (6.64) ds v(s, x) − v(s, 0) − ∂ x v(s, 0) → w(x) − − x. 2 2 2 2 4 2 t

6.2 The Maximum Principle and its Applications Next, for the same reason, x dy(v(s, y) − v(s, 0)) → 0

x

 1 dy. 2

(6.65)

dzF(w(z)).

(6.66)

 w(y) −

0

87

Finally,



t+1

ds t



x

dy 0

0

y

dzF(v(s, z)) →



x

y

dy 0

0

Since all of (6.63) tends to zero, the term involving m (s) must be independent of t, asymptotically. This means that t+1 dsm (s) = m(t + 1) − m(t) → λ, (6.67) t

for some λ ∈ R. Putting all this together, we get that  x x y 1 1 a 1 w(x) − − x + λ dy + dy dzF(w(z)) = 0. (6.68) w(y) − 2 4 2 2 0 0 0 Differentiating twice with respect to x yields that w satisfies the travelling wave √ 2. But we already have equation (5.34) for some value of λ ≥ √ an upper bound √ on m which implies that λ ≤ 2, which leaves us with λ = 2. This concludes the proof of the theorem.  The maximum principle has several important consequences for the asymptotics of solutions that will be exploited later. The first is a certain monotonicity property. Lemma 6.10 (Lemma 3.1 in [34]) Let u be a solution of the F-KPP equation as before. Assume that the initial conditions satisfy u(0, x) = 0, for x ≥ M, for some real M. Then, for any t, for all y ≥ x ≥ M, u(t, y) ≤ u(t, 2x − y),

(6.69)

∂ x u(t, x) ≤ 0.

(6.70)

and therefore

Proof Let ui , i = 1, 2, be solutions on R+ ×[x, ∞) with initial data u1 (0, y) = 0, for y ≥ x, and u2 (0, y) = u(0, 2x − y) for y ≥ x, and with boundary conditions ui (t, x) = u(t, x), for all t ≥ 0. By the maximum principle, u1 ≤ u2 , for all times. Since x ≥ M, and hence u1 (t, y) = u(t, y), for y ≥ x. On the other hand, by reflection symmetry, u2 (y, y) = u(y, 2x − y). Hence (6.69) holds. Since this inequality implies that u(t, x + ) ≤ u(t, x − ), the claim on the derivative at x is immediate. 

88

Bramson’s Analysis of the F-KPP Equation

The next lemma gives a continuity result for the solutions as functions of the initial conditions. Lemma 6.11 (Lemma 3.2 in [34]) Let ui be solutions of the F-KPP equation with F  (u) ≤ 1, and assume that they are bounded over finite time. Then, if the initial conditions satisfy u2 (0, x) − u1 (0, x) ≤ ,

for all x,

(6.71)

for all t and x.

(6.72)

then u2 (t, x) − u1 (t, x) ≤ et ,

The same holds for the absolute values of the differences. Proof Set v = u2 − u1 . Then 1 2 F(u2 ) − F(u1 ) ∂x v + v. (6.73) 2 u2 − u 1 By assumption on f , the coefficient of v is at most 1. Now let v+ solve (6.73) with initial conditions v+ (0, x) = v(0, x) ∨ 0 then by the maximum principle, for all times t ≥ 0, v+ (t, x) ≥ v(t, x) ∨ 0 remains true. Next, let v¯ be a solution of the linear equation 1 (6.74) ∂t v = ∂2x v + v, 2 ∂t v =

with initial conditions v¯ (0, x) = . Clearly, v¯ (t, x) = et . On the other hand, by the maximum principle and the assumption on the initial conditions, v+ (t, x) ≤ v¯ (t, x). Hence (6.72) follows. The same argument can be made for the negative part of v.  An application of this lemma is the next corollary. Corollary 6.12 (Corollary 1 in [34]) Let u be a solution of the F-KPP equation satisfying the standard conditions, and assume that u(t, x + m(t)) → wλ (x),

as t ↑ ∞,

(6.75)

uniformly in x. Then, m(t + s) − m(t) → λs,

(6.76)

uniformly for s in compact intervals. Hence, m(t)/t → λ. Proof Since wλ is stationary, the previous lemma implies that, for all t ≥ 0 and s0 ≥ 0, uniformly in s ≤ s0 , ( ( ( ( sup ((u(t + s, x + λs + m(t)) − wλ (x)(( ≤ e s0 ((u(t, x + m(t)) − wλ (x)(( , (6.77) x∈R

6.2 The Maximum Principle and its Applications

89

where the right-hand side tends to zero by assumption. The next lemma states a quantitative comparison result. Thus sup |u(t + s, x + λs + m(t)) − u(t + s, x + m(t + s))| → 0,

(6.78)

x

as t ↑ ∞, which implies m(t + s) − m(t) → λs.



Lemma 6.13 (Lemma 3.3 in [34]) Let ui , i = 1, 2, be solutions of the F-KPP equation with F  (u) ≤ 1. If 0 ≤ ui (0, x) ≤ 1 for all x, and for x > 0 it holds that u1 (0, x) = u2 (0, x), then there exists a constant, C, such that (( 2 ( (u (t, x) − u1 (t, x)(( ≤ Ct−1/4 , for x >

(6.79)

(6.80)

√ 2t − 2−5/2 ln t.

Proof The proof is similar to that of Lemma 6.10. With the same definitions as in that proof, we see that, by the Feynman–Kac formula, the solution v¯ of the linear equation is given as √ 0 (x−y)2 et t − x2 et e− 2t dy ≤ √ (6.81) v¯ (t, x) = √ e 2t . 2πt −∞ 2πx √ For x = 2t − 2−5/2 ln t, the right-hand side is bound asymptotically equal to √  t−1/4 /(2 π). The next lemma is the first result linking the behaviour of the tail of the initial distribution to the convergence to a travelling wave. Lemma 6.14 (Lemma 3.4 in [34]) Let √ u be a solution of the F-KPP equation satisfying the standard setting. If λ = 2, assume, in addition, that 1 − F  (u) √= ρ O(u ) for some ρ > 0. If the initial conditions are such that, for some λ ≥ 2, and a function γ(x) ↑ 1, as x ↑ ∞, u(0, x) = γ(x)wλ (x), where wλ is a solution of the stationary equation with speed λ, then ( ( sup ((u(t, x + λt) − wλ (x)(( → 0, as t ↑ ∞,

(6.82)

(6.83)

x≥bλ (t)

√ where bλ (t) = ( 2 − λ)t − 18 ln t. Proof

Set

⎧ ⎪ ⎪ ⎨u(0, x), for x > N, u (0, x) ≡ ⎪ ⎪ ⎩wλ (x), for x ≤ N. N

(6.84)

90

Bramson’s Analysis of the F-KPP Equation

We know that wλ decays exponentially for x → ∞. So, due to (6.82), for any δ > 0, there is N < ∞, such that, for all x ∈ R, wλ (x + δ) ≤ uN (0, x) ≤ wλ (x − δ).

(6.85)

The maximum principle then implies that wλ (x + δ) ≤ uN (t, λt + x) ≤ wλ (x − δ).

(6.86)

On the other hand, for x ≥ b(t) + N, |uN (t, x) − u(t, x)| ≤ Ct−1/4 . Thus, at the expense of a little error of order t−1/4 , we can replace uN by u in (6.86) for such  x. But this gives (6.83), since wλ is uniformly continuous. The following proposition further strengthens the point that, if a solution approaches the travelling wave for large x, then it does so also for smaller x. This will be one main tool to turn the tail asymptotics of Theorem 6.2 into a proof of convergence to the travelling wave. Proposition 6.15 (Proposition 3.3 in [34]) Under the assumptions of Lemma 6.14, if for some N, for all x ≥ N, γ1−1 (x)wλ (x) − γ2 (t) ≤ u(t, x + m(t)) ≤ γ1 (x)wλ (x) + γ2 (t),

(6.87)

where γ1 (x) → 1, as x ↑ ∞, and γ2 (t) → 0, as t ↑ ∞, then there exists c(t) tending to −∞ as t ↑ ∞, such that ( ( sup ((u(t, x + m(t)) − wλ (x)(( → 0. as t ↑ ∞. (6.88) x≥c(t)

Proof The proof of this proposition is a relatively simple application of the maximum principle via the preceding lemma.  Some nice properties hold if F not only satisfies the standard conditions, but is in addition concave, so that F(u)/u is decreasing. This holds if F comes from BBM with binary branching, where F(u)/u = (1 − u). In other situations one can compare solutions with those corresponding to upper and lower concave hulls, F(u) ≡ u max(F(u )/u , u ≥ u), 





F(u) ≡ u max(F(u )/u , u ≤ u).

(6.89) (6.90)

Then the following lemma can be applied. Lemma 6.16 (Lemma 3.5 in [34]) Assume that F is concave and ui for i = 1, 2, 3 are solutions of the F-KPP equation satisfying the standard conditions. Then, u3 (0, x) ≤ u2 (0, x) + u1 (0, x),

for all x,

(6.91)

6.2 The Maximum Principle and its Applications

91

implies u3 (t, x) ≤ u2 (t, x) + u1 (t, x),

for all x and for all t ≥ 0.

(6.92)

Similarly, for any 0 < c ≤ 1, if u1 (0, x) ≥ cu2 (0, x),

(6.93)

u1 (t, x) ≥ cu2 (t, x)

(6.94)

for all x, then

holds for all t ≥ 0 for all x. Proof The proof is a straightforward application of the maximum principle.  The following proposition compares the convergence of general initial data to the Heaviside case that was studied in Kolmogorov’s theorem. Proposition 6.17 (Proposition 3.4 in [34]) Let u be a solution of the F-KPP equation that satisfies the standard conditions with concave F. Assume that, for some t0 ≥ 1 and η > 0, u(t0 , 0) ≥ η.

(6.95)

Then, for all 0 < θ < 1 and  > 0, there exists T (independent of η) and a constant cθ such that, for t ≥ T , √   (6.96) u(t + t0 − θ−1 ln η, x) > w 2 |x| − m1/2 (t) + cθ − . In particular, for all δ > 0, u(t + t0 − θ−1 ln η, x) → 1,

(6.97) √ uniformly in |x| ≤ ( 2 − δ)t. If 1 − F(u) ≤ uρ , then we may choose θ = 1. Proof

We need some preparation.

Lemma 6.18 (Lemma 3.6 in [34]) Assume that u(t0 , 0) ≥ η, for some t0 ≥ 1 and η > 0. Then, for either J = [0, 1] or J = [−1, 0], u(t0 , x) ≥ η/10,

for all x ∈ J.

(6.98)

Proof We bound the solution by that of the linear F-KPP equation and use the Feynman–Kac representation. This implies that 2 e−y /2 (6.99) u(t0 , 0) ≤ e u(t0 − 1, y) √ dy. R 2π

92

Bramson’s Analysis of the F-KPP Equation

This is bounded from above by ⎞ ⎛ 0 ∞ 2 2 ⎜⎜⎜ e−y /2 e−y /2 ⎟⎟⎟ u(t0 − 1, y) √ dy, u(t0 − 1, y) √ dy⎟⎠ . 2e max ⎝⎜ −in f ty 0 2π 2π

(6.100)

Let us say that the maximum is realised for the integral over the positive half line. Then use that, for ∈ [0, 1], max e+(y−x)

2

/2−y2 /2

y≥0

Thus

u(t0 , 0) ≤ e3/2

∞

= ex

2

/2

≤ e1/2 .

e−(x−y) /2 dy. √ 2π 2

u(t0 − 1, y)

0

Now use that ∂t u ≥ ∂2xx u to see that ∞ 2 e−(x−y) /2 u(t0 − 1, y) √ dy ≤ u(t0 , x). −∞ 2π

(6.102)

(6.103) 

This gives (6.98). Lemma 6.19

(6.101)

Let u(t, x) be as usual and ⎧ ⎪ ⎪ ⎨η, u(0, x) = ⎪ ⎪ ⎩0,

x ∈ J, x  J,

(6.104)

for some interval J. For any θ ∈ (0, 1), there is C1θ > 0 such that, for t ≤ θ−1 ln(C1θ /η), and all x ∈ R, θt



e−(x−y) /(2t) dy. √ 2πt 2

u(t, x) ≥ ηe

J

If (5.32) holds, there is C1 > 0, such that, for t < − ln η, −(x−y)2 /(2t) e u(t, x) ≥ C1 ηet dy. √ J 2πt

(6.105)

(6.106)

Proof The key idea of the proof is to exploit the fact that if u is small, then F(u)/u is close to one. An upper bound then shows that, for a short time, u(t, x) will grow as claimed. We introduce a comparison solution uθ with the same initial conditions as u and with F(u) replaced by F θ (u) ≡ F(u) ∧ (θu). Clearly F θ (u) ≤ F(u) and so, by the maximum principle, u(t, x) ≥ uθ (t, x), for all t ≥ 0 and x ∈ R. On the other hand, because ∂t uθ ≤

1 2 θ ∂ u + θuθ , 2 xx

(6.107)

6.2 The Maximum Principle and its Applications

93

we have the trivial upper bound uθ (t, x) ≤ ηeθt ,

(6.108)

for all t ≥ 0 and x ∈ R. Since F  (u) ≤ 1, for given θ < 1 there will be a C1θ > 0 such that, for all 0 ≤ u ≤ C1θ , F(u)/u ≤ θ, and hence F θ (u)/u = θ. By the bound (6.108), this is true at least up to time t∗ = θ−1 ln(C1θ /η). This gives the claimed lower bound. In the case when (5.32) holds, we can choose θ = 1 and

 uθ = u. Then we know that F(u(t, x))/u(t, x) ≥ 1 − C2 ηet ρ , which we can use up to t∗ = − ln η. Again using the Feynman–Kac formula, we get for, t < t∗ , −(x−y)2 /(2t) 6t e s ρ u(t, x) ≥ e− 0 (1−C2 (ηe ) ) ds dy. (6.109) √ J 2πt But t∗ (ηe s )ρ ds ≤ ρ−1 , (6.110) 0

and hence −C2 /ρ

u(t, x) ≥ e

ηe

e−(x−y) /(2t) dy, √ 2πt 2

t J

(6.111) 

which is what is claimed.

We now prove the proposition. We consider only the case when (5.32) holds and thus set θ = 1. Set t1 = − ln η, t2 = t0 + t1 . The two preceding lemmas give −(x−y)2 /(2t1 ) e u(t2 , x) ≥ C1 dy, (6.112) √ 2πt1 J with J either [−1, 0] or [0, 1]. Thus, for |x| ≤ 1, we get, again from Lemma 6.19, that u(t2 + 1, x) ≥ C4 .

(6.113)

We can compare u(t + t2 + 1, x) ≥ u1 (t, x), where the initial conditions for u1 are 1(−1,1] (x)C4 . Let u2 be the solution with Heaviside initial conditions. We get   u1 (t, x) ≥ C4 u2 (t, x) − u2 (t, x + 1) . (6.114) Now set x = z + m1/2 (t). Then the right-hand side of (6.114) converges to ' & √ √ (6.115) C4 w 2 (z) − w 2 (z + 1) . √

Using the tail behaviour of w 0 < c1 < ∞, and hence

2

√

, this is, for large z, w

u1 (t, z + m1/2 (t) − c1 ) ≥ γ1−1 (z)w

√

2

2

(z + c1 ), for some

(z) − γ2 (t).

(6.116)

94

Bramson’s Analysis of the F-KPP Equation

Then, for all  > 0, for large enough t, u1 (t, z + m1/2 (t) − c1 ) ≥ w

√

2

(z) − ,

(6.117)

if z ≥ c(t), where c(t) is from Proposition 6.15 and tends to −∞ as t ↑ ∞.. This also implies that, for large enough t, u1 (t, c(t) + m1/2 (t) − c1 ) ≥ w

√ 2

(c(t)) − /2 ≥ 1 − .

(6.118)

Since u1 (t, x) is decreasing in x for x ≥ 0, we have that, for 0 ≤ x ≤ m1/2 (t) + c(t) − c1 , u1 (t, x) ≥ 1 −  ≥ w

√

2

(x − m1/2 (t) + c1 ) − .

(6.119)

On the other hand, for x ≥ m1/2 (t) + c(t) − c1 , it follows from (6.117) that u1 (t, x) ≥ w

√

2

(x − m1/2 (t) + c1 ) − .

This implies that, for all x ≥ 0, u(t + t0 − ln η + 1, x) ≥ u1 (t, x) ≥ w

√

2

(x − m1/2 (t) + c1 ) − .

(6.120)

(6.121)

This can be written as u(t + t0 − ln η, x) ≥ u1 (t, x) ≥ w

√

2

(x − m1/2 (t) + c) − ,

(6.122)

with a slightly changed constant c. The case when x < 0 follows similarly. √ √ Finally, since m1/2 (t) ∼ 2t and w 2 (z) ↑ 1, as z ↓ −∞, it follows that u(t + t0 − ln η, x) ↑ 1, √ uniformly in t if |x| ≤ ( 2 − δ)t, which is (6.97).

(6.123) 

The final proposition of this section asserts that the scaling function m1/2 (t) is essentially concave. Proposition 6.20 (Proposition 3.5 in [34]) Let u be a solution of the F-KPP equation satisfying the standard conditions and Heaviside initial conditions. Assume further that F is concave. Then there exists a constant M > 0 such that, for all s ≤ t, s m1/2 (s) − m1/2 (t) ≤ M. (6.124) t Proof Eqution (6.124) will follow from the fact that, for all t1 ≤ t2 and s ≥ 0, m1/2 (t1 + s) − m1/2 (t1 ) ≤ m1/2 (t2 + s) − m1/2 (t2 ) + M.

(6.125)

Namely, assume that (6.124) fails for some s ≤ t. Define T = inf{r ∈ [s, t] : m1/2 (r)/r = α},

(6.126)

6.3 Estimates on the Linear F-KPP Equation

95

where α ≡ inf q∈[s,t] m1/2 (q). Let t1 = sup{r ∈ [0, s] : m1/2 (r)/r = α}. Put t2 = s, and S = T − s. Then m1/2 (t2 ) > αt2 + M,

and

m1/2 (t2 + S ) = α(t2 + S ).

(6.127)

Hence m1/2 (t2 + S ) − m1/2 (t2 ) < αS − M.

(6.128)

Furthermore, m1/2 (t1 ) = αt1 , and m1/2 (t1 + S ) ≥ α(t1 + S ), so that m1/2 (t1 + S ) − m1/2 (t1 ) ≥ αS . Equations (6.128) and (6.129) contradict (6.125). It remains to prove (6.125). But, by Corollary 6.54, for t2 ≥ t1 , ⎧ ⎪ ⎪ ⎨u(t1 , x + m1/2 (t1 )), if x > 0, u(t2 , x + m1/2 (t2 )) ≥ ⎪ ⎪ ⎩1/2, if x ≤ 0.

(6.129)

(6.130)

In particular, u(t2 , x + m1/2 (t2 )) ≥

1 u(t1 , x + m1/2 (t1 )). 2

(6.131)

By the concavity result (6.93) in Lemma 6.16, this extends to u(t2 + s, x + m1/2 (t2 )) ≥

1 u(t1 + s, x + m1/2 (t1 )). 2

(6.132)

Since u is monotone, it follows that m1/4 (t2 + s) − m1/2 (t2 ) ≥ m1/2 (t1 + s) − m1/2 (t1 ). Moreover, it follows from the second assertion of Corollary 6.9, that u drops from 1/2 to 1/4 uniformly faster than w1/2 , so that m1/4 (t) − m1/2 (t) is bounded uniformly in t. This proves (6.125). 

6.3 Estimates on the Linear F-KPP Equation The linear F-KPP equation, ∂t φ =

1 2 ∂ φ + φ, 2 x

(6.133)

has already served as a reference in the previous sections. We now derive some precise estimates of the behaviour of its solutions from the explicit representation ∞ (x−y)2 et φ(t, x) = √ (6.134) u(0, y) e− 2t dy. 2πt −∞

96

Bramson’s Analysis of the F-KPP Equation

This representation reduces the work to get good bounds into standard estimates on Gaussian integrals, i.e. the Gaussian tail bounds (1.12) and cutting up the range of integration. √ Recall that λ, b are in the relation b = λ − λ2 − 2. We first prove an upper bound. Lemma 6.21 (Lemma 4.1 in [34]) Assume that, for some h > 0 and for some x0 , for all x ≥ t0 , x(1+h) u(0, y)dy ≤ e−bx . (6.135) x

Then, for each δ ≥ 0, and a constant C, (x−y)2 et 2 u(0, y)e− 2t dy ≤ t−1 eC e−b(x−λt)−δ t/3 , √ 2πt J c

(6.136)

for all t > 1 and for all x, where λ=

1 b + , b 2

(6.137)

and J = (x − (b + δ)t, x − (b − δ)t). Proof



The proof of this bound is elementary. See [34].

Bramson states a variation on this bound as the following: Corollary 6.22 (Corollary 1 in [34]) Assume that, for h > 0, for some 0 < √ b ≤ 2,  x(1+h)  −1 u(0, y)dy ≤ −b. (6.138) lim sup x ln x↑∞

x

Then, for δ ≥ 0 and 0 <  < b, 2 e−(x−y) /2t 2 et u(0, y) √ dy ≤ e−b(x−λt)+ x−δ t/4 , Jc 2πt

(6.139)

for all x and t large enough. J is as in the previous lemma. In particular, ln φ(t, x) ≤ −b(x − λt) +  x.

(6.140)

We need a sharper bound on φ(t, x). Under the hypothesis of Lemma 6.21, for all x, ∞ t0 −x 2 y2 1 1 − y2t φ(t, x) ≤ √ e−by− 2t dy. e dy + K √ 2πt t0 −x 2πt −∞

Lemma 6.23

(6.141)

97

6.3 Estimates on the Linear F-KPP Equation In particular, for x ≥ λt, x2

xt0

et− 2t + t + Ke−(x−λ)t . φ(t, x) ≤ + √ 2πx/ t

(6.142)

Proof We change coordinates so that the integral in (6.134) takes the form ∞ u(x + y) exp(−y2 /(2t))dy. (6.143) −∞

We split the integral in (6.134) into the part from −∞ to t0 − x and the rest. In the first part, we bound u by 1. In the second, we split the integral into pieces of length one, i.e. we write ∞ ∞ n+1+t0 −x  2 −y2 /(2t) u(0, x + y)e dy = u(0, x + y)e−y /(2t) dy t0 −x

≤ ≤

∞  n=0 ∞  n=0

 exp −  exp −

n=0

n+t0 −x

min

y∈[n+t0 −x,n+1+t0 −x]

min

y∈[n+t0 −x,n+1+t0 −x]



y /(2t) 2

n+1+t0 −x

n+t0 −x

u(0, x + y)dy



y /(2t) − b(n + t0 ) . 2

(6.144)

We would like to reconstruct the integral from the last line. To do this, note that, for any n, max

y∈[n+t0 −x,n+1+t0 −x]

y2 /(2t) −

min

y∈[n+t0 −x,n+1+t0 −x]

y2 /(2t)5 ≤ (2(n + t0 − x ± 1) + 1)/2t,

(6.145) where ± depends on whether n+t0 − x is positive or negative. For all n such that 2|n+t0 − x| ≤ Ct, this is bounded by a constant, while otherwise, summands are anyway smaller than exp(−Ct), which is negligible if C is chosen large enough. Hence   ∞ ∞  2 2 C −bx exp − min y /(2t) − b(n + t0 ) ≤ e e e−by e−y /(2t) dy. n=0

y∈[n+t0 −x,n+1+t0 −x]

t0 −x

(6.146) From here (6.141) follows. The last integral is bounded from above by √ 2 (6.147) e−bx+b t/2 2πt. Combining this with a standard Gaussian estimate for the first integral in (6.141) and recalling the definition of λ gives (6.142).  The next lemma provides a lower bound.

98

Bramson’s Analysis of the F-KPP Equation

Lemma 6.24

Assume that, for some h > 0,  x(1+h)  −1 u(0, y)dy ≥ −b, lim inf x ln x↑∞

(6.148)

x

√ with 0 < b < 2. Let M > λ be fixed. Then, for all  > 0, there is T > 0 such that, for (t, x) ∈ AT and x ≤ Mt, ln φ(t, x) ≥ b(x − λt) −  x,

(6.149)

where AT = {(t, x) : t ≥ T, x ≥ b0 t} ∪ {(t, x) : 1 ≤ t ≤ T, x ≥ b0 t}. Corollary 6.25 Let mφ (t) be defined by φ(t, mφ (t)) = 1/2. If  x(1+h)  −1 lim x ln u(0, y)dy = −b, x↑∞

for 0 < b
0,   P ∃0≤s≤t : zt (s) ≥ (sy1 + (t − s)y2 )/t = e−2y1 y2 /t . (ii) For x > 0 and 0 < s0 < t, 

P ∃0≤s≤s0

√ 2 s0 −x2 /(2s0 ) : z (s) ≥ x ≤ . e x t



(6.152)

(6.153)

6.4 Brownian Bridges

99

(iii) For y1 , y2 > 0 and 0 ≤ x ≤ y2 /2, (6.154) p(t, x; ∀0≤s≤t : zts ∈ [−y1 , y2 ]) ≥ 8   2 2 2 y1 e−x /(2t) x − (y1 + 2y2 ) ey2 /(2t) − y1 (x + y1 )2 /t , 3 πt where p(t, x; A) denotes the density of the Brownian motion restricted to the event A. (iv) For y1 , y2 > 0 and c0 , let zty1 ,y2 be a Brownian bridge from y1 to y2 in time t. Then   (6.155) P ∀0≤s≤t : zty1 ,y2 (s) > c0 (s ∧ (t − s)) 8 ∞    2 2 1 − e−4y1 x/t 1 − e−4y2 x/t e−2(x+c0 t/2−(y1 +y2 )/2) /t dx. = πt 0 Proof The proof is a nice exercise using the reflection principle (see Figure 6.1). We first prove (i). It is obvious that the probability in question is the same as   P ∃0≤s≤t : zt0,y2 −y1 (s) ≥ y2 . (6.156)

Figure 6.1 Use of the reflection principle.

But all paths that do not stay below y2 have a reflected counterpart that ends in y2 + y1 instead of in y2 − y1 . Thus the probability of BM ending in y2 − y1 and crossing the level y2 is the same as the probability of ending in y2 + y1 , hence

100

Bramson’s Analysis of the F-KPP Equation

(6.156) equals p(t, y2 + y1 ) = e−2y1 y2 /t , p(t, y2 − y1 ) where p(t, x) is the heat kernel. This proves (i). To prove (ii), Bramson uses that  

 P ∃ s≤s0 : zts > x ≤ 2P ∃ s≤s0 : Bs > x .

(6.157)

(6.158)

This can be shown as follows. Since the law of the Brownian bridge at time t/2 is the same as that of Bt/2 , the probability of the bridge exceeding a level x on an interval [0, s1 ] for s1 ≤ t/2 is the same as that of Brownian motion. This gives the result if s0 ≤ t/2. If s0 > t/2, we have       P ∃ s≤s0 : zts > x ≤ P ∃ s≤t/2 : zts > x + P ∃ s∈[t/2,s0 ] : zts > x . (6.159) For the second probability, we have that       P ∃ s∈[t/2,s0 ] : zts > x = P ∃ s∈[t−s0 ,t/2] : zts > x ≤ P ∃ s∈[0,t/2] : zts > x . (6.160) As both probabilities are the same as that, for Brownian motion, we get a rather crude bound with the factor 2 in (6.158). Applying the reflection principle to the Brownian motion probability and using the standard Gaussian tail asymptotics then yields the claimed estimate. To prove (iii), one uses the reflection principle to express the restricted density as p(t, x; ∀0≤s≤t : zt (s) ∈ [−y1 , y2 ])

(6.161)

= p(t, x) − p(t, x + 2y1 ) − p(t, x − 2y2 ) + p(t, x − 2y1 − 2y2 ). Inserting the explicit expression for the heat kernel, this equals  1  −x2 /2t 2 2 2 − e−(x+2y1 ) /2t − e−(x−2y2 ) /2t + e−(x−2y1 −2y2 ) /2t e √ 2πt 2  e−x /2t  1 − e−2y1 (x+y1 )/t − e−2y2 (y2 −x)/t + e−2(y1 +y2 )(y1 +y2 −x)/t = √ 2πt 2   e−x /2t  1 − e−2y1 (x+y1 )/t − e−2y2 (y2 −x)/t 1 − e2y1 (x−2y2 −y1 )/t . = √ 2πt (6.162) Using that, for u > 0, e−u ≥ 1 − u and e−u ≤ 1 − u + u2 /2, the right-hand side is bounded from below by  2e−x /2t  y1 (x + y1 ) − y1 (x + y1 )2 /t + e−2y2 (y2 −x)/t (x − 2y2 − y1 ) . √ 3 2πt 2

(6.163)

101

6.4 Brownian Bridges

Using the assumption x ≤ y2 /2 to simplify this expression yields the assertion of (iii). We skip the proof of part (iv), which is a simple application of the estimate (i).  Lemma 6.27 Let 12 < γ < 1. Let zt be a Brownian bridge from 0 to 0 in time t. Then, for all  > 0, there exists r large enough such that, for all t > 3r,   P ∃s ∈ [r, t − r] : |zt (s)| > (s ∧ (t − s))γ < . (6.164) More precisely, ∞    1 2γ−1 k 2 −γ e−k /2 P ∃s ∈ [r, t − r] : |zt (s)| > (s ∧ (t − s))γ < 8

(6.165)

k=r

(see Figure 6.2).

Figure 6.2 The Brownian bridge stays with high probability in the shaded area.

Proof The probability in (6.164) is bounded from above by t−r 

  P ∃s ∈ [k − 1, k] : |zt (s)| > (s ∧ (t − s))γ

k=r

≤2

t/2 

  P ∃s ∈ [k − 1, k] : |zt (s)| > (s ∧ (t − s))γ ,

(6.166)

k=r

by the reflection principle for the Brownian bridge. This in turn is now bounded from above by 2

t/2  k=r

  P ∃s ∈ [0, k] : |zt (s)| > (k − 1)γ .

(6.167)

102

Bramson’s Analysis of the F-KPP Equation

Using the bound of Lemma 6.26 (ii), we have   1 2γ−1 P ∃s ∈ [0, k] : |zt (s)| > (k − 1)γ ≤ 4(k − 1) 2 −γ e−(k−1) /2 .

(6.168)

Using this bound for each summand in (6.167), we obtain (6.165). Since the sum on the right-hand side of (6.165) is finite, (6.164) follows.  An important fact in the analysis of Brownian bridges is that hitting probabilities can be approximated by looking at discrete sets of times. This is basically a consequence of continuity of Brownian motion. Let : [0, t] → R be a function that is bounded from above. Let S k ≡ {sk,1 , . . . , sk,2k } ⊂ [0, t] be such that, for all k, j, sk, j ∈ [( j − 1)t2−k , jt2−k ], and that Set S n ≡

9 k≤n

(sk, j ) ≥ sup{ (s), s ∈ [( j − 1)t2−k , jt2−k ]} − 2−k .

(6.169)

S k , and set

⎧ ⎪ ⎪ ⎨ (x), (s) ≡ ⎪ ⎪ ⎩−∞, Sn

if s ∈ S n , otherwise.

(6.170)

Lemma 6.28 (Lemma 2.3 in [34]) With the notation above, the event A ≡ {z(s) > (s), ∀0 ≤ s ≤ t} is measurable, and   n (6.171) lim P z(s) > S (s), ∀0 ≤ s ≤ t = P(A). n↑∞

We will not give the proof which is fairly straightforward.

6.5 Hitting Probabilities of Curves We will need to compare hitting probabilities of different curves. In particular, we want to show that these are insensitive to slight deformations. The intuitive reason is related to what is called entropic repulsion: if a Brownian motion cannot touch a curve, it will actually stay quite far way from it, with large probability. Let Lt be a family of functions defined on [0, t]. For C > 0 and 0 < δ < 1/2, define ⎧ ⎪ ⎪Lt (s) − Csδ , if 0 ≤ s ≤ t/2, ⎨ (6.172) Lt (s) ≡ ⎪ ⎪ ⎩Lt (s) − C(t − s)δ , if s/2 ≤ s ≤ t, and

⎧ δ ⎪ ⎪ ⎨Lt (s) + Cs , Lt (s) ≡ ⎪ ⎪ ⎩Lt (s) + C(t − s)δ ,

if 0 ≤ s ≤ t/2, if s/2 ≤ s ≤ t.

(6.173)

6.5 Hitting Probabilities of Curves Lemma 6.29 (Lemma 6.1 in [34]) for some r0 > 0, Lt (s) ≤ 0,

103

Let Lt be a family of functions such that, ∀s ∈ [r0 , t − r0 ].

(6.174)

Then, for all  > 0, there exists r large enough such that, for all r > r and t > 3r, ((  ((  (( P zt (s) > Lt (s), ∀r ≤ s ≤ t − r (( ((   − 1(( < . (6.175) (( P zt (s) > L (s), ∀r ≤ s ≤ t − r (( t Proof In the proof of this lemma, Bramson uses the Girsanov formula to transform the avoidance problem for one curve into that for another curve by just adding the appropriate drift to the BM. In fact, if γt is one curve and γ˜ t is another curve, both starting and ending in 0, then clearly   P zt (s) > γt (s), ∀r ≤ s ≤ t − r   = P zt (s) − γt (s) + γ˜ t (s) > γ˜ t (s), ∀r ≤ s ≤ t − r . (6.176) But zt − γt + γ˜ t is a Brownian bridge under the measure # P, where # P is absolutely continuous with respect to the law of BM with Radon Nikod´ym derivative  t  d# P 1 t (∂ s β(s))2 ds , = exp ∂ s β(s)dzts − (6.177) dP 2 0 0 where β(s) = γ˜ t (s) − γt (s). We would be tempted to use this formula directly with Lt and Lt , but the singularity of sδ would produce a diverging contribution. Since the condition on the Brownian bridge involves only the interval (r, t − r), we can modify the drift on the intervals [0, r] and [t − r, t]. Bramson uses ⎧ ⎪ ⎪ if 0 ≤ s ≤ r, 2Crδ−1 s, ⎪ ⎪ ⎪ ⎪ ⎪ δ ⎪ ⎪ 2Cs , if r ≤ s ≤ t/2, ⎨ (6.178) βr,t (s) ≡ ⎪ ⎪ δ ⎪ ⎪ if t/2 ≤ s ≤ t − r, 2C(t − s) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩2Crδ−1 (t − s), if t − r ≤ s ≤ t. The discussion above clearly shows that   P zt (s) > Lt (s), ∀r < s < t − r ) 6t * 6 t 1 t 2 = E e− 0 ∂s βr,t (s)dzs − 2 0 (∂s βr,t (s)) ds 1zt (s)>Lt (s),∀r 0, ⎧ ⎪ if r ≤ s ≤ t/2, L(s + sδ ) + Csδ , ⎪ ⎪ ⎪ ⎪ ⎨ δ δ (6.182) θr,t ◦ L(s) ≡ ⎪ L(s + (t − s) ) + C(t − s) , if t/2 ≤ s ≤ t − 2r, ⎪ ⎪ ⎪ ⎪ ⎩L(s), otherwise. Then the following holds. Proposition 6.30 (Proposition 6.2 in [34]) For all  > 0, there exists r large enough such that, for all r > r and all t > 3r, ((  ((  −1 (( P zt (s) > (θr,t ◦ Lt (s)) ∨ (θr,t (( ◦ L(s)) ∨ L(s), ∀r ≤ s ≤ t − r (( (( <  − 1 (( (( P (zt (s) > Lt (s), ∀r ≤ s ≤ t − r) (6.183) and (( (( (( P zt (s) > L (s), ∀r ≤ s ≤ t − r (( t ((   − 1(( < . (6.184) (( P zt (s) > θ−1 ◦ L (s), ∀r ≤ s ≤ t − r (( t r,t Finally, Bramson proves the following monotonicity result. Proposition 6.31 (Proposition 6.3 in [34]) Let i , i = 1, 2, be upper semicontinuous at all but finitely many values of 0 ≤ s ≤ t, and let 1 (s) ≤ 2 (s), for all 0 ≤ s ≤ t. Let z x,y denote the Brownian bridge from x to y in time t. Then

P z x,y (s) > 2 (s), ∀0≤s≤t

(6.185) P z x,y (s) > 1 (s), ∀0≤s≤t

105

6.6 Asymptotics of Solutions of the F-KPP Equation is monotone increasing in x and y.

6.6 Asymptotics of Solutions of the F-KPP Equation We are now moving closer to proving Theorem 6.2. We still need some estimates on solutions. The first statement is a regularity estimate. Proposition 6.32 (Proposition 7.1 in [34]) Let u be a solution of the F-KPP equation satisfying the standard conditions. Assume that, for some h > 0,  t(1+h)  √ u(0, y)dy ≤ − 2. (6.186) lim sup t−1 ln t↑∞

t

Then lim sup (u(t, x2 ) − u(t, x1 )) = 0. t↑∞ 0≤x1 ≤x2

If, moreover, for some η > 0, N, M > 0, for x < −M, x+N u(0, y)dy ≥ η,

(6.187)

(6.188)

x

then (6.187) holds also if the condition in the supremum is relaxed to x1 ≤ x2 . √ Proof Equation (6.97) from Proposition 6.17 implies that, for x ∈ [0, ( 2 − δ)t], u(t, x) → 1, for any δ > 0. By the Feynman–Kac representation (6.2), using Condition (6.188), for all x < 0, ∞ (x−y)2 1 u(1, x) ≥ √ e− 2 u(0, y)dy (6.189) 2π −∞ x+N−M 1 2 2 e− max(M ,(N−M) )/2 u(0, y)dy ≥ η > 0. ≥ √ 2π x−M Hence u(t, x) → 1 also for x ∈ (−∞, 0], uniformly. Thus √ we only need to show that (6.187) holds with the supremum restricted to t( 2 − δ) ≤ x1 ≤ x2 . To do so, define, for s ≥ 0 and 0 < δ1 < 1, u2 as the solution with initial data ⎧ ⎪ ⎪ ⎨u(0, x), if x ≤ δ1 s, (6.190) u s (0, x) = ⎪ ⎪ ⎩0, if x > δ1 s. Now Lemma 6.10 implies that, for δ1 t ≤ x1 ≤ x2 , ut (t, x2 ) ≤ ut (t, x1 ).

(6.191)

106

Bramson’s Analysis of the F-KPP Equation

If we set v s (t, x) ≡ u(t, x) − u s (t, x), then ⎧ ⎪ ⎪ ⎨0, s v (0, x) = ⎪ ⎪ ⎩u(0, x),

if x ≤ δ1 s, if x > δ1 s.

(6.192)

But v s satisfies an equation ∂t v s =

1 2 s ∂ v ) + k s (t, x)v s , 2 x

(6.193)

where k s is some function of the solutions u, u s . The important point is, however, that k s (t, x) ≤ 1. Thus the Feynman–Kac representation yields the bound (x−y)2 1 t ∞ t v (t, x) ≤ √ e u(0, y) e− 2t dy. (6.194) δ1 s 2πt Using the growth bound (6.186) on the initial condition, it follows from Corol√ lary 6.22 that vt (t, x) → 0, uniformly in x ≥ t( 2 − δ21 /6). Using this together with (6.191) implies the desired convergence and proves the proposition.  The following propositions prepare for a more sophisticated use of the Feynman–Kac representation (6.6). This used in particular the comparison estimates of the hitting probabilities of different curves by the Brownian bridges. We use the notation of the previous section and fix the reference curve s (6.195) Lr,t (s) = m1/2 (s) − m1/2 (t)t−1 (t − s)α(r, t), t where m1/2 is defined in Eq. (6.50); α(r, t) = o(t) will be specified later. Next we introduce the functions Mr,t (s) ≡ Lr,t (s) + st m1/2 (t) +

t−s t α(r, t)

= m1/2 (s) + Lr,t (s) − Lr,t (s) (6.196)

t−s t α(r, t)

= m1/2 (s) + Lr,t (s) − Lr,t (s). (6.197)

and Mr,t (s) ≡ Lr,t (s) + st m1/2 (t) +

At this point we can extend the rough estimate on m1/2 (t) given in (6.53) for Heaviside initial conditions to initial conditions satisfying (6.186). Lemma 6.33√ Assume that u(0, x) satisfies (6.186). Then (6.53) holds, i.e. m1/2 (t)/t → 2. Proof Recall that, by the maximum principle, u(t, x) ≤ φ(t, x) with φ defined as (6.134). On the other hand, it is fairly straightforward to show that, if the φ of m1/2 (t) defined for φ initial data satisfy √ (6.186), then m (t), i.e. the analogue √ behaves like 2t. Hence lim supt m1/2 (t)/t ≤ 2. The bound (6.97) in Propo√  sition 6.17 implies, on the other hand, that lim inf t m1/2 (t)/t ≥ 2.

107

6.6 Asymptotics of Solutions of the F-KPP Equation

The strategy of using the Feynman–Kac formula relies on controlling the term k(y, s) appearing in the exponent in appropriate regions of the plane, and then establishing probabilities that the Brownian bridge passes through those. The next proposition states that k is close to one above the curves M. Proposition 6.34 (Proposition 7.2 in [34]) Let u be a solution of the F-KPP equation satisfying the standard assumptions and with initial data satisfying (6.186) for some h > 0. Then, there is a constant C2 such that, for r large enough and y > Mr,t (s), one has that ⎧ −ρsδ ⎪ ⎪ if r ≤ s ≤ t/2, ⎨1 − C2 e , (6.198) k(s, y) ≥ ⎪ δ ⎪ −ρ(t−s) ⎩1 − C2 e , if t/2 ≤ s ≤ t − r. Thus

6 t−r

e3r−t e 2r

k(t−s,x(s))ds

→ 1,

as r ↑ ∞,

(6.199)

if x(s) > Mr,t (t − s) for s ∈ [2r, t − r]. Proof We give a proof of of this proposition under the slightly stronger hypothesis of Lemma 6.133. The strategy is the following: (i) Prove (6.199) with m1/2 (s) replaced by any function m(s) √ √ for Mt,t defined such that 2s ≥ m(s) ≥ 2s − C ln s, for√any C < ∞. (ii) Use this bound to control u(t, x) for x ≥ 2t − C ln t. (iii) Deduce from this that √ 3 m1/2 (t) ≥ 2t − √ ln t + M, (6.200) 2 2 for M < ∞. We begin with step (i) and assume that the m1/2 in the definition of M is stated in item (i). By definition, for s ∈ [r, t/2], after some tedious computations, Mr,t (s) ≥ m1/2 (s + sδ ) + (4 + (α(r, t) − m1/2 (t))/t)sδ .

(6.201)

By Lemma 6.33, for large enough r < t, the quantity in the brackets is positive and hence Mr,t (s) ≥ m1/2 (s + sδ ).

(6.202)

We use that u(s, x)√ ≤ φ(s, x), and that, by Lemma 6.23, under the assumption (6.138) with b = 2, δ

φ(s, m1/2 (s) + sδ + z) ≤ e s−(m1/2 (s+s )) /(2s)+t0 ≤ et0

√

√ 2 − 2sδ

e

2

+ Ke−

√

2

δ

+ Ke−(m1/2 (s+s )−

√ δ 2s

.

√ 2s

(6.203)

108

Bramson’s Analysis of the F-KPP Equation

Hence, for y > Mr,t (s) and s ∈ [r, t/2], u(y, s) ≤ C3 e−

√ δ 2s

.

(6.204)

The analogous result follows in the same way for s ∈ [t/2, t − r]. Using that F satisfies the standard conditions then implies (6.198). The conclusion (6.199) is obvious. We now come to step (ii). √ Lemma√6.35 Let u be as in the proposition. Let x = x(t) ≡ 2t − Δ(t), where Δ(t) ≤ 2t − C ln t, for some C < ∞. Then √ ∞ 2 √ e 2y e−y /2t −1 2 2Δ(t) u(0, y) √ dy. (6.205) u(x, t) ≥ Ct e y0 2πt Proof We know from step (i) that, for our choices of M¡ for z x,y (s) > Mr,t (s) on [2r, t − r],  t−r  exp k(s, z x,y (s)ds ≥ C3 et , (6.206) 2r

C3 depends on r but not on t. Now choose y0 small enough such that 6where ∞ u(0, y)dy > 0. Now we use the Feynman–Kac representation in the form y0 (6.7) (with r = 0) to get the bound ∞ 2 * 6 t−r e−(x−y) /2t ) u(0, y) √ u(x, t) ≥ E 1zx,y (s)>Mr,t (t−s),∀s∈[2r,t−r] e 2r k(s,zx,y (s)ds dy −∞ 2πt ∞ 2  e−(x−y) /2t  t u(0, y) √ P z x,y (s) > Mr,t (t − s), ∀ s∈[2r,t−r] dy. ≥ C3 e y0 2πt (6.207) The curves M, M can be chosen with α(r, t) = y0 , with y0 fixed (usually negative). One can always choose m(s) such that m(s) − st m(t) ≤ M < ∞. Then, for large enough r, the probability in the last line can be bounded from below using our comparison results for hitting probabilities of Brownian bridges, which imply that     P z x,y (s) > Mr,t (t − s), ∀ s∈[r,t−r] ≥ C1 P z x,y (s) > Mr,t (t − s), ∀ s∈[r,t−r] 

(6.208) ≥ C1 P z(s) > 0, ∀ s∈[r,t−r] , where in the last line z ≡ z0,0 . Using (5.8) and the fact that there is a positive probability that, at times r and t − r, the Brownian bridge is larger than, say, 1, one sees that there is a constant, C, depending only on r, such that P (z(s) > 0, ∀s ∈ [r, t − r]) ≥ C/t.

(6.209)

6.6 Asymptotics of Solutions of the F-KPP Equation Hence we have that u(t, x) ≥ C4 t−1 et



∞

2

u(0, y) y0

Inserting x = (6.205).

e−(x−y) /2t . √ 2πt

109

(6.210)

√ 2t − Δ(t) and dropping terms that tend to zero, we arrive at 

It is now easy to get the desired lower bound on m1/2 . Proposition 6.36 (Proposition 8.1 in [34]) Under the hypothesis stated above, there exists a constant C0 < ∞ and t0 < ∞ such that, for all t ≥ t0 , √ 3 (6.211) m1/2 (t) ≥ 2t − √ ln t + C0 ≡ n(s). 2 2 Proof We insert into (6.205) Δ(t) = u(t,

√

2t − Δ(t)) ≥ Ct

D √ 2 2

−3/2 ln t)

ln t − z. Then we see that



√

∞

e

u(0, y) y0

≥ Ct−3/2 eD ln t−)



e

2y −y2 /2t

e √ 2π

A

u(0, y) e y0

√ 2y −y2 /2t

e

(6.212) dy

√

˜ y0 )t D−3/2 e−2 2z , ≥ C(A, 6A for some A < ∞ chosen such that y u(0, y)dy > 0, where we have just re0 placed the exponentials in the integral by their minima on [y0 , A]. Clearly, this can be made equal to 1/2 by setting D = 3 and choosing finite z appropriately. This implies the assertion for m1/2 .  Having shown that the true m1/2 satisfies the hypothesis made in step (i), we see that the assertion (6.198) holds for M defined with the true m1/2 . This concludes the proof of Proposition 6.34.  Remark Bramson proves this proposition under the weaker hypothesis (6.186). His argument goes as follows. First, by definition, u(s+σδ , m1/2 (s+ sδ )) = 1/2. Next set v(t0 , 0) ≡ u(s, m1/2 (s + sδ )). Fix s  0. Then v(t, x) solves the F-KPP equation and v(sδ , 0) = u(s, m1/2 (s + sδ )). But by (6.97) in Proposition 6.17, if v(t0 , 0) = η, then v(t + t0 − ln η, x) ↑ 1, as t ↑ ∞. But if sδ + ln η ↑ ∞, then v(sδ , 0) ↑ 1, as s∞ , in contradiction to the fact that v(sδ , 0) = 1/2. Hence we d must have that ln η + sδ < ∞, or η ≤ e−σ . This result already suggests that the contributions in the Feynman–Kac representation should come from Brownian bridges that stay above the curves M and thus enjoy the full effect of the k. To show this, one first shows that bridges staying below the curves M do not contribute much.

110

Bramson’s Analysis of the F-KPP Equation

Define the sets

  G x,y (r, t) ≡ z x,y : ∃s ∈ [2r, t − r] : z x,y (s) ≤ Mr,t (t − s)

(6.213)

  G x,y (r1 ; r, t) ≡ z x,y : ∃s ∈ [2r ∨ r1 , t − r1 ] : z x,y (s) ≤ Mr,t (t − s) .

(6.214)

and

Proposition 6.37 (Proposition 7.3 in [34]) Let u be a solution that satisfies (6.186). Assume that α(r, t) = o(t) and that ⎧ s t−s δ ⎪ ⎪ if s ∈ [r, t/2], ⎨ t m1/2 (t) + t α(r, t) + 8s , m1/2 (s) ≤ ⎪ (6.215) ⎪ ⎩ s m1/2 (t) + t−s α(r, t) + 8(t − s)δ , if s ∈ [t/2, t − r]. t t Then, for y > α(r, t) and x ≥ m1/2 (t), for r large enough, ) 6 t−r *   e3r−t E e 2r k(t−s,zx,y (s)ds 1G x,y (r,t) ≤ r−2 P Gcx,y (r, t) .

(6.216)

Proof The proof of this proposition is quite involved and will only be sketched. We need some localising events for the Brownian bridges. Set I j ≡ [ j, j + 1) ∪ (t − j − 1, t − j], for j = 0, . . . , j0 − 1, with j0 < t ≤ j0 + 1, and I j0 = [ j0 , t − j0 ]. Define the exit times   S 1 (r, t) ≡ sup s : 2r ≤ s ≤ t/2, z x,y (s) ≤ Mr,t (t − s) (6.217)   S 2 (r, t) ≡ inf s : t/2 ≤ s ≤ t − r, z x,y (s) ≤ Mr,t (t − s) , (6.218) and

⎧ 1 ⎪ ⎪ ⎨S (r, t), S (r, t) ≡ ⎪ ⎪ ⎩S 2 (r, t),

if S 1 (r, t) + S 2 (r, t) > t, if S 1 (r, t) + S 2 (r, t) ≤ t.

(6.219)

Then define A j (r, t) ≡ {z x,y : S (r, t) ∈ I j },

j = 0, . . . , j0 ,

(6.220)

and further, A1j (r, t) ≡ {z x,y ∈ Ar,t : z x,y (s) > −(s∧(t− s)+ys/t+ x(t− s)/t. ∀s ∈ I j }. (6.221) Finally, A2j (r, t) = A j (r, t) − A1j (r, t). First one shows that the probability of not crossing below M in the interval [r1 , t − r1 ] is controlled by the probability of not crossing in [2r, t − r],    r1  (6.222) P Gcx,y (r1 ; r, t) ≤ C3 P Gcx,y (r, t) , r for y ≥ α(r, t) and x ≥ m1/2 (t). The proof of this fact is an application of the reflection principle i for bridges. The next step is more complicated. It consists in showing that if a bridge

6.6 Asymptotics of Solutions of the F-KPP Equation

111

realises the event A1j (r, t) for large j, i.e. if Mr,t is crossed near the centre, then this will give a small contribution to u. More precisely, * ) 6 t−r   δ (6.223) e3r−t E e 2r k(t−s,zx,y (s))ds 1A1j (r,t) ≤ C4 e− j /4 P A1j (r, t) . The point is that, around the place where the curve M is crossed, u(s, z(s) is roughly u(s, m1/2 − sδ ). Again by Proposition 6.17, at this position u will be very close to 1 , and hence k will be very small. Fiddling around, this produces δ a loss by a factor e− j /4 . Finally, the event on the left-hand side of (6.216) is decomposed into the union of the events A j (r, t), and, using the estimates (6.223), one obtains a bound of the form (6.216), where the choice of the factor 1/r2 is quite arbitrary; in fact one could get any negative power of r.  The next proposition is a reformulation of Proposition 6.30. Proposition 6.38 (Proposition 7.4 in [34]) Let u be as above and assume that (6.215) holds as in the previous proposition. For a constant M1 > 0, assume that x ≥ m1/2 (t) + M1 and y ≥ α(r, t) + M1 . Then,   P z x,y (s) > Mr,t (t − s), ∀s ∈ [r, t − r]   →1 (6.224) P z x,y (s) > m1/2 (t − s), ∀s ∈ [r, t − r] and

  P z x,y (s) > m1/2 (t − s), ∀s ∈ [r, t − r]   → 1, P z x,y (s) > Mr,t (t − s), ∀s ∈ [r, t − r]

(6.225)

uniformly in t, as r ↑ ∞. So far the probabilities concern only the bulk part of the bridges. To control the initial and final phases we need to slightly deform the curves. Define ⎧ ⎪ ⎪ if s ∈ [0, t − 2r], x ⎨Mr,t (s), (6.226) Mr,t (s) ≡ ⎪ ⎪ ⎩(x + m1/2 (t))/2, ifs ∈ [t − 2r, t], ⎧ ⎪ Mr,t (s), ⎪ ⎪ ⎪ ⎪ x,y ⎨ Mr,t (s) ≡ ⎪ y/2, ⎪ ⎪ ⎪ ⎪ ⎩(x + m (t))/2, 1/2 and

⎧ ⎪ ⎪ ⎨Mr,t (s), M r,t (s) ≡ ⎪ ⎪ ⎩−∞, 

if s ∈ [r, t − 2r], if s ∈ [0, r],

(6.227)

if s ∈ [t − 2r, t],

if s ∈ [r, t − 2r], otherwise.

(6.228)

112

Bramson’s Analysis of the F-KPP Equation

Proposition 6.39 (Proposition 7.5 in [34]) Let u be as above and assume that (6.215) holds as in the previous proposition. Then,   x0 P z x,y (s) > Mr,t (t − s), ∀ s∈[0,t−r]  → 1,  (6.229) P z x,y (s) > M r,t (t − s), ∀ s∈[0,t−r] uniformly in t ≥ 8r, x ≥ x0 ≥ m1/2 (t) + 8r and y ≥ α(r, t), as r ↑ ∞. Moreover,   x0 ,y0 P z x,y (s) > Mr,t (t − s), ∀s ∈ [0, t]   → 1, (6.230) P z x,y (s) > M r,t (t − s), ∀s ∈ [0, t] and uniformly in t ≥ 8r, x ≥ x0 ≥ m1/2 (t) + 8r and y ≥ y0 ≥ max(α(r, t), 8r), as r ↑ ∞.

6.7 Convergence Results We have now finished the preparatory steps and come to the main convergence results. In this section we always take u(t, x) to be a solution of F-KPP satisfying the standard conditions and we assume that (6.186) and (6.188) hold. We will also assume that the finite mass conditions (6.8) holds. Moreover, we will assume mostly that there exist y0 such that ∞ u(0, y)dy > 0. (6.231) y0

Now set y¯ ≡ y − α(r, t),

z = x − m1/2 (t),

z¯ ≡ x − n(t).

(6.232)

We need to consider the analogues of the curves introduced above when the function m1/2 (s) is replaced by n(s). We indicate this by adding a superscript n. We want an upper bound on the probability that a Brownian bridge lies above M. Lemma 6.40 (Corollary 1 in Chapter 8 of [34]) Assume that α(r, t) = O(ln r). Then, if x ≥ n(t) + 1, y ≥ α(r, t) + 1, there exists a constant C > 0 such that, for r large enough,     P z x,y (s) > Mr,t (t − s), ∀s ∈ 2r, t − r] ≤ Cr 1 − e−2¯yz¯/t . (6.233) Proof Playing around with the different curves, Bramson arrives at the upper bound   CP zz¯,¯y (s) > 0, ∀s ∈ [2r, t − 2r] . (6.234)

113 √ Now it is plausible that, with positive probability, zz¯,¯y (2r) ∼ zz¯,¯y (t − 2r) ∼ r. Then the probability of staying positive is (1 − e−r/(t−4r) ), which for large t is about what is claimed.  6.7 Convergence Results

We now move to the lower bound on m1/2 . Proposition 6.41 (Proposition 8.2 in [34]) Under the general assumptions of this section, for all x ≥ m1/2 (t) + 1 and large enough t, ∞ 2  e−(x−y) /2t  t u(t, x) ≤ C2 e (6.235) u(0, y) √ 1 − e−2¯yz/t dy, y0 2πt where C2 depends on initial data but not on t. Hence u(t, x) ≤ C3 ze− and m1/2 (t) ≤

√

2t −

√ 2z

3 √ ln t + C4 , 2 2

(6.236)

(6.237)

for some constant C4 . Proof Again we assume F concave. We may set y0 = 0 and, by the maximum principle, we may assume u(0, y) = 1, for y ≤ 0. The main step in the proof of (6.235) is to use the Feynman–Kac representation and to introduce a one in the form 1G x,y (r,t) + 1Gcx,y (r,t) . Then we use Proposition 6.37 and Lemma 6.40. This yields that ) 6t *     E e− 0 k(t−s,zx,y (s))ds ≤ et P Gcx,y (r, t) + et r−2 P Gcx,y (r, t) (6.238)   ≤ C5 et 1 − e−2y¯z/t , for some constant C5 , for y ≥ 1 and x ≥ n(t) + 1. Using monotonicity arguments, one derives, with some fiddling at the upper bound, ∞ 2  e−(x−y) /2t  u(t, x) ≤ C6 et (6.239) u(0, x) √ 1 − e−2¯yz¯/t dy, 0 2πt for x ≥ m1/2 (t) + 1. To conclude, we must show that z¯ can be replaced by z for which we√want to show that n(t) and m1/2 (t) differ by at most a constant. Set z2 = x − 2t. Then the left-hand side of (6.239) equals ∞ √ √ e−(z2 −y)2 /2t   C6 e− 2z2 u(0, x) e 2y √ (6.240) 1 − e−2¯yz¯/t dy 0 2πt 8 ∞ √ √ √ 2 −3/2 y e 2y u(0, y)dy ≤ C7 t−3/2 z¯e− 2z2 ≤ C8 z¯e− 2¯z . z¯ ≤ C6 t π 0

114

Bramson’s Analysis of the F-KPP Equation

Since n(t) ≤ m1/2 (t), this implies (6.236). But then u(t, m1/2 (t)) ≤ C8 (m1/2 (t) − n(t)) e−

√ 2(m1/2 (t)−n(t))

.

(6.241)

Since the right-hand side is bounded from below uniformly in t, this implies that m1/2 − n(t) is bounded uniformly in t, hence (6.237).  Proposition 6.41 gives the following upper bound on the distribution of the maximum of BBM that will be used in the next chapter. Lemma 6.42 (Corollary 10 in [9]) There exist numerical constants, ρ, t0 < 0, such that for all x > 1, and all t ≥ t0 ,     √ 3x ln t x2 + √ . (6.242) P max xk (t) − m1/2 (t) ≥ x ≤ ρx exp − 2x − k≤n(t) 2t 2 2 t For Heaviside initial conditions and using y0 = −1, (6.235) gives 0 −(x+m(t)−y)2 /2t   e u(t, x + m1/2 (t)) ≤ C2 et (6.243) 1 − e−2(y+1)x/t dy. √ −1 2πt

Proof

Using that 1 − e−u ≤ u, for u ≥ 0, and writing out what m1/2 (t) is, the result follows.  Note that this bound establishes the tail behaviour for the probability of the maximum, Eq. (5.53), that was used in Section 5.6. We are now closing in on the final step in the proof. Before we do that, we need one more a priori bound, which is the converse to (6.235). Corollary 6.43 (Corollary 1 in [34]) Under the assumptions as before, for x ≥ m1/2 (t) and all t, ∞ 2  e−(x−y) /2t  u(t, x) ≥ C5 et u(0, y) √ (6.244) 1 − e−2¯yz/t dy, α(r,t) 2πt if α(r, t) = O(ln r). C5 does not depend on x or t. Proof The main point is that we can bound the probability that the Brownian bridge stays above Mr,t on [2r, t − r] (up to constants) by the probability that zz,¯y stays positive on [r, t − r], which in turn is bounded by a constant times (1 − exp(−2¯yz/t)). The rest of the proof is then obvious.  By now we control our functions up to multiplicative constants. The reason for the errors is essentially that we do not control the integrals in the exponent in the Feynman–Kac formula well outside of the intervals [2r, t − r]. In the next step we remedy this. Now the representation (6.7) comes into play. This avoids having to control what happens up to time r. We now come to the final step, the proof of Theorem 6.2.

115

6.7 Convergence Results

Proof of Theorem 6.2 As advertised before, everything reposes on the use of the Feynman–Kac representation in the form (6.7). Inserting a condition for x the bridge to stay above a curve Mr,t , we get the lower bound

* e−(x−y) /(2(t−r)) ) 6 t−r k(t−s,zt−r x,y (s))ds 1 x E e0 u(r, y) √ dy. zt−r x,y (s)>Mr,t (t−s),∀s∈[0,t−r] 2π(t − r) −∞ (6.245) We want to show that, under the condition on the path, the non-linear term in the integral in the exponent can be replaced by 1. This is provided by Proposition 6.39 for the part of the integral between 2r and t − r, since on this segment x Mr,t (s) = Mr,t (t − s). Now we are interested in x > m1/2 (t). For 0 ≤ s ≤ 2r, x Mr,t (t − s) = (x + m1/2 (t))/2. Thus, for x = m1/2 (t) + z with z large enough, by x Proposition 6.41, for x(s) > Mr,t (t − s), ∞

u(t, x) ≥

2

u(t, x(s)) ≤ C3 ze−z

√ 2

.

(6.246)

x

But for x(s) > Mr,t (t − s) = 12 (x + m1/2 (t)) = m1/2 (t) + z/2, we have that u(t − s, x(s)) ≤ C7 ze−z/

√ 2

.

(6.247)

Hence, k(t − s.x(s)) ≥ 1 − C8 e−ρz/

√ 2

,

(6.248)

which tends to 1 nicely, as z ↑ ∞, and so −2r

e

 exp

2r

   k(t − s, x(s))ds ≥ exp −2rC8 e−C8 z .

(6.249)

0

If x − m1/2 (t) ≥ r, this tends to 1 uniformly in t > r, as r ↑ ∞. Therefore we have the lower bound

x u(r, y)e−(x−y) /(2(t−r)) t−r P z x,y (s)) > Mr,t (t − s), ∀ s∈[0,t−r] dy, √ 2π(t − r) −∞ (6.250) for x ≥ m1/2 (t) + r, and C1 (r) ↑ 1, as r ↑ ∞, uniformly in t. This is (6.11). To prove a matching upper bound, one first wants to show that, in the Feynman–Kac representation (6.2), one may drop the integral from −∞ to − ln r. Indeed, by simple change of variables, and using that initial conditions are in u(t, x) ≥ C1 (r)e

t−r

∞

2

116

Bramson’s Analysis of the F-KPP Equation

[0, 1], and F(u)/u is decreasing in u, we get that % $  t − ln r −(x−y)2 /(2t) e k(t − s, z x,y (s))ds dy u(0, y)E exp √ 0 −∞ 2πt %  t − ln r −(x+ln r−y)2 /(2t) $ e k(t − s, z x,y−ln r (s))ds dy E exp ≤ √ −∞ 0 2πt %  t − ln r −(x+ln r−y)2 /(2t) $ e k(t − s, z x+ln r,y (s))ds dy E exp ≤ √ −∞ 0 2πt H (6.251) = u (t, x + ln r). Our rough upper bound (Eq. (6.239)) and lower bound (Eq.(6.244)) suffice to show that, for x ≥ m1/2 (t), uniformly in t > 3r, uH (t, x + ln r) ↓ 0, as r ↑ ∞. u(t, x) This shows that u(t, x) ≥ C2 (r)



∞ − ln r

u(0, y) e−

(x−y)2 2t

) 6t * E e 0 k(t−s,zx,y (s))ds dy,

(6.252)

(6.253)

where C2 (r) → 1, as r ↑ ∞, if x ≥ m1/2 (t), uniformly in t > 3r. Now we insert into the Feynman–Kac representation a 1 in the form   + 1∃s∈[0,t]:zt−r . 1 = 1zt−r x,y (s)>Mr,t (t−s),∀s∈[0,t] x,y (s)≤Mr,t (t−s)

(6.254)

Note that, by definition of M , the second indicator function is equal to the indicator function of the event G x,y (r, t). We want to use Proposition 6.37 to show that the contribution coming from the second indicator function is negligible. To be able to do this, we had to make sure that we can drop the contributions from y < − ln r. Now we get readily − ln r −(x−y)2 /(2t) ) 6t * e u(0, y)E e 0 k(t−s,zx,y (s))ds 1G x,y (r,t) dy √ −∞ 2πt − ln r −(x−y)2 /(2t) ) 6 t−r * e u(0, y)E e 2r k(t−s,zx,y (s))ds 1G x,y (r,t) dy ≤ e3r √ −∞ 2πt − ln r −(x−y)2 /(2t)   e (6.255) u(0, y)P Gcx,y (r, t) . ≤ r−2 et √ −∞ 2πt Now Lemma 6.40 provides a bound for the probability in the last line. Inserting this, and observing that we have the lower bound (6.244), we see that this is at most a constant times u(t, x)/r, and hence negligible as r ↑ ∞.

117

6.7 Convergence Results

We have arrived at the bound ∞ ) 6t * (x−y)2  u(0, y) e− 2t E e 0 k(t−s,zx,y (s))ds 1zt−r u(t, x) ≤ C2 (r) dy. (s))≥M (t−s),∀s∈[0,t] x,y r,t −∞

(6.256) Since by definition, Mr,t imposes no condition on the interval [0, 2r], we move the initial conditions forward to time r, using that ∞

6r (y−y )2 u(r, y) = u(0, y)e− 2r E e 0 k(r−s,zy,y (s))ds dy , (6.257) −∞

and the Chapman–Kolmogorov equation for the heat kernel, to get from (6.256) the bound, for x > m1/2 (t), u(t, x) ≤ C3 (r)e

t−r



∞

e−(x−y) /(2(t−r)) u(r, y) √ 2π(t − r) −∞ 2

(6.258)

 × P zt−r x,y (s)) > Mr,t (t − s), ∀s ∈ [0, t − r] dy,

with C3 (r) tending to 1, as r ↑ ∞. This is (6.12). We have now proven the bounds (6.11) and (6.12). We still need to prove (6.15). Here we need to show that for the Brownian bridge staying above M x is just as unlikely as staying6above M . These issues were studied in Section ∞ 6.5. Let y0 ≤ 0 be such that y u(0, y)dy > 0. The bounds we have on m1/2 (s) 0 readily imply that (6.215) holds with α(r, t) = − ln(r) when r is large enough. Thus we can use Proposition 6.39 to get that   x P ztx,y0 (s) > Mr,t (t − s), ∀ s∈[0,t−r]   = 1 − (r), (6.259) P ztx,y0 (s) > Mr,t (t − s), ∀ s∈[0,t−r] for t ≥ 8r, x − m1/2 (t) ≥ 8r and r large enough, with (r) ↓ 0, as r ↑ ∞. We want to push this to a bound that works for all y. This requires two more pieces of input. The first is a consequence of Proposition 6.41 and Corollary 6.43. Corollary 6.44 (Corollary 2 in [34]) Let x2 ≥ x1 ≥ 0 and let y0 be such that (6.231) holds. Then, for t large enough, e−(x2 −y0 ) /2t u(t, x2 ) ≥ C6 −(x −y )2 /2t . u(t, x1 ) e 1 0 2

(6.260)

Here C6 > 0 does not depend on t or xi . Proof Just combine the upper and lower bounds we have already proven and use the monotonicity properties of the exponentials that appear there. 

118

Bramson’s Analysis of the F-KPP Equation

Next we need the following consequence of (a trivial version of) the so-called FKG inequality. Lemma 6.45 Let h, g : R → R be increasing measurable functions. Let X be a real valued random variable. Then   E h(X)g(X) ≥ E[h(X)]E[g(X)]. (6.261) Let μ denote the law of X. Then   1 μ(dx) μ(dy) (h(x) − h(y)) (g(x) − g(y)) . E h(X)g(x) −E[h(X)]E[g(X)] = 2 (6.262) Since both f and g are increasing, the two terms in the product have the same sign, and hence the right-hand side of (6.262) is non-negative. This implies the assertion of the lemma.  Proof

Lemma 6.46 (Lemma 8.1 in [34]) Let h : R → [0, 1] be increasing and let g : R → R+ be such that, for x1 ≤ x2 , it holds that g(x1 ) ≤ Cg(x2 ), for some C > 0. Ff X is a real valued random variable, for which E[h(X)] ≥ 1 − , then E[g(X)h(X)] ≥ (1 − C)E[g(X)].

(6.263)

Proof Set g1 (x) ≡ sup{g(y) : y ≤ x}. Then g1 is increasing and positive, so the FKG inequality applies and states that E[g1 (X)h(X)] ≥ E[g1 (X)]E[h(X)] ≥ (1 − )E[g1 (X)].

(6.264)

By assumption, for any x, g1 (x) ≤ Cg(x). Hence E[g(X)(1 − h(X)] ≤ CE[g1 (X)(1 − h(X))]

(6.265)

≤ CE[g1 (X)] − C(1 − )E[g1 (X)] = CE[g1 (X)]. Hence E[g(X)h(X)] ≥ E[g(X) − Cg1 (X)] ≥ (1 − )E[g(X)].

(6.266) 

We use Lemma 6.46 in the following construction. Let us denote by b(y; x, y0 , t, r) the probability density that the Brownian bridge from ztx,y0 passes through y at time t − r. This can be expressed in terms of the heat kernel p(t, x) as b(y; x, y0 , t, r) =

p(r, y − y0 )p(t − r, x − y) . p(t, x − y0 )

(6.267)

6.7 Convergence Results

119

Noting that the condition on the bridge in the denominator of (6.259) has bearing only on the time interval [0, t − r], we see that, using the Markov property,   x P ztx,y0 (s) > Mr,t (t − s), ∀ s∈[0,t−r] (6.268)   x = dyb(y; x, y0 , t, r)P zt−r x,y (s) > Mr,t (t − s), ∀ s∈[0,t−r] . We now prepare to use the FKG inequality in the form of Lemma 6.46. For this we represent the left-hand side of (6.259) as

  x P zt−r x,y (s) > Mr,t (t − s), ∀ s∈[0,t−r]  dyb(y; x, y0 , t, r)  P ztx,y0 (s) > Mr,t (t − s), ∀ s∈[0,t−r]  P zt−r (s) > M x (t − s), ∀ s∈[0,t−r] r,t x,y   =  P zt−r x,y (s) > Mr,t (t − s), ∀ s∈[0,t−r]    P zt−r x,y (s) > Mr,t (t − s), ∀ s∈[0,t−r]  b(y; x, y0 , t, r)dy ×  P ztx,y0 (s) > Mr,t (t − s), ∀ s∈[0,t−r] x x (y)μr,t (dy). (6.269) ≡ hr,t

x x is a probability measure, and hr,t is increasing. Hence, the hyNotice that μr,t x potheses of Lemma 6.46 are satisfied for the expectation of the function hr,t x with respect to the probability μr,t . It remains to choose a function g such that the expectations in the left- and right-hand sides of (6.263) turn into the integrals appearing in (6.11) and (6.12). For this we choose

gr (y) ≡

u(r, y) 1y≥0 . p(r, y − y0 )

(6.270)

Corollary 6.44 ensures that gr satisfies the hypothesis on g needed in Lemma 6.46 with a constant C = 1/C6 . With these choices, x x gr (y)hr.t (y)μr,t (dy) (6.271)   6y x u(r, y)p(t − r, x − y)P zt−r x,y (s) > Mr,t (t − s), ∀ s∈[0,t−r] dy 0   = p(t, x − y0 )P ztx,y0 (s) > Mr,t (t − s), ∀ s∈[0,t−r]   6y  t−r u(r, y)p(t − r, x − y)P z (s) > M (t − s), ∀ s∈[0,t−r] x,y r,t 0   ≥ (1 − C) . p(t, x − y0 )P ztx,y0 (s) > Mr,t (t − s), ∀ s∈[0,t−r]

120

Bramson’s Analysis of the F-KPP Equation

This yields

∞ −∞

(x−y)2

 x u(r, y) e− 2(t−r)  t−r P z x,y (s) > Mr,t (t − s), ∀ s∈[0,t−r] dy (6.272) √ 2π(t − r) (x−y)2 ∞  u(r, y) e− 2(t−r)  t−r P z x,y (s) > Mr,t (t − s), ∀ s∈[0,t−r] dy, ≥ (1 − C) √ 2π(t − r) 0

with  = (r) from Eq. (6.259). Here we have used that the integrals from −∞ to zero are non-negative. Finally, one shows that, for the values of x in question, these integrals are also negligible compared to those from 0 to ∞, and hence the integral on the right-hand side can also be replaced by that over the entire real line. This yields the upper bound in (6.15). Since the lower bound is trivial, we have proven (6.15) and this concludes the proof of Theorem 6.2.  We can now prove the special case of Bramson’s convergence theorem, Theorem 5.8, that is of interest to us. Theorem 6.47 Let u be a solution of the F-KPP equation with F satisfying the standard conditions. Assume that the finite mass condition (6.8) holds, and that, for some η, M < N > 0, x+N u(0, y)dy > η, for x ≤ −M. (6.273) x

Then u(t, x + m(t)) → w

√

2

(x),

as

t ↑ ∞,

(6.274)

uniformly in x ∈ R, where m(t) =

√ 2t −

3 √ ln t. 2 2

(6.275)

Proof The proof uses Proposition 6.15. To show that the hypotheses of the proposition are satisfied, we show that, for x large enough, Cx e−

√ 2x −1 γ1 (x)

− γ2 (t) ≤ u(t, x + m(t)) ≤ Cx e−

√ 2x

γ1 (x) + γ2 (t), √ − 2x

(6.276)

is precisely where γ1 (x) → 1, as x ↑ √∞ and γ2 (t) → √ 0, as t ↑ ∞. But Cx e the tail asymptotics of w 2 , so that w 2 can be chosen (remember that we are free to shift the solution to adjust the constant) such that the hypotheses of Proposition 6.15 are satisfied, and the conclusion of the theorem follows. Equation 6.276 follows from Theorem 6.2 via the following corollary, whose proof we postpone to the next chapter, where various further tail estimates will be proven.

6.8 Bibliographic Notes

121

Corollary 6.48 Let u and m(t) be as in Theorem 6.47. Then √

ex 2 u(t, x + m(t)) = C, lim lim x↑∞ t↑∞ x

(6.277)

for some 0 < C < ∞. From here we get the desired control on the tail of u and hence convergence to the travelling wave solution. 

6.8 Bibliographic Notes 1. The F-KPP equation was first introduced by Fisher [56] and be Kolmogorov, Petrovsky and Piscounov [80] in 1937. The motivation came from genetics. The fact, that there is a travelling wave and that solutions converge to it in the case of Heaviside initial conditions was already discovered in [80]. Aronson and Weinberger [12] later discussed the equation in a variety of different contexts. Further results on the initial conditions for convergence to travelling waves with different speeds were obtained by Uchiyama [112]. 2.1 Bramson [34]obtained the precise form of m(t). The present chapter is based on the comprehensive treatise given in this monograph.

7 The Extremal Process of BBM

In this chapter we discuss the construction of the extremal process of branching Brownian motion following the paper [11]. This construction has two parts. The first is the proof of convergence of Laplace functionals of the point process of BBM as t ↑ ∞. This will turn out to be doable using estimates on the asymptotics of solutions of the F-KPP equation that are the basis of Bramson’s work. The ensuing representation for Laplace functionals is rather indirect. In the second part, we give an explicit description of the point process that has this Laplace functional as a cluster point process.

7.1 Limit Theorems for Solutions The bounds in (6.10) have been used by Chauvin and Rouault to compute the probability of deviations of the maximum of BBM (see Lemma 2 in [40]). We will use their arguments in this slightly more general setting. Our aim is to control limits when t and later r tend to infinity. √ We will need to analyse solutions u(t, x + 2t), where x = x(t) depends on t in different ways. First we look at x fixed. This corresponds to particles that are just 2 √3 2 ln t ahead of the natural front of BBM. We start with the asymptotics of the function ψ defined in (6.9).

Proposition 7.1 Let the assumptions of Theorem 6.2 be satisfied, and assume in addition that y0 ≡ sup{y : u(0, y) > 0} is finite. Then the following hold for any z ∈ R. 122

123

7.1 Limit Theorems for Solutions (i)

8 ∞ √ √ 2 lim e y ey 2 u(r, y + 2r)dy ≡ C(r). t↑∞ π 0 (7.1) (ii) limr↑∞ C(r) ≡ C exists and C ∈ (0, ∞). (iii) √ √ t3/2 (7.2) lim ez 2 3 u(t, z + 2t) = C. t↑∞ √ ln t 2 2 √ z 2

√ t3/2 ψ(r, t, z + 2t) = 3 √ ln t 2 2

Proof Consider first the limit t ↑ ∞. Recall the definition of ψ from (6.9). On the left-hand side of (7.1) we have√ an integral √ where, for fixed z, the inte3 y 2 √ grand converges pointwise to 2 π y e u(r, y + 2r). To see this, note that the integrand is given by ⎛ 3 ln t) ⎞ (z+ √ √ √ (y−z)2 ⎜ ⎟⎟⎟ t3/2 2y − 2(t−r) ⎜ −2y 2t−r2 ⎜ ⎜ ⎟⎠ u(r, y + 2r) e e √ ⎝1 − e 3 √ 2π(t − r) 2 2 ln t ' & 8 2t3/2 z + 2 √3 2 ln t √ √ √ √ 2 2y ∼ , u(r, y + 2r)ye 2y . ∼ u(r, y + 2r)ye π 2π(t − r)3 √3 ln t 2 2

(7.3) To deduce convergence of the integral, we need to show that we can use dominated convergence. We see from the first line in (7.3) that the integrand in (6.9) after multiplication with the prefactor in (7.1) is bounded from above by √ √ (7.4) Ky e 2y u(r, y + 2r), where K is some finite constant. Thus we need an a priori bound on u. By the Feyman-Kac representation and the fact that u(0, y) = 0, for y > y0 , implies that y0 1 − (y−x)2 r (7.5) u(r, x) ≤ e e 2r dy √ −∞ 2πr ⎧ ⎪ ⎪ x ≤ y0 , ⎨Cr , ≤⎪ ⎪C e−(y0 −x)2 /2r , ⎩ x≥y . 0

r

√ Setting x = y + 2r and inserting this into (7.4), we can bound the integrand in (6.9) by ye

√ 2y

1y≤y0 − √2r + e

√

√ 2y −(y+ 2r−y0 )2 /2r

e

1y≥y0 − √2r

≤ C(r, y0 )1y≤y0 − √2r + C(r, y0 )y e−y

2

/2r+yy0 /r

1y≥y0 − √2r ,

(7.6)

124

The Extremal Process of BBM

which is integrable in y and independent of t. Hence Lebesgue’s dominated convergence theorem applies and the first part of the proposition follows. The more interesting task is to show now that C(r) converges to a non-trivial limit, as r ↑ ∞. By Theorem 6.2, for r large enough, lim sup ez

√ 2

t↑∞

√ 3/2 √ √ t3/2 z 2t ψ(r, t, z + 2t) ≤ γ(r) lim e 2t) u(t, z + 3 t↑∞ ln t √ ln t 2 2

= C(r)γ(r)

(7.7)

and lim inf ez t↑∞

√ 2

√ t3/2 √ √ t3/2 ψ(r, t, z + 2t) u(t, z + 2t) ≥ γ(r)−1 lim ez 2 t↑∞ ln t ln t

3 √ 2 2

= C(r)γ(r)−1 .

(7.8)

These bounds hold for all r, and since γ(r) → 1, we can conclude that the left-hand side is bounded from above by lim inf r↑∞ C(r) and from below by lim supr↑∞ C(r). So the lim sup ≤ lim inf and hence limr↑∞ C(r) = C exists. As a by-product, we also see that the left-hand sides of (7.7) and (7.8) must agree, and thus we obtain (7.2). A very important aspect is that the bounds (7.7) and (7.8) hold for any r. But for large enough finite r, the constants C(r) are strictly positive and finite by the representation (7.1), and this shows that C is strictly positive and finite.  We will make a slight stop on the way and show how the sharp asymptotics for the upper tail of solutions follow from what we just did. We begin by proving Corollary 6.48. Proof of Corollary 6.48 Note that & √ ψ(r, t, z + m(t)) = ψ r, t, 2t + z −

3 √ 2 2

' ln t .

(7.9)

Then, by the same arguments as in the proof of Proposition 7.1, we get that √ 8 ∞ √ √ √ ez 2 2 ψ(r, t, z + 2t) = y ey 2 u(r, y + 2r)dy = C(r). (7.10) lim t↑∞ z π 0 Here C(r) is the same constant as before. Now we can choose z > 8r in order to be able to apply Theorem 6.2 and then let r ↑ ∞. This yields (6.277).  The next lemma is a variant of the previous proposition for the case when √ x ∼ t.

125

7.1 Limit Theorems for Solutions

Lemma 7.2 Let u be a solution of the F-KPP equation (5.22) with initial conditions satisfying the assumptions of Theorem 5.8 and y0 ≡ sup{y : u(0, y) > 0} < ∞ .

(7.11)

Then, for a > 0, and y ∈ R, √

lim

t→∞

e

√ 2a t 3/2

t √ a t

√ √ √ 2 ψ(r, t, y + a t + 2t) = C(r)e− 2y e−a /2

(7.12)

where C(r) is the constant from Eq. (7.10). Moreover, the convergence is uniform for a in compact sets. Proof The structure of the proof is the same as in the proposition. Compute the pointwise limit and produce an integrable majorant from a bound using the linearised F-KPP equation.  It will be very important to know that, as r ↑ ∞, in the integral representing √ C(r), only the y’s that are of order r give a non-vanishing contribution. The precise version of this statement is the following lemma. Lemma 7.3 Let u be a solution of the F-KPP equation (5.22) with initial conditions satisfying the assumptions of Theorem 5.8 and y0 ≡ sup{y : u(0, y) > 0} < ∞ . Then, for any z ∈ R,



√ A1 r

lim lim sup

A1 ↓0

r↑∞

lim lim sup

A2 ↑∞

r↑∞

u(r, z + y +

√

(7.13)

2r) y ey

√ 2

dy = 0,

(7.14)

dy = 0 .

(7.15)

0



∞ √ A2 r

u(r, z + y +

√

2r) y ey

√ 2

Proof Again it is clear that we need some a priori information on the behaviour of u. This time it is obtained by comparing u to a solution of the F-KPP equation with Heaviside initial conditions, which we know are probabilities for the maximum of BBM. Namely, by assumption, u(0, y) ≤ 1{y y0 + 1 + z + y . (7.16) k≤n(t)

For the probabilities, we have a sharp estimate, given in Lemma 6.42 of Chapter 6. √ We use this lemma with z ∼ t, so the z2 /t term is relevant. The last term involving ln t/t can, however, be safely ignored. The lemma follows in a rather

126

The Extremal Process of BBM

straightforward and purely computational way once we insert this bound into the integrals. The details can be found in [11]. Note that the precise bound (6.242) is needed, and the trivial bound obtained by comparing the probability in (7.16) to that of the case of iid random variables would not suffice.  + 6 √ √ ∞ The following lemma shows how limr↑∞ π2 0 u(r, y+ 2r) y ey 2 dy behaves when the spatial argument of u is shifted. Lemma 7.4 Let u be a solution of the F-KPP equation (5.22) with initial conditions satisfying the assumptions of Theorem 5.8 and y0 ≡ sup{y : u(0, y) > 0} < ∞ .

(7.17)

Then, for any z ∈ R, 8 ∞ √ √ 2 lim yey 2 u(r, z + y + 2r)dy r↑∞ π 0 8 ∞ √ √ √ √ 2 − 2z lim yey 2 u(r, y + 2r)dy = Ce− 2z . =e r↑∞ π 0

(7.18)

Proof The proof is obvious from the fact proven in Lemma 7.3. Namely, changing variables,

∞

yey

√ 2

u(r, z + y +

√ √ 2r)dy = e− 2z

0

∞

(y − z)ey

√ 2

u(r, y +

√ 2r)dy. (7.19)

z

Now the part of the integral from z to 0 cannot contribute, as it does not involve √ y of the order of r, and for the same reason replacing y by y − z changes nothing in the limit.  The following is a slight generalisation of Lemma 7.4. Lemma 7.5 Let h(x) be continuous function that is bounded and is zero for x small enough. Then 8 0 ) √ √ * 2 lim E h(y + max xi (t) − 2t) (−y)e− 2y dy i t↑∞ π −∞ ∞ √ √ h(z) 2Ce− 2z dz, =

(7.20)

−∞

where C is the constant appearing in the law of the maximum, Theorem 5.10.

7.2 Existence of a Limiting Process Proof

127

Consider first the case when h(x) = 1[b,∞) (x), for b ∈ R. 8 0 ) √ √ * 2 E h(y + max xi (t) − 2t) (−y)e− 2y dy lim i t↑∞ π −∞ 0 & '+ √ √ 2 = lim P y + max xi (t) − 2t > b (−y)e− 2y dy π i t↑∞ −∞ ∞ & √ ' + 2 √2y P max xi (t) > b + y + 2t dy = lim π ye i t↑∞ 0 ∞ + √ √ = lim u(t, b + y + 2t) π2 ye 2y dy, (7.21) t↑∞

0

where u solves the F-KPP equation (5.22) with Heaviside initial conditions u(0, x) = 1 xu ,

(7.26)

=1

with c > 0 and u ∈ R. Set g(x) ≡ 1 − e−φ(−x) . Clearly g(x) vanishes for k x > − min (u ), while for x < − max (u ) we have that g(x) ≥ 1 − e− =1 c > 0. This implies that g(x) satisfies both hypotheses for the initial conditions of Theorem 5.8. On the other hand, if u(t, x) solves the F-KPP equation with initial conditions u(0, x) = g(x), ⎡ n(t) ⎤ ⎢⎢⎢ k ⎥⎥⎥ − φ(x (t)−x) ⎢ ⎥⎥⎥ i =1 1 − u(t, x) = E ⎢⎢⎣ (7.27) e ⎦ i=1

is the Laplace functional for our choice of φ. Thus, if we can show that u(t, x + m(t)) converges to a non-trivial limit, we obtain the following theorem.  Theorem 7.6 The point process Et = k≤n(t) δ xk (t)−m(t) converges in law to a point process E. Proof First, the sequence of processes Et is tight by Corollary 2.24, since, for any bounded interval, B ⊂ R, lim lim P (Et (B) ≥ n) = 0, n↑∞ t↑∞

(7.28)

so the limiting point process must be locally finite. (Basically, if (7.28) failed then the maximum of BBM would have to be much larger than it is known to be). We now show (7.28). Clearly it is enough to show this for B = [y, ∞). Assume that (7.28) does not hold. Then there exists  > 0 and a sequence tn ↑ ∞ such that, for all n < ∞, 

(7.29) P Etn ([y, ∞)) ≥ n ≥ . On the other hand, we know that, for any δ > 0, we can find aδ ∈ R such that   lim P max xk (t) ≤ m(t) + aδ ≥ 1 − δ. (7.30) t↑∞

k≤n(t)

Now choose δ = /2. It follows that, for n large enough,  ; : P {Et (B) ≥ n} ∩ max xk (tn + 1) ≤ m(tn + 1) + a/2 ≥ /2. k≤n(tn +1)

(7.31)

But this probability is bounded from above by the probability that the offspring

7.2 Existence of a Limiting Process

129

of the n particles at time tn that must be above m(tn ) + y remain below m(tn + 1) + a/2 , which is smaller than   n  (( ( j) ( P x j (tn ) + max xk (1) ≤ m(tn + 1) + a/2 x j (tn ) ≥ m(tn ) + y i=1

k≤n j (1)

$  %n ≤ P max xk (1) ≤ m(tn + 1) − m(tn ) + a/2 − y k≤n(1) $  %n √ ≤ P max xk (1) ≤ 2 + a/2 − y . k≤n(1)

(7.32)

But the probability in the last line is certainly smaller than 1, and hence, choosing n large enough, the expression in the last line is strictly smaller than /2, which leads to a contradiction. This proves (7.28). For any φ of the form (7.26), we know from Theorem 5.8 that u(t, x + m(t)) converges, as t ↑ ∞, to a travelling wave w √2 . Hence lim ψt (φ) ≡ ψ(φ) t↑∞

(7.33)

exists, and is strictly smaller than one. But $

 % ψt (φ) = E exp − φ dEt ⎡ ⎤ ⎢⎢⎢   ⎥⎥⎥

 = E ⎢⎢⎣⎢ exp − φ xk (t) − m(t) ⎥⎥⎥⎦ k≤n(t) ⎤ ⎡ ⎥⎥⎥ ⎢⎢⎢  ⎢ g(m(t) − xk (t))⎥⎥⎦⎥ = u(t, 0 + m(t)), = E ⎢⎣⎢

(7.34)

k≤n(t)

and therefore limt↑∞ ψt (φ) ≡ ψ(φ) exists, which proves (7.33).



Remark The particular choice of the functions φ for which we prove convergence of the Laplace functionals is made because they satisfy the hypothesis of Theorem 5.8. This implies, of course, convergence for all φ ∈ Cc+ (R) (and more). The functions do not satisfy the assumption (ii) in Bramson’s theorem, so convergence of the corresponding solutions cannot be uniform in x, but that is not important. In [11], convergence was proved directly for φ ∈ Cc+ (R) using a truncation procedure. Essentially, it makes no difference whether the support of φ is bounded from above or not, since the maximum of the point process Et is finite almost surely, uniformly in t.

130

The Extremal Process of BBM

Flashback to the Derivative Martingale. Having established the tail asymptotics for solutions of the F-KPP equation, one might be tempted to compute the law of the maximum (or Laplace functionals) by recursion at an intermediate time r, e.g. % $  6 (

E exp − φ(−x + y)Et (dy) = E E e− φ(−x+y)Et (dy) ((Fr ⎤⎤ ⎡ ⎡ n(r) ⎢⎢⎢ ⎢⎢⎢ 6 (( ⎥⎥⎥⎥⎥⎥ (i) − φ(−x+x (r)−m(t)+m(t−r)+y)E (dy) ⎢ ⎢ i t−r (Fr ⎥⎥⎦⎥⎥⎥⎦⎥ e = E ⎢⎣⎢E ⎢⎣⎢ i=1 ⎤⎤ ⎡ ⎡ n(r) ⎢⎢⎢ ⎢⎢⎢  √  (( ⎥⎥⎥⎥⎥⎥ ⎢ ⎢ (7.35) 1 − v(t − r, −x + xi (r) − 2r) (Fr ⎥⎥⎥⎦⎥⎥⎥⎦ , ≈ E ⎢⎢⎣E ⎢⎢⎣ i=1

E(i) s

are iid copies on BBM and 6 √ √ (i) v(s, x − xi (r) + 2r + m(s)) ≡ 1 − e− φ(−x+xi (r)− 2r+y)Es (dy) . (7.36) √ We have used that m(t) − m(t − r) = 2r + O(r/t). We want to use the fact (see Theorem 5.2) that the particles that will show φ at  √ up in√the √support of √  time t must come from particles in an interval 2r − c1 r, 2r − c2 r , if r is large enough and c1 > c2 > 0 become large. Hence we may assume that the conditional expectations that appear in (7.35) can be replaced by their tail asymptotics, i.e. √ √ √ √ v(t − r, x − xi (r) + 2r + m(t − r)) ∼ C(x − xi (r) + 2r) e− 2(x−xi (r)+ 2r) . (7.37) where

Hence we would get $  % lim E exp − φ(x + y)Et (dy) t↑∞ ⎞⎤ ⎡ ⎛ n(r) √ √ ⎟ ⎢⎢⎢ ⎜⎜⎜  √ ⎟⎟⎥⎥⎥ (x − xi (r) + 2r) e− 2(x−xi (r)+ 2r) ⎟⎟⎟⎠⎥⎥⎥⎦ = lim E ⎢⎢⎣⎢exp ⎜⎜⎝⎜−C r↑∞ i=1 ) & √ √ '* = lim E exp −CZr e− 2x − CYr x e− 2x , (7.38) r↑∞

with Zt and Yt the martingales from Section 2.4. The constant C will of course depend in general on the initial data for ν, i.e. on φ. Now one could argue, like Lalley and √ Sellke [83], that Yr must converge to a non-negative limit, implying that 2r − xi (r) ↑ ∞ if this limit is strictly positive. Then again, this implies that Yr → 0, a.s., and hence $  % ) √ * − 2x lim E exp − φ(x + y)Et (dy) = lim E e−CZr e . (7.39) t↑∞

r↑∞

7.2 Existence of a Limiting Process

131

The argument above is not quite complete. In the following subsection we will show that the final result is nonetheless true.

A representation for the Laplace Functional In the preceding subsection we showed convergence of the Laplace functional. The following proposition exhibits the general form of the Laplace functional of the limiting process. In this section we will always assume that φ is a function of the form (7.26). Proposition 7.7 Let Et be the process (7.23). For φ as given in (7.26) and any x ∈ R, $  % ) & √ '* lim E exp − φ(y + x)Et (dy) = E exp −C(φ)Ze− 2x , (7.40) t↑∞

where, for u(t, y) the solution of the F-KPP equation with initial conditions u(0, y) = 1 − e−φ(−y) , 8 ∞ √ √ 2 u(t, y + 2t)y e 2y dy (7.41) C(φ) = lim t↑∞ π 0 is a strictly positive constant depending only on φ and Z is the derivative martingale. Proof The initial conditions u(0, y) satisfy the hypotheses that were required in the previous results on sharp approximation. The existence of the constant C(φ) then follows from Proposition 7.1. On the other hand, it follows from Theorem 5.8 and the Lalley–Sellke representation that ) √ * − 2x , (7.42) lim u(t, x + m(t)) = 1 − E e−C(φ)Ze t↑∞

for some constant C(φ) (recall that the relevant solution of (5.34) is unique up to shifts (see Lemma 5.7), which translates into uniqueness of the representation (7.40) up to the choice of the constant C). Clearly, from the asymptotic of solutions we know that ) √ * − 2x 1 − E e−CZe = C. (7.43) lim √ x↑∞ x e− 2x Thus all that is left is to identify this constant with C(φ). But from Corollary 6.48 it follows that C must √ be C(φ). This is essentially done by rerunning Proposition (7.1) with 2t replaced by the more precise m(t). This proves (7.40). 

132

The Extremal Process of BBM

7.3 Interpretation as Cluster Point Process In the preceding section we gave a full construction of the Laplace functional of the limiting extremal process of BBM and gave a representation for it in Proposition 7.7. Note that all the information on the limiting process is contained in how the constant C(φ) depends on the function φ. The characterisation of this dependence via a solution of the F-KPP equation with initial conditions given in terms of φ does not appear very revealing at first sight. In the following we will remedy this by giving explicit probabilistic descriptions of the underlying point process.

Interpretation via an Auxiliary Process We will construct an auxiliary class of point processes that a priori have nothing to do with the real process of BBM. Let (ηi ; i ∈ N) denote the atoms of a Poisson point process on (−∞, 0) with intensity measure 8 √ 2 (−x) e− 2x dx . (7.44) π For each i ∈ N, consider independent BBMs {xk(i) (t), k ≤ n(i) (t)}. Note that, for each i ∈ N, √ (7.45) max xk(i) (t) − 2t → −∞, a.s. k≤n(i) (t)

This follows, e.g., from the fact that the martingale Y(t) defined in (5.51) converges to zero, a.s. The auxiliary point process of interest is constructed from these ingredients as Πt ≡

(i) (t)  n

i

k=1

δ √1

2

√ ln Z+ηi +xk(i) (t)− 2t ,

(7.46)

where Z has the same law as the limit of the derivative martingale. The existence and non-triviality of the process in the limit t ↑ ∞ is not obvious, especially in view of (7.45). We will show that it not only exists, but has the same law as the limit of the extremal process of BBM. Theorem 7.8

Let Et be the extremal process (7.23) of BBM. Then law

lim Et = lim Πt . t↑∞

t↑∞

(7.47)

Proof The proof of this result just requires the computation of the Laplace transform, which we are already quite skilled at.

7.3 Interpretation as Cluster Point Process

133

The Laplace functional of Πt using the form of the Laplace functional of a Poisson process, reads $  % E exp − φ(x)Πt (dx) ⎡ ⎛ ⎞⎞⎤ ⎛ (i) (t)  ⎢⎢⎢ ⎜⎜⎜  ⎜⎜⎜n ⎟⎟⎟⎟⎟⎟⎥⎥⎥  √ √ (i) ⎟⎟⎟⎟⎟⎟⎟⎟⎥⎥⎥⎥ ⎜⎜⎜⎜ = E ⎢⎢⎢⎢⎣exp ⎜⎜⎜⎜⎝− φ η + ln Z/ 2 + x − 2t i k ⎠⎠⎦ ⎝ i k=1 ⎡ ⎛ ⎞⎤ ⎢⎢ ⎜⎜  (i) ⎟⎟⎟⎥⎥⎥ = E ⎢⎢⎢⎣exp ⎜⎜⎜⎝− Θt (ηi ⎟⎟⎠⎥⎥⎦ , (7.48) i

where we set Θ(i) t (x)

≡

(i) n (t)

 √  √ φ x + ln Z/ 2 + xk(i) − 2t .

(7.49)

k=1

Now we want to compute the expectation. Using the independence of the Cox process and the processes x(i) (t), and using the explicit form of the Laplace functional of Poisson processes, we see that (7.48) is equal to *'* ) & )  (( 1 (7.50) E exp E e−Θt (x) − 1 ((σ(Z) $  0 ) % *+ √  (( 1 − 2x 2 = E exp E e−Θt (x) − 1 ((σ(Z) (−x) e dx . π −∞

Note that the conditional expectation in the exponent is just the expectation with respect to one BBM, x(1) . The outer expectation is just the average with respect to the derivative martingale Z. Now the exponent is explicitly given as ⎤ ⎞+ 0 ⎛⎜ ⎡⎢ √ ⎟⎟ ⎜⎜⎜ ⎢⎢⎢ n(t) & −φ(x+Z/ √2+xk (t)− √2t) '⎥⎥⎥⎥ ⎥⎥⎦ − 1⎟⎟⎟⎟⎠ π2 (−x) e− 2x dx ⎜⎜⎝E ⎢⎢⎣ e −∞ k=1 0 √ √ √ + u(t, −x + 2t − Z/ 2) π2 (−x) e− 2x dx =− −∞ ∞ √ √ √ + u(t, x + 2t − Z/ 2) π2 x e 2x dx, (7.51) =− 0

where u is a solution of the F-KPP equation with initial conditions 1 − e−φ(−x) . Hence, by (7.18), 8 ∞ √ √ 2 1 u(t, x + 2t − √ ln Z)xe 2x dx (7.52) lim t↑∞ π 0 2 8 ∞ √ √ 2 u(t, x + 2t)xe 2x dx = ZC(φ), lim =Z π t↑∞ 0

134

The Extremal Process of BBM

where the last equality follows by Proposition 7.7. Thus we finally arrive at $  %

lim E exp − φ(x)Πt (dx) = E e−ZC(φ) . t↑∞

This implies that the Laplace functionals of limt↑∞ Πt and of the extremal process of BBM are equal. and proves the proposition. 

The Poisson Process of Cluster Extremes We will now give another interpretation of the extremal process. In the paper [9], the following result was shown. Consider the ordered enumeration of the particles of BBM at time t, x1 (t) ≥ x2 (t) ≥ · · · ≥ xn(t) (t).

(7.53)

Fix r > 0 and construct the equivalence classes of these particles such that each class consists of particles that have a common ancestor at time s > t − r. For each of these classes, we can select as a representative its largest member. This can be constructed recursively as follows: i1 = 1, ik = min( j > ik−1 : q(ii , j) < t − r, ∀ ≤ k − 1).

(7.54)

This is repeated to exhaustion and provides the desired decomposition. We denote the resulting number of classes by n∗ (t). We can think of the xik (t) as the heads of families at time t. We then construct the point processes Θrt

≡

n∗ (t) 

δ xik (t)−m(t) .

(7.55)

k=1

In [9] the following result was proven. Theorem 7.9 Let Θrt be defined above. Then lim lim Θrt = Θ, r↑∞ t↑∞

(7.56)

where convergence is in law and√ Θ is√ the random Poisson process (Cox process) with intensity measure CZ 2e− 2x dx, Z being the derivative martingale and C the constant from Theorem 5.10. Remark The thinning out of the extremal process is removing enough correlation to Poissonise the process. This is rather common in the theory of extremal processes (see, e.g. [15, 16]).

7.3 Interpretation as Cluster Point Process

135

I will not give the proof from [9] here. Rather, I will show how this result links up to the representation for the extremal process of BBM given above. In the following subsection I will also indicate how this result can be proven using a nice idea from Biskup and Louidor [22]. Proposition 7.10 Let



Πext t ≡

i

δ √1

2

√ ln Z+ηi +Mi (t)− 2t} ,

(7.57)

where M (i) (t) ≡ maxk xk(i) (t), i.e. the point process obtained by retaining from Πt the maximal particles of each of the BBMs. Then & √ ' √ law − 2x = P = PPP Z 2Ce dx (7.58) lim Πext Z t t↑∞

as a point process on R, where C is the same constant appearing the law of the maximum. In particular, the maximum of limt↑∞ Πext t has the same law as the limit law of the maximum of BBM. Proof Consider the Laplace functional of our thinned auxiliary process, ⎞⎤ ⎡ ⎛ √ ⎟⎟⎟⎥⎥⎥ ⎢⎢⎢ ⎜⎜⎜  (i) ⎢ ⎜ φ(ηi + M (t) − 2t)⎟⎟⎠⎥⎥⎦ . (7.59) E ⎢⎣exp ⎜⎝− i

Since the M (i) s are iid, and denoting by Fη the σ–algebra generated by the Poisson point process of the η’s, we get that ⎡ ⎛ ⎞⎤ √ ⎟⎟⎟⎥⎥⎥ ⎢⎢⎢ ⎜⎜⎜  (i) ⎢ ⎜ E ⎢⎣exp ⎜⎝− φ(ηi + M (t) − 2t)⎟⎟⎠⎥⎥⎦ i ⎤ ⎡  √  (( ⎥⎥⎥ ⎢⎢⎢  (i) E exp −φ(ηi + M (t) − 2t) (Fη ⎥⎥⎦ = E ⎢⎢⎣

(7.60)

i

⎞⎤ ⎡ ⎛ ⎟⎟⎥⎥ ⎢⎢⎢ ⎜⎜⎜  g(ηi )⎟⎟⎟⎠⎥⎥⎥⎦ , = E ⎢⎢⎣exp ⎜⎜⎝− i

where g(i) t (z) ≡ φ(z + M(t) −

√

2t).

Proceeding as in the proof of Theorem 7.8 , ⎞⎤ ⎡ ⎛ ⎟⎟⎥⎥ ⎢⎢⎢ ⎜⎜⎜  g(ηi )⎟⎟⎟⎠⎥⎥⎥⎦ E ⎢⎢⎣exp ⎜⎜⎝− i

⎡ ⎛ ⎢⎢⎢ ⎜⎜ ⎢ = E ⎢⎣exp ⎜⎜⎜⎝−

0

−∞

)

√ √ −g(y+ln Z/ 2+M(t)− 2t)

E M(t) e

(7.61)

(7.62) ⎞⎤ * 82 √ ⎟⎥⎥ − 2y ⎟ (−y) e −1 dy⎟⎟⎟⎠⎥⎥⎥⎦ , π

136

The Extremal Process of BBM

where E M(t) denotes expectation with respect to the max of BBM at time t. Using Lemma 7.4 with h(x) = 1 − e−φ(−x) , we see that 8 0 ) √ √ * √ 2 −φ(y+ln z/ 2+M(t)− 2t) (−y) e− 2y dy E M(t) 1 − e lim t↑∞ −∞ π ∞ √  √ 1 − e−φ(a) C 2e− 2a da, (7.63) = −∞

which is the Laplace functional of the PPP with intensity ZCe− The assertion of Proposition 7.10 follows.

√ 2a

da on R. 

So this is nice. The process of the maxima of the BBMs in the auxiliary process is the same Poisson process as the limiting process of the heads of families. This gives a clear interpretation of the BBMs in the auxiliary process.

Excursion: Universality of the Poissonian Nature of Cluster Extremes The fact that local maxima of Gaussian processes tend to be Poissonian is much more general and does not a priori hinge on the underlying tree structure. All that is required is a sufficient decay of correlations. A classical example is provided by the stationary Gaussian processes on the real line with covariances that decay with the square of the distance [84]. Biskup and Louidor [22] have devised a nice method of proof that should give rise to the following theorem. Theorem 7.11 [Universality conjecture] Let X be a Gaussian process indexed by a family of finite sets S n with |S n | ↑ ∞ and endowed with the covariance metric dn (x, y) = cov(X x , Xy ). Assume that there exists a sequence mn such that   P max X x (n) − mn ≤ u → G(u), x∈S n

as n ↑ ∞,

(7.64)

where G(u) is a distribution function. Further assume the following: (i) Balls of finite radius contain finitely many points, i.e. supn #{y ∈ S n : dn (x, y) < r} < ∞. (ii) For any λ ∈ R,   lim sup P ∃ x,y∈S n : d(x, y) ∈ (r, 1 − r), X x − mn > λ, Xy − mn > λ = 0. r↑∞ n↑∞

(7.65) Then, for any sequence rn such that rn ↑ ∞, rn /n ↓ 0, the point process  Pn ≡ δXx (n)−mn 1Xx (n)=maxy:dn (x,y)≤rn Xy (n) (7.66) x∈S n

7.3 Interpretation as Cluster Point Process

137

converges to a Cox process with intensity of the form We−cu du, where W is a positive random variable and c is a positive constant. Moreover,

−cu G(u) = E e−We . (7.67) Proof This is a sketch of the proof from [22] transposed into this abstract setting. I do not claim that this is a complete proof, and some further assumptions may have to be made. We begin with some preliminary observations. The parmn ticular form of (7.64) seems to impose that lim EX 2 = a for some finite, strictly x positive a. (Maybe we need to assume this...). The key idea is old, and goes back to proofs of Gaussian concentration of measure theorems at least. The Gaussian process X can be realised as X(1), where X(t) is family of Brownian motions with correlation structure



E X x (t)Xy (s) = (t ∧ s)E X x Xy . (7.68) Thus we can decompose, for  > 0, # X(1) = X(1 − ) + X(),

(7.69)

# = X(1) − X(1 − ) is independent of X(1 − ). One now chooses where X()  = 1/mn such that

#x ()2 = E X x2 = 1/a = O(1). E X (7.70) √ Notice that the maxima of X(1 − ) are of order mn 1 −  ∼ mn − mn /2 = mn − 1/2. Therefore, whenever x is in the level set near the maxima of X(1 − ), #x () is in the level set near the then there is a good chance that X x (1 − ) + X maximal level, and in fact there is a chance it will be a local extreme. A simple covariance estimate shows that ) ) 2 * 2 * /x () − X /y () = E X x − Xy E X (7.71) ) 2 * E X x − Xy dn (x, y) , = ∼   mn E X x2 which will be o(1) if dn (x, y) < r. More precisely, we have that, for any r < ∞ and any δ > 0,   δ2 P max |X x − Xy | > δ ≤ #{y : dn (x, y) ≤ r}e−mn 2r ↓ 0. (7.72) y:dn (x,y)≤r

On the other hand, the differences of the variables X x (1 − ) within such a ball are O(1). This implies that, if x is a local maximum of extreme height within a ball of radius r for the process X(1 − ), then this remains the case for the process X(1). Moreover, if x is a local maximum of X(1 − ) and of extreme

138

The Extremal Process of BBM

level, then X x will be of extreme level. Because of the assumption (ii), there can be no further point at the extreme level in an annulus between distance r and rn (where rn is as in the theorem, with probability tending to 1 as n ↑ ∞ and then r ↑ ∞. But this implies that x will be a local maximum also of X within a ball of radius rn . This implies the following distributional invariance property for the limiting process, P:   δηi = δηi −a/2+gi , (7.73) P≡ i

i

where gi are iid Gaussian random variables, a = Eg2i and equality holds in distribution. A result of Liggett [86] then asserts that this characterises P as Cox process and yields the assertion of the theorem.  Remark Note that this proof surely works for BBM. But it gives less information than the proof in [9], as it does not identify the random variable W. The gist of the argument of Biskup and Louidor is that Poisson follows if one can break the random field into two parts such that the ’small part’ has variance of order one, is virtually constant over a single cluster, and independent for different clusters. Then the extreme local maxima as a set are not altered by the small fields, while their values are reshuffled, and the invariance under this reshuffling implies Poisson.

More on the Law of the Clusters Let us continue to look at the auxiliary process. We know that it is very hard √ for any of the M (i) (t) to exceed 2t and thus to end up above a level a as t ↑ ∞. Therefore, many of the atoms ηi of the Poisson process will not have relevant offspring. The following proposition states that there is a narrow window deep below zero from which all relevant particles come. This is analogous to the observation in Lemma 7.13. Proposition 7.12 For any y ∈ R and ε > 0, there exist 0 < A1 < A2 < ∞ and t0 depending only on y and ε, such that  √ √ √  sup P ∃i,k : ηi + xk(i) (t) − 2t ≥ z, ∧ ηi  −A1 t, −A2 t < ε.

(7.74)

t≥t0

Proof

Throughout the proof, the probabilities are conditional on Z. Clearly

7.3 Interpretation as Cluster Point Process we have

139

 √ √ P ∃i,k : ηi + xk(i) (t) − 2t ≥ z, but ηi ≥ −A1 t ⎤ ⎡  (( ⎥⎥⎥ √ ⎢⎢⎢ (i) √ ≤ E ⎢⎢⎣ 1ηi ∈[−A1 t,0] P M (t) − 2t + ηi ≥ y(Fη ⎥⎥⎦ , =

√

=

i 0

−A1 t √ A1 t

0

√   √ P M (i) (t) − 2t + ηi ≥ z C(−y) e− 2y dy



 √ 3 P M(t) − m(t) ≥ z + y + √ ln t Cy e 2y dy. (7.75) 2 2

Inserting the bound on the probability from Lemma 6.42, we get, on the domain of integration,   √ 3  √  √ 3 − 2( √ ln t+z+y) −y2 /2t 2 2 e . P M(t) − 2t ≥ z + y + 2t ≤ ρ √ ln t + z + y e 2 2 (7.76) Inserting this into (7.75), this is bounded by  A1 √t  3 2  −3/2 (7.77) ρ t √ ln t + z + y e−y /2t dy ≤ ρ A31 . 0 2 2 Similarly, we have that  √ √ P ∃i,k : ηi + xk(i) (t) − 2t ≥ z, but ηi ≤ −A2 t ∞  √ √ √  ≤ 2t ≥ z + y + 2t Cy e 2y dy √ P M(t) − A2 t ∞ ∞ 2 2 −y2 /2t ≤ ρt ˜ −3/2 y e dy = ρ ˜ y2 e−y /2 dy, √ A2 t

(7.78)

A2

which manifestly tends to zero as A2 ↑ ∞.



We see that there is a close analogy to Lemma 7.3. How should we interpret the auxiliary process? Think of a very large time t, and go back to time t − r. At of particles which all lie well below √ that time, there is a certain distribution √ 2(t−r) by an amount of the order of √r. From those a small fraction will have √ offspring that reach the excessive size 2r + r and thus contribute to the extremes. The selected particles form the Poisson process, while their offspring, which are now BBMs conditioned to be extra large, form the clusters.

The Extremal Process Seen from the Cluster Extremes We finally come to yet another description of the extremal process. Here we start with the Poisson process of Proposition 7.10 and look at the law of the

140

The Extremal Process of BBM

clusters ‘that made it up to there’. Now we know √ that the points in the auxiliary √ process all came from the t window around − √2t. Thus we may expect the clusters to look like BBM that was bigger than 2t. Therefore we define the process  Et = δ xk (t)− √2t . (7.79) k≤n(t)

Obviously, the limit of such a process √ must be trivial, since the probability that the maximum of BBM shifted by − 2t does not√drift to −∞ is vanishing. However, conditionally on the event {maxk xk (t) − 2t ≥ 0}, the process Et  does converge to a well-defined point process E = j δξ j as t ↑ ∞. We may then define the point process of the gaps  δ xk (t)−max j≤n(t) x j (t) (7.80) Dt ≡ k

and D=



δΔ j ,

Δ j ≡ ξ j − max ξi , i

j

(7.81)

where ξi are the atoms of the limiting process E. Note that D is a point process on (∞, 0] with an atom at 0. Theorem 7.13 Let PZ be as in (7.58) and let {D(i) , i ∈ N} be a family of independent copies of the gap process (7.81). Then the point process Et =  k≤n(t) δ xk (t)−m(t) converges in law, as t ↑ ∞, to a Poisson cluster point process E given by  law E ≡ lim Et = δ pi +Δ(i) . (7.82) t↑∞

i, j

j

This theorem looks quite reasonable. The only part that may be surprising is that all the Δ j have the same law,√since a priori we only know that they come √ from a t neighbourhood below 2t. This is the content of the next theorem. √ Theorem 7.14 Let x ≡ −a t + b for some a > 0, b ∈ R. The point process  δ x+xk (t)− √2t (7.83) k≤n(t)

 ( √ converges in law under P ·(({x + maxk xk (t) − 2t > 0} , as t ↑ ∞ to a welldefined point process E. The limit does not depend on a or b, and the maximum of √ E shifted by x has the law of an exponential random variable of parameter 2.

7.3 Interpretation as Cluster Point Process √ Proof Set max Et ≡ maxi xi (t) − 2t. We first show that, for z ≥ 0, & ' (( √ lim P x + max Et > z (( x + max Et > 0 = e− 2z , t↑∞

141

(7.84)

for X > 0. This follows from the fact that the conditional probability is just   P x + max Et > X  , (7.85) P x + max Et > 0 and the numerator √ be well approximated by the functions √ and denominator can ψ(r, t, X − x + 2t) and ψ(r, t, −x + 2t), respectively. Using Lemma 7.2 for these, we get the assertion (7.84). Second, we show that, for any function φ that is continuous and has support bounded from below, the limit of ( % $  (( (7.86) E exp − φ(x + z)Et (dz) ( x + max Et > 0 exists and is independent of x. The proof is just a tick more complicated than before, but relies again on the properties of the functions ψ. I will skip the details.  We now prove Theorem 7.13. Proof of Theorem 7.13 The Laplace functional of the process in (7.82) clearly is given by $  % ∞ 6 √

√ E exp −CZ E 1 − e− φ(y+z)D(dz) 2e− 2y dy (7.87) −∞

for the point process D defined in (7.81). We show that the limiting Laplace functional of the auxiliary process can be written in the same form. Then Theorem 7.13 follows from Theorem 7.8. The Laplace functional of the auxiliary process is given by ⎡ ⎛ ⎞⎤ ⎢⎢⎢ ⎜⎜⎜  √ ⎟⎟⎟⎥⎥⎥ (i) 1 ⎢ ⎜ lim φt (φ) = lim E ⎢⎢⎣exp ⎜⎜⎝− φ(ηi + √2 ln Z + xk (t) − 2t)⎟⎟⎟⎠⎥⎥⎥⎦ . (7.88) t↑∞ t↑∞ i,k

Using the form for the Laplace transform of a Poisson process, we have for the right-hand side ⎡ ⎛ ⎞⎤ ⎢⎢⎢ ⎜⎜⎜  √ ⎟⎟⎟⎥⎥⎥ (i) 1 ⎢ ⎜ φ(ηi + √2 ln Z + xk (t) − 2t)⎟⎟⎟⎠⎥⎥⎥⎦ (7.89) lim E ⎢⎢⎣exp ⎜⎜⎝− t↑∞ i,k ⎡ ⎛ ⎞⎤ 0 ) * 82 6 √ ⎢⎢⎢ ⎜⎜⎜ ⎟⎥⎥ − 2y ⎟ − φ(x+y)Et (dx) ⎢ ⎜ = E ⎢⎣exp ⎜⎝−Z lim (−y) e dy⎟⎟⎟⎠⎥⎥⎥⎦ . E 1−e t↑∞ −∞ π

142

The Extremal Process of BBM

Define Dt ≡



δ xi (t)−max j≤n(t) x j (t) .

(7.90)

i≤n(t)

The integral on the right-hand side equals $  % 8 √ 2 φ(z + y + max Et )Dt (dz) lim (−y) e− 2y dy, E f t↑∞ −∞ π

0

(7.91)

with f (x) ≡ 1 − e−x . By Proposition 7.12, there exist A1 and A2 such that % 8 0 $  √ 2 φ(z + y + max Et )Dt (dz) (−y) e− 2y dy E f π −∞ % 8 −A1 √t $  √ 2 φ(z + y + max Et )Dt (dz) = (−y) e− 2y dy √ E f π −A2 t (7.92) + Ωt (A1 , A2 ), where the error term satisfies lim

sup Ωt (A1 , A2 ) = 0.

A1 ↓0,A2 ↑∞ t≥t0

Let mφ be the minimum of the support of φ. Note that   φ(z + y + max Et )Dt (dz) f

(7.93)

(7.94)

is zero when y + max Et < mφ , and that the event {y + max Et = mφ } has probability zero. Therefore, $  % E f φ(z + y + max Et )Dt (dz)  % $  φ(z + y + max Et )Dt (dz) 1{y+max Et >mφ } =E f $  ( % ( φ(z + y + max Et )Dt (dz) ((y + max Et > mφ =E f   (7.95) × P y + max Et > mφ . One can show (see Corollary 4.12 in [11]) that the conditional law of the pair (Dt , y + max Et ) given {y + max Et > mφ } converges, as t ↑ ∞, to a pair of independent random √ variables, where the limit of a + max Et is exponential with parameter 2 (see (7.84)). Moreover, the convergence is uniform in √ √ y ∈ [−A1 t, −A2 t]. This implies the convergence of the random variable

7.3 Interpretation as Cluster Point Process

143

6

φ(z + y + max Et )Dt (dz). Therefore, writing the expectation over the exponential random variable explicitly, we obtain $  ( % ( φ(z + y + max Et )Dt (dz) ((y + max Et > mφ lim E f t↑∞ % ∞ $  √ √ − √2s φ(z + s)D(dz) E f 2e ds. (7.96) = e 2mφ mφ

On the other hand,

√ −A1 t





8

√ √ 2 (−y) e− 2y dy = Ce− 2mφ + Ωt (A1 , A2 ) π −A2 t (7.97) by Lemma 7.5, where limA1 ↓0,A2 ↑∞ lim supt↑∞ Ωt (A1 , A2 ) = 0. Using the same approximation as in (7.92). Combining (7.97), (7.96) and (7.95), one sees that (7.89) converges to % $  ∞ 6 √

√ − φ(y+z)D(dz) − 2y E 1−e 2e dy , (7.98) E exp −CZ √

P y + max Et > mφ

−∞

which is, by (7.88), the limiting Laplace transform of the extremal process of branching Brownian motion: this shows (7.87) and concludes the proof of Theorem 7.13.  The properties of BBM conditioned to exceed their natural threshold were already described in detail by Chauvin and √ Rouault [40]. There will be one branch (the spine) that exceeds √ the level 2t by an exponential random variable of of parameter 2 (see √the preceding proposition). The spine is very close to a straight line of slope 2. From this spine, ordinary BBMs branch off at Poissonian times. Clearly, all the branches that split off at times later than r before the end-time will reach at most the level √ √ 2(t − r) + 2r −

√ 3 3 √ ln r = 2t − √ ln r. 2 2 2 2

(7.99)

√ Seen from the top, i.e. 2t, this tends to −∞ as r ↑ ∞. Thus only branches that are created ‘a finite time’ before the end-time t remain visible in the extremal process. This does, of course, correspond perfectly to the observation in Theorem 7.9 that implies that all the points visible in the extremal process have a common ancestor at a finite time before the end. As a matter of fact, from the observation above one can prove Theorem 7.9 easily.

144

The Extremal Process of BBM

7.4 Bibliographic Notes 1. The main results on the convergence of the extremal process were obtained independently by Arguin et al. [11] and A¨ıd´ekon et al. [5]. In this chapter we have presented the proof from [11] with some modifications. A¨ıd´ekon et al. use a different and more probabilistic approach that employs the spine decomposition and uses some of the results first proven by Arguin et al. [8, 9]. The work of Chauvin and Rouault on conditional BBM [40] provides important input to both approaches. 2. Lalley and Sellke conjecture in [83] the convergence of the extremal process (with the correct random rescaling) but do not give a characterisation of the limiting process. 3. The nice approach to the Poisson nature of the process of cluster extremes is used by Biskup and Louidor [22] for the Gaussian free field in dimension two. It is based on a result of Liggett [86].

8 Full Extremal Process

In this chapter we present an extension of the convergence of the extremal process of BBM. We have seen in Chapter 2 that in extreme value theory (see, e.g. [84]) it is customary to give a description of extremal processes that also encode the locations of the extreme points (‘complete Poisson convergence’). We want to obtain an analogous result for BBM. Now our ‘space’ is the Galton– Watson tree, or more precisely its ‘leaves’ at infinity. In the case of deterministic case with binary branching at integer times, this can easily be mapped to the unit interval since the boundary of the tree is simply the infinite binary sequences σ ≡ (σ1 σ2 · · · σn · · · ), with σ ∈ {0, 1}. These sequences can be naturally mapped into [0, 1] via σ →

∞ 

σ 2− −1 ∈ [0, 1].

(8.1)

=1

We now want to do the same for a general Galton–Watson tree.

8.1 The Embedding Our goal is to define a map γ : {1, . . . , n(t)} → R+ in such a way that it encodes the genealogical structure of the underlying supercritical Galton–Watson process. To this end, recall the labelling of the Galton–Watson tree through multi-indices, presented in Section 4.5. Then define γt (i) ≡ γt (i(t)) ≡

w(t) 

i j (t)e−t j .

(8.2)

j=1

Recall that w(t) is the total number of branching events in the tree up to time t. For a given i, the function (γt (i), t ∈ R+ ) describes a trajectory of an individual 145

146

Full Extremal Process

in R+ . Thus to any individual at time t we can now associate the position on R × R+ , (xk (t), γt (ik (t))). The important point is that, for any given multi-index in ∂T, the function γt (i) converges, as t ↑ ∞, to some point γ(i) ∈ R+ , almost surely. Hence the sets γt (τ(t)) also converge, for any realisation of the tree, to some (random) set γ(τ(∞)). It is easy to see that γ(τ(∞)) is a Cantor set. Using the embedding γ defined above, we now state the extended convergence theorem. We use the notation from the previous chapters. # on R+ × R, #t ≡ n(t) δ(γ (ik (t)),x (t)−m(t)) → E Theorem 8.1 The point process E t k k=1 as t ↑ ∞, where  #≡ δ(qi ,pi )+(0,Δ(i) ) , (8.3) E i, j

j

where (qi , pi )i∈N √are the atoms of a Cox process on R+ × R with intensity measure Z(dv)×Ce− 2x dx, where Z(dv) is a random measure on R+ , characterised in Lemma 8.2, and Δ(i) j are the atoms of the independent and identically distributed point processes Δ(i) from Theorem 7.13. #t is that it allows us to visualise Remark The nice feature of the process E (i) the different clusters Δ corresponding to the different point of the Poisson  process of cluster extremes. In the process n(t) k=1 δ xi (t)−m(t) considered in earlier work, all these points get superimposed and cannot be disentangled. In other # encodes both the values and the (rough) genealogical words, the process E structure of the extremes of BBM. The measure Z(dv) is an interesting object in itself. For v, r ∈ R+ and t > r, we define  √ √ √ ( 2t − x j (t)) e 2(x j (t)− 2t) 1γr (i j (r))≤v , (8.4) Z(v, r, t) = j≤n(t)

which is a truncated version of the usual derivative martingale Zt . In particular, observe that Z(∞, r, t) = Zt . Lemma 8.2

For each v ∈ R+ Z(v) ≡ lim lim Z(v, r, t) r↑∞ t↑∞

(8.5)

exists almost surely. Moreover, 0 ≤ Z(v) ≤ Z, where Z is the limit of the derivative martingale and Z(v) is monotone increasing in u, almost surely. Finally, we show that the measure Z(dv) has no atoms. Lemma 8.3

The function Z(v) defined in (8.5) is continuous, almost surely.

8.2 Properties of the Embedding

147

The measure Z(v) is the analogue of the corresponding derivative martingale measure studied in Duplantier et al. [49, 50] and Biskup and Louidor [22, 21] in the context of the Gaussian free field. For a review, see Rhodes and Vargas [100]. The objects are examples of what is known as multiplicative chaos that was introduced by Kahane [73].

8.2 Properties of the Embedding We need the three basic properties of γ, which are stated in the following three lemmas. They all concern properties of particles that are extremal at time t and thus appear in the extremal process. Lemma 8.4 states that the map γt is well approximated by γr , when r is large, but small compared to t, provided t is large enough. This is crucially needed in the proof of Lemma 8.2. Lemma 8.4 Let D = [D, D] ⊂ R be a compact interval. Define, for 0 ≤ r < t < ∞, the events   (8.6) Aγr,t (D) = ∀k with xk (t) − m(t) ∈ D : γt (ik (t)) − γr (ik (r)) ≤ e−r/2 . For any  > 0 there exists 0 ≤ r(D, ) < ∞ such that, for any r > r(D, ) and t > 3r, c   (8.7) P Aγr,t (D) < . Lemma 8.5 ensures that the positions of extremal particles in the tree are not atomic in the sense that γ maps particles with a low probability to a very small neighbourhood of a fixed a ∈ R. Let a ∈ R+ and let D be as in Lemma 8.4. Define the event   (8.8) Bγr,t (D, a, δ) = ∀k with xk (t) − m(t) ∈ D: γr (ik (r))  [a − δ, a] .

Lemma 8.5

For any  > 0 there exists δ > 0 and r(a, D, δ, ) such that, for any r > r(a, D, δ, ) and t > 3r, c   (8.9) P Bγr,t (D, a, δ) < . Lemma 8.5 asserts that any two points that get close to the maximum of BBM will have distinct images under the map γ, unless their genealogical distance is small.

148

Full Extremal Process

Lemma 8.6 Let D ⊂ R be as in Lemma 8.4. For any  > 0, there exists δ > 0 and r(δ, ) such that, for any r > r(δ, ) and t > 3r,  P ∃i, j≤n(t):d(xi (t),x j (t))≤r : xi (t), x j (t) ∈ m(t) + D,  (8.10) and |γt (ii (t)) − γt (i j (t))| ≤ δ < . The three properties are rather intuitive, although the proofs are slightly painful and not that intuitive. We do not reproduce them here but refer to [29]. We are now ready to prove Lemma 8.2. Proof of Lemma 8.2 For v, r ∈ R+ fixed, the process Z(v, r, t) defined in (8.4) is a martingale in t > r (since Z(∞, r, t) is the derivative martingale and 1γr (ii (r))≤v does not depend on t). To see that Z(v, r, t) converges, a.s., as t ↑ ∞, note that Z(v, r, t) =

n(r) 

1γr (ii (r))≤v e

√

√ 2(xi (r)− 2r)

 n(i) (t−r) 

i=1

√  √ (i) √ 2r − xi (r) e 2(x j (t−r)− 2(t−r))

j=1

+

n(i) (t−r) 

 √2(x(i) (t−r)− √2(t−r)) √ j 2(t − r) − x(i) j (t − r) e



j=1

=

n(r) 

1γr (ii (r))≤v e

√ √ 2(xi (r)− 2r)

 (i) √ 2r − xi (r) Yt−r

i=1

+

n(r) 

1γr (ii (r))≤v e

√ √ 2(xi (r)− 2r) (i) Zt−r .

(8.11)

i=1

Here Zt(i) , i ∈ N, are iid copies of the derivative martingale, and Yt(i) are iid (i) → 0, copies of the McKean martingale. From Section 5.6 we know that Yt−r (i) (i) (i) a.s., while Zt−r → Z , a.s., where Z are non-negative random variables. Hence n(r) √  √ e 2(xi (r)− 2r) Z (i) 1γr (ii (r))≤v , (8.12) lim Z(v, r, t) ≡ Z(v, r) = t↑∞

i=1

where Z , i ∈ N are iid copies of Z. To show that Z(v, r) converges, as r ↑ ∞, we go back to (8.4). Note that, for fixed v, 1γr (ii (r))≤v is monotone  in  √decreasing r. On the other hand, we have seen in Eq. (5.52) that mini≤n(t) 2t − xi (t) → +∞, almost surely, as t ↑ ∞. Therefore, the part of the √ sum in (8.4) that involves negative terms (namely those for which xi (t) > 2t) converges to zero, almost surely. The remaining part of the sum is decreasing in r, and this implies that the limit, as t ↑ ∞, is monotone decreasing, almost surely. Moreover, 0 ≤ Z(v, r) ≤ Z, a.s., where Z is the almost sure limit of the derivative martingale. Thus limr↑∞ Z(v, r) ≡ Z(v) exists. Finally, 0 ≤ Z(v) ≤ Z and Z(v) is an (i)

149

8.3 The q-Thinning

increasing function of v because Z(v, r) is increasing in v, a.s., for each r. This proves Lemma 8.2. 

8.3 The q-Thinning  i The proof of the convergence of n(t) i=1 δ(γ(i (t)),xi (t)−m(t)) involves two main steps. In the first step, one shows that the local extrema converge to a Cox process. This result is an extension of the Poisson convergence result from Section 7.3. Recall the process Θrt defined √ in (7.55). For rd ∈ R+ and t > 3rd , we write, with Rt = m(t) − m(t − rd ) − 2rd = o(1), D

d) ≡ Θ(r t

n(r d) 

δ x j (rd )− √2rd +M j (t−rd )−Rt ,

(8.13)

j=1

where M j (t − rd ) ≡ maxk≤n j (t−rd ) xk( j) (t − rd ) − m(t − rd ) and x( j) , j ∈ N, are independent BBMs. Then Proposition 8.7

Let xik (t) denote the atoms of the process Θrt d . Then lim lim

rd ↑∞ t↑∞

n∗ (t) 

D

δ(γt (iik (t)),xik (t)−m(t)) =



/ δ(qi ,pi ) ≡ E,

(8.14)

i

k=1

/ with intensity measure where (qi , pi√)i∈N are the points of the Cox process E − 2x dx, with the random measure Z(dv) defined in (8.5). Moreover, Z(dv) × Ce lim lim

r↑∞ rd ↑∞

n(r d) 

D / δ(γr (x j (r)),x j (rd )− √2rd +M j ) = E,

(8.15)

j=1

where M j are iid and distributed with the limit law of the maximum of BBM. The proof of Proposition 8.7 relies on Lemma 8.2 and uses the properties of the map γ obtained in Lemmas 8.4 and 8.5. In particular, we use that, in the limit as t ↑ ∞, the image of the extremal particles under γ converges, and that essentially no particle is mapped too close to the boundary of any given compact set. Then we use the same procedure as in the proof of Proposition 5 in [9].. Proof of Proposition 8.7 We show the convergence of the Laplace functionals. Let φ : R+ × R → R+ be a measurable function with compact support. For simplicity we start by looking at simple functions of the form φ(x, y) =

N  i=1

ai 1Ai ×Bi (x, y),

(8.16)

150

Full Extremal Process

where Ai = [Ai , Ai ] and Bi = [Bi , Bi ] for N ∈ N, i = 1, . . . , N, ai , Ai , Ai ∈ R+ and Bi , Bi ∈ R. The extension to general functions φ then follows by monotone convergence. For such φ, we consider the Laplace functional ⎡ ⎞⎤ ⎛ n∗ (t) ⎢⎢⎢ ⎜⎜⎜   ⎟⎟⎟⎥⎥⎥ i ⎢ ⎜ k φ γt (i (t)), x¯ik (t) ⎟⎟⎠⎟⎥⎥⎦⎥ . Ψt (φ) ≡ E ⎢⎣⎢exp ⎜⎝⎜− (8.17) k=1

The idea is that the function γ only depends on the early branchings of the particle. To this end we insert the identity 1 = 1Aγr,t (suppy φ) + 1Aγ (supp φ)c r,t

(8.18)

y

into (8.17), where Aγr,t is defined in (8.6), and by suppy φ we mean the support of φ with respect to the second variable. By Lemma 8.4, we have that, for all  > 0, there exists r > 0 such that, for all r > r ,  c  P Aγr,t (suppy φ) < , (8.19) uniformly in t > 3r. Hence it suffices to show the convergence of ⎞ ⎤ ⎡ ⎛ n∗ (t) ⎥⎥⎥ ⎢⎢⎢ ⎜⎜⎜   ⎟⎟⎟ i φ γt (i k (t)), x¯ik (t) ⎟⎟⎠⎟ 1Aγr,t (suppy φ) ⎥⎥⎦⎥ . E ⎢⎢⎣⎢exp ⎜⎜⎝⎜−

(8.20)

k=1

We introduce another one into (8.20) in the form  γ  "  γ c , (8.21) 1"i=1 N N Br,t (suppy φ,Ai )∩Bγr,t (suppy φ,Ai ) + 1 i=1 Br,t (supp φ,Ai )∩Bγr,t (supp φ,Ai ) y

y

where we use the shorthand notation Bγr,t (suppy φ, Ai ) ≡ Bγr,t (suppy φ, Ai , e−r/2 )

(8.22)

(recall (8.8)). By Lemma 8.5 there exists, for all  > 0, r¯ such that, for all r > r¯ and uniformly in t > 3r,  c   γ N P ∩i=1 Br,t (suppy φ, Ai ) ∩ Bγr,t (suppy φ, Ai ) < . (8.23) Hence we only have to show the convergence of ⎞ ⎛ n∗ (t) % $ ⎜⎜⎜   ⎟⎟⎟ i    ⎟ ⎜ k " φ γt (i (t)), x¯ik (t) ⎟⎟⎠ 1Aγ (supp φ)∩ N Bγ (supp φ,A )∩Bγ (supp φ,Ai ) . E exp ⎜⎜⎝− r,t r,t r,t i=1 i y y y k=1

(8.24) Observe that on the event in the indicator function in the last line the following holds: if for any i ∈ {1, . . . , N}, γt (ik (t)) ∈ [Ai , Ai ] and x¯k (t) ∈ suppy φ then also

8.3 The q-Thinning

151

γr (ik (r)) ∈ [Ai , Ai ], and vice versa. Hence (8.24) is equal to ⎞ ⎛ n∗ (t) $ % ⎜⎜⎜   ⎟⎟⎟ i    ⎟ ⎜ k " φ γr (i (r)), x¯ik (t) ⎟⎟⎠ 1Aγ (supp φ)∩ N Bγ (supp φ,A )∩Bγ (supp φ,Ai ) . E exp ⎜⎜⎝− r,t r,t r,t i=1 i y y y k=1

(8.25) Now we apply again Lemmas 8.4 and Lemma 8.5 to see that the quantity in (8.25) is equal to ⎞⎤ ⎡ ⎛ n∗ (t) ⎢⎢⎢ ⎜⎜⎜   ⎟⎟⎟⎥⎥⎥ i ⎢ ⎜ k E ⎢⎢⎣exp ⎜⎜⎝− φ γr (i (r)), x¯ik (t) ⎟⎟⎟⎠⎥⎥⎥⎦ + O(). (8.26) k=1

Introducing a conditional expectation given Frd , we get (analogous to (3.16) in [9]), as t ↑ ∞, that (8.26) is equal to ⎞⎤ ⎡ ⎛ n∗ (t) ⎢⎢⎢ ⎜⎜⎜   ⎟⎟⎟⎥⎥⎥ i ⎢ ⎜ k lim E ⎢⎢⎣exp ⎜⎜⎝− φ γr (i (r)), x¯ik (t) ⎟⎟⎟⎠⎥⎥⎥⎦ t↑∞ k=1

$ n(r % d)  ( j max x( j) (t−rd )−m(t−rd )) ( (Frd E e−φ(γr (i (r)),x j (rd )−m(t)+m(t−rd ) e i≤n( j) (t−rd ) i = lim E t↑∞

j=1

⎤ ⎡n(r ) d ⎢⎢⎢   (( ⎥⎥⎥ √ j E exp −φ(γr (i (r)), x j (rd ) − 2rd + M) (Frd ⎥⎥⎥⎦ , = E ⎢⎢⎢⎣

(8.27)

j=1

where M is the limit of the rescaled maximum of BBM. The last expression is completely analogous to Eq. (3.17) in [9]. Following the analysis of this expression up to Eq. (3.25) in [9], we find that (8.27) is equal to √ ) 

− √2 B − √2 B  * − 2y j (rd ) N ai j i i −e i=1 (1−e )1Ai (γr (i (r))) e , (8.28) crd E e−C j≤n(rd ) y j (rd ) e √ where y j (rd ) = x j (rd ) − 2rd and limrd ↑∞ crd = 1, and C is the constant from law of the maximum of BBM. Using Lemma 8.2 we can show that (8.28) converges, as rd ↑ ∞ and r ↑ ∞, to ⎡ ⎤ ⎞ ⎛ N √ √ ⎢⎢⎢ ⎥⎥ ⎟⎟ ⎜⎜⎜  ai − 2Bi − 2 Bi ⎟ ⎢ ⎟ ⎜ E ⎢⎣exp ⎜⎝−C (1 − e ) e −e ⎟⎠ (Z(Ai ) − Z(Ai ))⎥⎥⎥⎦ i=1

% $   √  √ −φ(x,y) − 2y − 1 Z(dx) 2Ce dy . = E exp e

(8.29)

/ which proves Proposition 8.7. This is the Laplace functional of the process E  We can now pause and prove Lemma 8.3.

152

Full Extremal Process

Proof of Lemma 8.3 To show that Z(du) is non-atomic, fix , δ > 0 and let D ⊂ R be compact. By Lemma 8.6 there exists r1 (, δ) such that, for all r > r1 (, δ) and t > 3r,  P ∃i, j≤n(t) : d(xi (t), x j (t)) ≤ r, xi (t), x j (t) ∈ m(t) + D,  (8.30) and |γt (ii (t)) − γt (i j (t))| ≤ δ < . Rewriting (8.30) in terms of the thinned process E(r/t) (t) gives   P ∃ik ,ik : x¯ik , x¯ik ∈ m(t) + D, |γt (iik (t)) − γt (iik (t))| ≤ δ ≤ .

(8.31)

Assuming for the moment that E(r/t) (t) converges as claimed in Proposition 8.7, this implies that, for any  > 0, for small enough δ > 0,   (8.32) P ∃δ > 0 : ∃i  j : |qi − q j | < δ < . This could not be true if Z(du) had an atom. This proves Lemma 8.2, provided  we can show convergence of E(r/t) (t). Proposition 8.7 is the key step in the proof of Theorem 8.1. To pass from the convergence of the process of the cluster extremes to the full extremal process, we just need to use the convergence of the decoration process Dt and the fact that the points in the decoration process correspond to recent relatives of the cluster extremes, and that therefore their position in the tree under γ is exactly the same as that of the cluster extremes. For a formal proof, see [28].

8.4 Bibliographic Notes 1. The material in this chapter is taken from Bovier and Hartung [29], where more details can be found. It is motivated by a result analogous to Theorem 8.1 for the discrete Gaussian free field that was obtained by Biskup and Louidor [23].

9 Variable Speed BBM

We have seen that BBM is somewhat related to the GREM, where the covariance is a linear function of the distance. It is natural to introduce versions of BBM that have more general covariance functions A(x). This can be achieved by changing the speed (= variance) of the Brownian motion with time.

9.1 The Construction The general model can be constructed as follows. Let A : [0, 1] → [0, 1] be a right-continuous increasing function. Fix a time horizon t and let Σ2 (u) = tA(u/t).

(9.1)

Note that Σ2 is almost everywhere differentiable and denote by σ2 (s) its derivative wherever it exists. We define Brownian motion with speed function Σ2 as a time change of ordinary Brownian motion on [0, t] as BΣs = BΣ2 (s) .

(9.2)

Branching Brownian motion with speed function Σ2 is constructed like ordinary branching Brownian motion except that, if a particle splits at some time s < t, then the offspring particles perform variable speed Brownian motion with speed function Σ2 , i.e. their laws are independent copies {BΣr − BΣs }t≥r≥s , all starting at the position of the parent particle at time s. We denote by n(s) the number of particles at time s and by {xi (s); 1 ≤ i ≤ n(s)} the positions of the particles at time s. If we denote by d(xk (t), x (t)) the time of the most recent common ancestor of the particles i and k, then a simple computation shows that E [xk (s)x (s)] = Σ2 (d(xk (s), x (s))) . 153

(9.3)

154

Variable Speed BBM

Moreover, for different times, we have E [xk (s)x (r)] = Σ2 (d(xk (t), x (t) ∧ s ∧ r)) .

(9.4)

Remark Strictly speaking, we are not talking about a single stochastic pro+ cess, but about a family {xk (s), k ≤ n(s)}t∈R s≤t of processes with finite time horizon, indexed by that horizon, t. The case when A is a step function with finitely many steps corresponds to Derrida’s GREMs, with the only difference that the binary tree is replaced by a Galton–Watson tree. The case we discuss here corresponds to A being a piecewise linear function. The case when A is arbitrary has been dubbed CREM in [31] (and treated for binary regular trees). In that case the leading order of the maximum was obtained; this analysis carries over mutatis mutandis to the BBM situations. Fang and Zeitouni [52] have obtained the order of the correction (namely t1/3 ) in the case when A is strictly concave and continuous, but there are no results on the extremal process or the law of the maximum. This result has very recently strengthened by Maillard and Zeitouni [88], who proved convergence of the law of the maximum to some travelling wave and computed the next order of the correction (which is logarithmic).

9.2 Two-Speed BBM Understanding the piecewise linear case seems to be a prerequisite to getting the full picture. The simplest case is two-speed BBM, i.e. ⎧ 2 ⎪ ⎪ 0 ≤ s < bt ⎨σ1 , 0 < b ≤ 1, (9.5) σ2 (s) = ⎪ ⎪ ⎩σ2 , bt ≤ s ≤ t, 2

with the normalisation σ21 b + σ22 (1 − b) = 1. Note that in the case b = 1, σ2 = ∞ is allowed. Fang and Zeitouni [51] showed that ⎧√ ⎪ ⎪ 2t − 2 √1 2 ln t + O(1), ⎪ ⎪ ⎪ ⎪ ⎨√ max xk (t) = ⎪ 2t(σ1 b + σ2 (1 − b)) ⎪ ⎪ k≤n(t) ⎪ ⎪ ⎪ ⎩− √3 (σ1 + σ2 ) ln t + O(1), 2 2

(9.6)

if σ1 < σ2 , (9.7) if σ1 > σ2 .

The second case has a simple interpretation: the maximum is achieved by adding to the maxima of BBM at time tb the maxima of their offspring at time t(1 − b) later, just as in the analogous case of the GREM. The first case looks

155

9.2 Two-Speed BBM

like the REM, which is not surprising, as we have already seen in the GREM that correlations have no influence as long as A(u) < u. But when looking at the extremal process, we will find that here things are a lot more subtle. The main result of [27] is the following. Theorem 9.1 ([27]) Let xk (t) be branching Brownian motion with variable speed as given in (9.5). Assume that σ1 < σ2 and b ∈ (0, 1). Then (i)





) √ *  − 2x lim P max xk (t) − m(t) ˜ ≤ x = E e−C Ye , t↑∞

(9.8)

k≤n(t)

√ where m(t) ˜ = 2t − 2 √1 2 ln t, C  is a constant and Y is a random variable that is the limit of a martingale (but different from Z!). (ii) The point process  # #t ≡ δ xk (t)−m(t) → E, (9.9) E ˜ k≤n(t)

as t ↑ ∞, in law, where #= E



δηk +σ2 Δ(k) ,

(9.10)

j

k, j

where ηk is the kth √ atom of a mixture of Poisson point process with intensity measure C  Ye− 2x dx, with C  and Y as in (i), and Δ(k) i are the atoms of independent and identically distributed point processes Δ(k) , which are the limits in law of  δ xi (t)−max j≤n(t) x j (t) , (9.11) j≤n(t)

where x(t) is BBM of speed 1 conditioned on max j≤n(t) x j (t) ≥

√

2σ2 t.

The picture is completed by the limiting extremal process in the case σ1 > σ2 . This result is much simpler and could be guessed from known facts about the GREM. Theorem 9.2 ([27]) Let xk (t) be as in Theorem 9.1, but with σ2 < σ1 . Again b ∈ (0, 1). Let E ≡ E0 and E(i) , i ∈ N be independent copies of the extremal process of standard branching Brownian motion. Let √ 3 3 m(t) ≡ 2t(bσ1 + (1 − b)σ2 ) − √ (σ1 + σ2 ) ln t − √ (σ1 ln b + σ2 ln(1 − b)), 2 2 2 2 (9.12) and set  #t ≡ E δ xk (t)−m(t) . (9.13) k≤n(t)

156

Variable Speed BBM

Then #t = E # lim E

(9.14)

t↑∞

exists, and #= E



δσ1 ei +σ2 e(i) ,

(9.15)

j

i, j

(i) where ei , e(i) j are the atoms of the point processes E and E , respectively.

We just remark on the main steps in the proof of Theorem 9.1. The first step is a localisation of the particles that will eventually reach the top at the time of the speed change. The√following Proposition 9.3 says that these are in a √ t–neighbourhood of σ21 t 2b, which is much √ smaller then the position of the leading particles at time tb, which is near σ1 t 2b . Thus, the faster particles in the second half-time must make up for this. It is not very hard to know everything about the particles after time tb. The main problem one is faced with is to control their initial distribution at this time. Fortunately, this can be done with the help of a martingale. Define, for s ∈ R+ . Ys =

n(s) 

e−s(1+σ1 )+ 2

√ 2xi (s)

.

(9.16)

i=1

When σ1 < 1, one can show that Y is a uniformly integrable positive martingale with mean value one. The martingale Y will take over the role of the derivative martingale in the standard case.

Position of Extremal Particles at Time bt The key to understanding the behaviour of the two-speed BBM is to control the positions of particles at time bt which are in the top at time t. This is done using straightforward Gaussian estimates. Proposition 9.3 Let σ1 < σ2 . For any d ∈ R and any  > 0, there exists a constant A > 0 such that, for all t large enough, √ √ √ ˜ − d and x j (bt) − 2σ21 bt  [−A t, A t] ≤ . P ∃ j≤n(t) s.t. x j (t) > m(t) (9.17) Proof

Using a first order Chebyshev inequality, we bound (9.17) by √ √ √ √   t/(2 2t−d/ t √ et E 1{w1 − √2σ1 √bt[−A ,A ]} Pw2 w2 > 2t−σ1 bwσ1 −ln , 1−b

(9.18)

2

with w1 , w2 independent N(0, 1)-distributed random variables, A =

√ 1 A, bσ1

157

9.2 Two-Speed BBM

Pw2 denotes the law of the variable w2 . Inserting into the expectation in (9.18) a factor 1 in the form 1 = 1{ √2t−σ1 √bw1 z + 2r e( 2+a)z 1 − e−2az dz, C(a) = lim r→∞

0

k≤n(r)

(9.53)

164

Variable Speed BBM

where { x¯k (t), k ≤ n(t)} are the particles of a standard BBM. Proof Denote by {xi (bt), 1 ≤ i ≤ n(bt)} the particles of a BBM with variance σ1 at time bt, and by Fbt the σ–algebra generated this BBM. Moreover, for 1 ≤ i ≤ n(bt), let {xij ((1 − b)t), 1 ≤ j ≤ ni ((1 − b)t)} denote the particles of independent BBM with variance σ2 at time (1 − b)t. By second moment estimates one can easily show that the order of the maximum is m(t), ˜ i.e. for any  > 0, there exists d < ∞ such that $ % P max xk (t) − m(t) ˜ ≤ −d ≤ /2. (9.54) 1≤k≤n(t)

Therefore,

% $ % $ ˜ ≤ y ≤ P max xk (t) − m(t) ˜ ≤y P −d ≤ max xk (t) − m(t) 1≤k≤n(t) 1≤k≤n(t) $ % ≤ P −d ≤ max xk (t) − m(t) ˜ ≤ y + /2 (9.55) 1≤k≤n(t)

For details see [51]. On the other hand, by Proposition 9.3, all the particles that may contribute to the event in question are localised at time bt in the narrow window described by Gbt,A, 12 . Thus we obtain % $ ˜ ≤y (9.56) P max xk (t) − m(t) 1≤k≤n(t) $ % = P max max xi (bt) + xij ((1 − b)t) − m(t) ˜ ≤y 1≤i≤n(bt) 1≤ j≤ni ((1−b)t) ⎡ (( %⎤ $ ⎢⎢⎢  ⎥⎥⎥ ( = E ⎢⎢⎢⎣ P max xij ((1 − b)t) ≤ m(t) ˜ − xi (bt) + y(( Fbt ⎥⎥⎥⎦ ( 1≤ j≤ni ((1−b)t) 1≤i≤ni (bt) (( %% $ $  ( P max xij ((1 − b)t) ≤ m(t) ˜ − xi (tb) + y(( Ftb + . ≤E ( 1≤ j≤ni ((1−b)t) 1≤i≤n(bt)

xi ∈Gbt,A, 1 2

The corresponding lower bound holds without the . We write the probability in (9.56) as (( % $ ( i −1 ˜ − xi (tb) + y) (( Ftb 1−P max x¯ j ((1 − b)t) > σ2 (m(t) ( 1≤ j≤ni ((1−b)t)   −1 ˜ − xi (bt) + y) , (9.57) = 1 − u (1 − b)t, σ2 (m(t) where x¯ij ((1 − b)t) are the particles of a standard BBM and u is the solution of the F-KPP equation with Heaviside initial conditions. By Proposition 9.8 and setting √ √ 2 Ct (x) ≡ e 2x+x /2t t1/2 u(t, x + 2t), (9.58)

165

9.2 Two-Speed BBM we have that   ˜ − xi (bt) + y) u (1 − b)t, σ−1 2 (m(t)   √ ˜ − xi (bt) + y) − t 2(1 − b)) ((1 − b)t)−1/2 = C(1−b)t σ−1 2 (m(t) ' '2 & √ & m(t)−x √ √ ˜ m(t)−x ˜ 1 i (bt)+y − 2(1−b)t i (bt)+y − 2(1−b)t − 2 − 2(1−b)t σ σ

×e

e

2

2

Now all the xi (bt) that appear are of the form xi (bt) =

√

.

(9.59)

√ 2σ21 bt + O( t), so that

  √ ˜ − xi (bt) + y) − 2(1 − b)t) C(1−b)t σ−1 2 (m(t) √ = C(1−b)t (a(1 − b)t + O( t)),

(9.60)

where, (using (9.6)) √ ⎛√ ⎞ ⎟⎟ √ 1 ⎜⎜⎜⎜ 2 − 2σ21 b √ − 2(1 − b)⎟⎟⎟⎠ = 2(σ2 − 1). a≡ ⎜ 1−b⎝ σ2

(9.61)

By Proposition 9.8,   √ ˜ − xi (bt) + y) − 2(1 − b)t) = C(a), lim C(1−b)t σ−1 2 (m(t)

(9.62)

t↑∞

with uniform convergence for all i appearing in (9.56) and where C(a) is the constant given by (9.53). After a little algebra we can rewrite the expectation in (9.56) as E

$  

1 − C(a)((1 − b)t)

−1/2

e

(1−b)t−

2 (m(t)+y−x ˜ i (bt)) 2(1−b)tσ22

%

(1 + o(1)) .

(9.63)

1≤i≤n(bt)

xi ∈Gbt,A,1/2

Using that xi (bt) − & exp (1 − b)t −

√

√ √ 2σ21 tb ∈ [−A t, A t], we have the uniform bounds

2 (m(t)+y−x ˜ i (bt)) 2(1−b)tσ22

'

 √ ≤ exp (1 − σ22 )bt + log t + A t ,

(9.64)

which is exponentially small in t since σ22 > 1. Hence (9.63) is equal to ⎡ ⎢⎢⎢ ⎢⎢  E ⎢⎢⎢⎢⎢ ⎢⎣

1≤i≤n(bt)

xi ∈Gbt,A,1/2

⎤ ⎛ ⎞⎥⎥ 2 (m(t)+y−x ˜ i (bt)) ⎜⎜⎜ ⎟ ⎟⎟⎥⎥⎥ (1−b)t− 2(1−b)tσ22 exp ⎜⎜⎝⎜−C(a)((1 − b)t)−1/2 e (1 + o(1))⎟⎟⎟⎠⎥⎥⎥⎥⎥ . (9.65) ⎥⎦

Expanding the square in the exponent and keeping only the relevant terms

166

Variable Speed BBM

yields (1 − b)t −

2 (m(t)+y−x ˜ i (bt)) 2(1−b)tσ22

√ √ = (1 − b)t − 2y − tσ22 (1 − b) + 2σ21 bt + 2xi (bt) − 12 ln t √ 2 2tσ21 b − xi (bt) + + o(1) 2(1 − b)σ22 t √ 2 2tσ21 b − xi (bt) √ √ 2 1 + o(1). = − 2y − bt(1 + σ1 ) + 2xi (bt) + 2 ln t − 2(1 − b)σ22 t (9.66)

The first four summands in the last line of (9.66) would nicely combine to produce the McKean martingale as coefficient of C(a). Namely, & ' √ √ 1 2 exp −C(a)((1 − b)t)−1/2 e− 2y−bt(1+σ1 )+ 2xi (bt)+ 2 ln t

 1≤i≤n(bt)

xi ∈Gbt,A,1/2

⎛ ⎜⎜⎜ ⎜⎜ C(a) − √2y e = exp ⎜⎜⎜⎜⎜− √ ⎜⎝ 1−b



⎞ ⎟

⎟⎟⎟ √ ⎟⎟ −bt(1+σ21 )+ 2σ1 x¯i (bt) ⎟

e

1≤i≤n(bt)

xi ∈Gbt,A,1/2

⎟⎟⎟ . ⎟⎠

(9.67)

However, the fifth summand in the last line of (9.66) is of order one and cannot √ be neglected. To deal with them, we split the process at time b t. We write, √ √ somewhat abusively, xi (bt) = xi (b t) + xl(i) (b(t − t)), where we understand √ √ that xi (b t) is the ancestor at time b t of the particle that at time t is labelled i √ if we think backwards from time t, while the labels of the particles at time b t √ run only over the different ones, i.e. up to n(b t), if we think in the forward direction. No confusion should occur if this is kept in mind. Using Proposition 9.4 and Proposition 9.5, we can further localise the path of the particle. Recalling the definition of G s,A,γ and Tr,s , we rewrite (9.65), up to a term of order , as $  $   E exp −C(a)((1 − b)t)−1/2 (9.68) E √ 1≤i≤n(b t)

xi ∈Gb √t,B,γ

&

√ (i) (b(t− t)) l

1≤l≤n

xi ∈Gbt,A, 1 ;xl(i) ∈Tb(t− √t),r

× exp (1 − b)t −

2

√ √ 2' (i) (m(t)+y−x ˜ i (b t)−xl (b(t− t))) (1 2 2(1−b)tσ2

%%  (( ( + o(1)) (( F √tb . (

√ √ √ √ √ Using that xi (b t) + xl(i) (b(t − t)) − 2σ21 tb ∈ [−A t, A t] and m(t) ˜ =

167

9.2 Two-Speed BBM √ 2t −

1 √ 2 2

log t, we can rewrite the terms multiplying C(a) in (9.68) as

 √ √ √ exp − (1 + σ21 )bt + 2(xi (b t) + xl(i) (b(t − t))) √ √ √ √ √  1 (x (b t)+xl(i) (b(t− t))− 2σ21 bt)2 + O(1/ t) − log(1 − b) − 2y − i 2t 2(1−b)σ 2 2 √ √ ≡ E(xi , xl(i) ) = E(xi (b t), xl(i) (b(t − t))). (9.69) Abbreviate



Xi ≡

√ (i) (b(t− t)) l (i) xi ∈Gbt,A, 1 ;xl ∈Tr,b(t− √t) 2

E(xi , xl(i) ).

(9.70)

1≤l≤n

Then (9.68) takes the form ⎡ ⎤ ⎢⎢⎢ ⎥ ) *⎥⎥⎥⎥ ⎢⎢⎢  (( ⎥ E ⎢⎢⎢⎢⎢ E XiC(A)(1 + o(1)) (( Fb √t ⎥⎥⎥⎥⎥ . ⎢⎢⎣ 1≤i≤n(b √t) ⎥⎥⎦ √ xi ∈Gb

(9.71)

t,B,γ

The idea is that the exponential in the conditional expectation can be replaced by its linear approximation, and that the linear term can be nicely computed. Performing some fairly straightforward algebra, we see that E[Xi |Fb √t ] ≤e

=e

√ √ b(σ21 t− t)− 2y



√ Kt +A t √ Kt −A t

e

√ √ √ √ (z+xi (b t)− 2σ2 bt)2 1 2(z+xi (b t))− 2

√ √ √ −(1+σ21 )b t+ 2xi (bt)− 12 log(1−b)− 2y

= e−(1+σ1 )b 2

2σ (1−b)t 2

&

σ22 (1−b) √ 1−σ21 b/ t

√ √ √ t+ 2xi (bt)− 2y

'1/2

√

e−z /2σ1 b(t− t) + √ dz 2πσ21 b(t − t)

√ B t √ −B t

2

e−w

2

2

/2t √dw 2πt

σ2 (1 + o(1)), (9.72) + √ where o(1) ≤ O(tγ−1 ) and B = A/σ1 σ2 b(1 − b/ t). Note that the inequality comes from the fact that we have dropped the tube conditions. However, by the fact that the Brownian bridge is independent of its endpoint and that the bridge verifies the tube condition with probability at least (1 − ) (see Lemma 6.27), it follows that the right-hand side of (9.72) multiplied by an additional factor (1 − ) is a lower bound. This is what we want. To justify the replacement of the exponential by the linear approximation, due to the inequalities 1 − x ≤ e−x ≤ 1 − x +

1 2 x , 2

x > 0,

(9.73)

168

Variable Speed BBM

we just need to control the second moment. But $& '2 % √ √ √ √

2 A √ , E x2 |Fb √t ≤ e−2(1+σ1 )b t+2 2xi (b t)−2 2y E Yb(t− t)

(9.74)

A √ is the truncated McKean martingale defined in (9.30). Note that where Yb(t− t) its second moment is bounded by D2 (r) (see (9.43)). Comparing this to (9.72), one sees that

√ √ √ √ E x2 |Fb √t 2 2 γ/2

≤ D2 (r)e−(1+σ1 )b t+ 2xi (b t) ≤ Ce−(1−σ1 )b t+0(t ) , (9.75) √ E x|Fb t

which tends to zero uniformly as t ↑ ∞. Thus the second moment term is negligible. Hence we only have to control ⎡ ⎤ ⎢⎢⎢ ⎥⎥ √ √ √ ⎥⎥⎥⎥ ⎢⎢⎢⎢   2 −(1+σ )b t+ 2x (bt)− 2y i 1 E ⎢⎢⎢⎢ σ2 ⎥⎥⎥⎥⎥ 1 − C(a) e ⎢⎢⎣ 1≤i≤n(b √t) ⎥⎥⎦ √ xi ∈Gb

t,B,γ

⎡ ⎛ ⎢⎢⎢ ⎜⎜⎜ ⎢⎢⎢ ⎜⎜⎜ ⎢ = E ⎢⎢⎢⎢exp ⎜⎜⎜⎜⎜− ⎢⎢⎣ ⎜⎜⎝

 √ 1≤i≤n(b t)

xi ∈Gb √t,B,γ

⎞ ⎤ ⎟⎟⎟ ⎥⎥⎥ ⎟⎟⎟ ⎥⎥⎥ √ √ √ ⎟ −(1+σ21 )b t+ 2xi (bt)− 2y C(a) e σ2 ⎟⎟⎟⎟ (1 + o(1))⎥⎥⎥⎥⎥ ⎟⎟⎠ ⎥⎥⎦

) & ' * √ = E exp −C(a)σ2 e− 2y Y˜ bB√t,γ (1 + o(1)) ,

(9.76)

where Y˜ bB√t,γ

=

√ n(b t) i=1

e−(1+σ1 )b 2

√ √ √ t+ 2xi (b t)

1 xi (b √t)− √2σ21 b √t∈[−Btγ/2 ,Btγ/2 ] .

(9.77)

Now from Lemma 9.7, Y˜ bB√t,γ converges in probability and in L1 to the random

variable Y, when we let first t and then B tend to infinity. Since YbB√t,γ ≥ 0 and C(a) > 0, it follows that ) & √ '* lim lim inf E exp −C(a)σ2 Y˜ bB√t,γ e− 2y B↑∞ t↑∞ ) & √ '* = E exp −σ2C(a)y e− 2y . (9.78) Finally, letting r tend to +∞, all the –errors (that are still present implicitly), vanish. This concludes the proof of Theorem 9.11. 

Existence of the Limiting Process The following existence theorem is the basic step in the proof of Theorem 9.5.

9.2 Two-Speed BBM Theorem 9.12 Let σ1 < σ2 . Then, the sequence of point processes  Et = δ xk (t)−m(t) ˜

169

(9.79)

k≤n(t)

converges in law to a non-trivial point process E. It suffices to show that, for φ ∈ Cc (R) positive, the Laplace functional $  % Ψt (φ) = E exp − φ(y)Et (dy) , (9.80) of the processes Et converges. The proof of this is essentially a combination of the corresponding proof for ordinary BBM and what we did when showing convergence of the maximum. The result is that   lim Ψt (φ) = E exp (−σ2C(a, φ)Y) (9.81) t↑∞

where C(a, φ) is given by

∞ √ √ 1 2 C(a, φ) = lim √ v(t, z + 2t) e( 2+a)z−a t/2 dz, (9.82) t→∞ 2π 0 where v(t, x) is the solution of the F-KPP equation with initial data v(0, x) = ¯ ¯ ≡ φ(σ2 yz). 1 − e−φ(−x) . Here φ(z)

The Auxiliary Process The final step is the interpretation of the limiting process. This is again very much analogous to the standard BBM case. Let (ηi ; i ∈ N) be the atoms of a Poisson point process η on (−∞, 0) with intensity measure √ 2 σ √ 2 e−( 2+a)z e−a t/2 dz. 2π

(9.83)

For each i ∈ N, consider independent standard BBMs, x¯i . The auxiliary point process of interest is the superposition of the iid BBMs with drift shifted by 1 ηi + √2+a log Y, where a is the constant defined in (9.61):  √ ' . δ& √ 1 (9.84) Πt = i i,k

ηi +

2+a

log Y+ x¯k (t)− 2t σ2

Remark The form of the auxiliary process is similar to the case of standard BBM, but with a different intensity of the Poisson process. In particular, the intensity decays exponentially with t. This is a consequence of the fact that particles √ at the time of the√speed change were forced to be O(t) below the line 2t, in contrast to O( t) in the case of ordinary BBM. The reduction of the intensity of the process with t forces the particles to be selected at these locations.

170

Variable Speed BBM

Theorem 9.13 Let Et be the extremal process of the two-speed BBM. Then law

lim Et = lim Πt .

t→∞

(9.85)

t→∞

As the proof is in nature similar to that in the standard case, we skip the details. The following proposition shows that, in spite of the different Poisson ingredients, when we look at the process of the extremes of each of the xi (t), we end up with a Poisson point process just like in the standard BBM case. Proposition 9.14 Define the point process  δ& √ 1 Πext t ≡ ηi +

i,k

2+a

√ ' . log Y+maxk≤ni (t) x¯ki (t)− 2t σ2

Then law

lim Πext = PY ≡ t

t→∞



(9.86)

δ pi ,

(9.87)

i∈N

where PY √is the√Poisson point process on R with intensity measure σ2C(a)Y 2e− 2x dx. Proof We consider the Laplace functional of Πext t . Let M (i) (t) = max x¯k(i) (t)

(9.88)

¯ = φ(σ2 z). We want to show and, as before, φ(z) ⎡ ⎛ ⎞⎤ √ ⎟⎟⎟⎥⎥⎥ ⎢⎢⎢ ⎜⎜⎜  (i) ¯ i + M (t) − 2t⎟⎟⎠⎥⎥⎦ lim E ⎢⎢⎣exp ⎜⎜⎝− φ(η t↑∞

i

 = exp −σ2C(a)

∞ −∞



1−e

−φ(x)

√

2e

√ − 2x

 dx .

Since ηi is a Poisson point process and the M (i) are iid, we have ⎞⎤ ⎡ ⎛ √ ⎟⎟⎟⎥⎥⎥ ⎢⎢⎢ ⎜⎜⎜  (i) ¯ i + M (t) − 2t⎟⎟⎠⎥⎥⎦ E ⎢⎢⎣exp ⎜⎜⎝− φ(η i

(9.89)

(9.90)

 ) √ * √ dz 2 ¯ E 1 − e−φ(z+M(t)− 2t) e−( 2+a)z−a t/2 √ , −∞ 2π

 = exp −σ2

0

where M(t) has the same distribution as one of the variables M (i) (t). Now we ¯ . Hence the result follows by using apply Lemma 9.10 with√h(x) = 1√− e−φ(z) ¯ = φ(σ2 z) and 2 + a = 2σ2 , together with the change of variables that φ(z)  x = σ2 z. The following proposition states that the Poisson points of the auxiliary process contributing to the limiting process come from a neighbourhood of −at.

171

9.2 Two-Speed BBM

Proposition 9.15 Let z ∈ R,  > √0. Let ηi be the atoms of a Poisson point 2 process with intensity measure Ce−( 2+a)x−a t/2 dx on (−∞, 0]. Then there exists B < ∞ such that  √ √ √  sup P ∃i, k : ηi + x¯ki (t) − 2t ≥ z, ηi  [−at − B t, −at + B t] ≤ . (9.91) t≥t0

Proof

By a first order Chebyshev inequality, we have  √ √ P ∃i, k : ηi + x¯k(i) (t) − 2t ≥ z, ηi > −at + B t 0 √   √ 2 ¯k (t) ≥ 2t − x + z e−( 2+a)x e−a t/2 dx ≤C √ P max x −at+B t √ at−B t

=C

  √ √ 2 P max x¯k (t) ≥ 2t + x + z e( 2+a)x e−a t/2 dx, (9.92)

0

by the change of variables x → −x. Using the asymptotics of Lemma 6.42 we can bound (9.92) from above by

√ at−B t

√

√

t−1/2 e− 2(x+z) e−(x+z) /2t e( 2+a)x e−a t/2 dx 0 −B √ √ −( 2+a)z −z2 /2 −( 2+a)z −B2 /2 e dz ≤ ρCe e , ≤ ρCe √

ρC

2

2

(9.93)

−a t

√ √ by changing variables x → x/ t − a t. For any z ∈ R, this can be made as small as we want by choosing B large enough. Similarly, one bounds √  √ √ 2 P ∃i, k : ηi + xki (t) − 2t ≥ z, ηi < −at − B t ≤ ρCe−( 2+a)z e−B /2 . (9.94) 

This concludes the proof.

The next proposition describes the law of the clusters x¯k(i) . This is analogous to Theorem 7.14 (Theorem 3.4 in [11]). Proposition 9.16 Let x = at( + o(t) and { x˜√ k (t), k ≤ n(t)} be a standard BBM under the conditional law P · (( {max x˜k (t) − 2t − x > 0} . Then the point process  δ x˜k (t)− √2t−x (9.95) k≤n(t)

(  √ converges in law under P ·(({max x˜k (t) − 2t − x > 0} as t → ∞ to a well¯ The limit does not depend on x − at, and the maximum defined point process E. ¯ of E shifted by x has the law of an exponential random variable with parameter √ 2 + a.

172

Variable Speed BBM

Proof Set E¯ t = that, for X > 0,

 k

δ x˜k (t)− √2t and max E¯ t = max x˜k (t) −

√ 2t. First we show

& ' (( √ lim P max E¯ t > X + x (( max E¯ t > x = e−( 2+a)X .

t→∞

(9.96)

P max E¯ >X+x To see this we rewrite the conditional probability as [P[max tE¯ >x] ] and use the t uniform bounds from Proposition 6.2. Observing that √ √ ψ(r, t, X + x + 2t) (9.97) = e−( 2+a)X , lim √ t→∞ Ψ(r, t, x + 2t)

where ψ is defined in Eq. (7.24), we get (9.96) by first taking t → ∞ and then r → ∞. The general claim of Proposition 9.16 follows from (9.96) in exactly the same way as Theorem 7.14.  Define the gap process Dt =



δ x˜k (t)−max j x˜ j (t) .

(9.98)

k

¯ i.e. E¯ ≡  j δξ j , and define Denote by ξi the atoms of the limiting process E,  D≡ δΔ j , Δ j = ξ j − max ξi . (9.99) j

i

D is a point process on (−∞.0] with an atom at 0. Corollary 9.17 Let x = −at + o(t). In the limit t → ∞, the random variables Dt and x + max E¯ t are conditionally independent on the event {x + max E¯ t > b} for any b ∈ R. More precisely, for any bounded function f, h and φ¯ ∈ Cc (R), $   % (( ( ¯ ¯ ¯ φ(z)Dt (dz) h(x + max E) ( x + max E > b lim E f t→∞ √ √ % 6 ∞ $  −( 2+a)z h(z)( 2 + a)e dz b ¯ φ(z)D(dz) =E f . (9.100) √ −( 2+a)b e Proof

The proof is essentially identical to the proof of Corollary 4.12 in [11]. 

Finally we come to the description of the extremal process as seen from the Poisson process of cluster extremes, which is the formulation of Theorem 9.5. Theorem 9.18 Let PY be as in (9.87) and let {D(i) , i ∈ N} be a family of independent copies of the gap process (9.99) with atoms Δ(i) j . Then the point

9.2 Two-Speed BBM

173

process Et converges in law as t → ∞ to a Poisson cluster point process E given by  law E = δ pi +σ2 Δ(i) . (9.101) j

i, j

The proof of this theorem is very close to that of Theorem 7.13, resp. Theorem 2.1 in [11] and will be skipped.

The Case σ1 > σ2 In this section we prove Theorem 9.1. The existence of the process E from (9.15) will be a byproduct of the proof. The point is that, in this case, if we rerun the computations to find the location of the particles that contribute to the maximum at time bt, the naive computation would place them above the level of the maximal particles at that time. But, of course, there are no such particles. Thus the particles that reach the highest level are those that have been maximal at time bt. The following lemma that is contained in the calculation of the maximal displacement in [51] makes this precise. Lemma 9.19 ([51]) For all  > 0, d ∈ R, there exists a constant D large enough such that, for t sufficiently large, P [∃k ≤ n(t) : xk (t) > m(t) + d and xk (bt) < m1 (bt) − D] < .

(9.102)

Proof of Theorem 9.1 First we establish the existence of √ a limiting process. Note that m(t) = m1 (bt) + m2 ((1 − b)t), where mi (s) = 2σi s − 2 √3 2 σi log s. Recall that ¯ = φ(σ2 z) φ(z)

(9.103)

g(z) = 1 − e−φ(−z) .

(9.104)

and ¯

Here we assume that φ(x) = 1 x>a , for a ∈ R. Using that the maximal displacement is m(t) in this case, we can proceed as in the proof of Theorem 9.12 and only have to control ⎡ ⎡ ( ⎤⎤ ⎢⎢⎢  ⎢⎢⎢   (m(t)−x (bt))  (( ⎥⎥⎥⎥⎥⎥ i i E ⎢⎢⎢⎣ g − x¯ j ((1 − b)t) (( Ftb ⎥⎥⎥⎦⎥⎥⎥⎦ , (9.105) 1 − Ψt (φ) = E ⎢⎢⎢⎣ σ2 ( i≤n(tb)

j≤ni ((1−b)t)

where x¯ij ((1 − b)t) are the particles of a standard BBM at time (1 − b)t, and xi (bt) are the particles of a BBM with variance σ1 at time bt. Using Lemma

174

Variable Speed BBM

9.19 and Theorem 1.2 of [51] as in the proof of Theorem 9.11 above, we obtain that (9.105), for t sufficiently large, equals ⎡ ⎤ ⎡ ⎢⎢⎢ ( ⎤⎥  ⎢⎢⎢   (m(t)−x (bt))  (( ⎥⎥⎥⎥⎥⎥⎥ ⎢⎢⎢ i E ⎢⎢⎢⎢ E ⎢⎢⎢⎣ g − x¯ij ((1 − b)t) (( Ftb ⎥⎥⎥⎦⎥⎥⎥⎥⎥ + O(). σ2 ( ⎢⎣ ⎥⎦ i≤n(bt) j≤n ((1−b)t) xi (bt)>m1 (bt)−D

i

(9.106) The remainder of the proof has an iterated structure. In a first step we show that conditioned on Fbt for each i ≤ n(bt) the points {xi (bt) + xij ((1 − b)t) − m(t)|xi (bt) > m1 (bt) − D} converge to the corresponding points of the point process xi (bt) − m1 (bt) + σ2 E˜ (i) , where E˜ (i) are independent copies of the extremal process of standard BBM. To this end, observe that ⎤ ⎡ ⎥⎥⎥ ⎢⎢⎢  i ⎢ g(z − x¯ j ((1 − b)t))⎥⎥⎥⎦ (9.107) u((1 − b)t, z) = E ⎢⎢⎣ j≤n((1−b)t)

solves the F-KPP equation with initial conditions u(0, z) = g(z). Moreover, the assumptions of Theorem 5.8 are satisfied. Hence (9.106) is equal to ⎡ ⎤ ⎢⎢⎢ ( %  $ ⎥⎥⎥ √ m (bt)−x (bt) (  ⎢⎢⎢ ⎥⎥⎥ − 2 1 σ i (( ¯ 2  + E ⎢⎢⎢⎢ E e−C(φ)Ze (9.108) (( Fbt (1 + o(1)) ⎥⎥⎥⎥ . ⎢⎣ ⎥⎦ i≤n(bt)

xi (bt)>m1 (bt)−D

¯ is from standard BBM, i.e. Here C(φ) + ∞ √ √ 2y ¯ = lim 2 C(φ) u(t, y + 2t)y e dy. π t↑∞

(9.109)

0

Note that already in (9.108) the concatenated structure of the limiting point process becomes visible. In a second step, we establish that the points xi (bt) − # Define m1 (t) that have a descendant in the lead at time t converge to E. & ⎧ ) √ σ1 '* ⎪ ⎪ ¯ − 2 σ2 y , ⎪ if σ1 y < D, ⎨E exp −C(φ)Ze hD (y) ≡ ⎪ (9.110) ⎪ ⎪1, ⎩ if σ y ≥ D. 1

Then the expectation in (9.108) can be written as (we ignore the error term o(1), which is easily controlled using that the probability that the number of terms in the product is larger than N tends to zero as N ↑ ∞, uniformly in t) ⎤ ⎡ ⎥⎥⎥ ⎢⎢⎢  ⎢ hδ,D (m1 (bt)/σ1 − x¯i (t))⎥⎥⎥⎦ , (9.111) E ⎢⎢⎣ i≤n(bt)

9.2 Two-Speed BBM where now x¯ is standard BBM. Defining ⎡ ⎤ ⎢⎢⎢  ⎥⎥⎥ ⎢ hD (z − x¯i (bt))⎥⎥⎥⎦ , vD (t, z) = 1 − E ⎢⎢⎣

175

(9.112)

i≤n(t)

vD is a solution of the F-KPP equation with initial condition vD (0, z) = 1 − hD (z). But these initial conditions satisfy the assumptions of Theorem 5.8, and therefore, ) √ * ˜ ¯ ˜ − 2x vD (t, m(t) + x) → E e−C(D,Z,C(φ))Ze , (9.113) where Z˜ is an independent copy of Z and + ∞ √ √ ˜ ¯ vD (t, y + 2t)y e 2y dy. C(D, Z, C(φ)) = lim π2 t↑∞

(9.114)

0

By the same arguments as in standard BBM setting, one obtains that + ∞ √ √ ˜ ˜ ¯ ¯ v(t, y + 2t)y e 2y dy, (9.115) C(Z, C(φ)) ≡ lim C(D, Z, C(φ)) = lim π2 D↑∞

t↑∞

0

where v is the solution of the F-KPP equation with initial conditions v(0, z) = 1 − h(z), with ) & √ σ1 '* ¯ − 2 σ2 z . h(z) = E exp −C(φ)Ze (9.116) Therefore, taking the limit first as D ↑ ∞ in the left-hand side of (9.113), we get that ) √ * ˜ ¯ ˜ − 2x . (9.117) lim Ψt (φ(· + x)) = lim lim vD (t, m(t) + x) = E e−C(Z,C(φ))Ze t→∞

D↑∞ t→∞

˜ C(φ)) ¯ are strictly positive, one uses that the To see that the constants C(Z, Laplace functionals Ψt (φ) are bounded from above by $   % E exp −φ max xi (bt) + max x1j ((1 − b)t) − m(t) . (9.118) i≤n(bt)

j≤n1 ((1−b)t)

Here we have used that the offspring of any of the particles at time bt has the same law. So the sum of the two maxima in the expression above has the same distribution as the largest descendant at time t of the largest particle at time bt. The limit of Eq. (9.118) as t ↑ ∞ exists and is strictly smaller than 1 by the convergence in law of the recentred maximum of a standard BBM. But this ˜ Hence a limiting point process exists. implies the positivity of the constants C. Finally, one may easily check that the right-hand side of (9.117) coincides with the Laplace functional of the point process defined in (9.15) by basically repeating the computations above. 

176

Variable Speed BBM

Remark Note that in particular, the structure of the variance profile is con˜ ¯ and also that the information on the structained in the constant C(D, Z, C(φ)) ture of the limiting point process is contained in this constant. In fact, we see that, in all cases we have considered in this chapter, the Laplace functional of the limiting process has the form ) & √ '* lim Ψt (φ(· + x)) = E exp −C(φ)Me− 2x , (9.119) t↑∞

where M is a martingale limit (either Y of Z) and C is a map from the space of positive continuous functions with compact support to the real numbers. This function contains all the information on the specific limiting process. This is compatible with the finding in [88] in the case where the speed is a concave function of s/t. The universal form (9.119) is thus misleading, and without knowledge of the specific form of C(φ), (9.119) contains almost no information. Remark There is no difficulty in extending this result to the case of multispeed BBM with finitely many decreasing speeds.

9.3 Universality Below the Straight Line It turns out that, in the case when the covariance stays strictly below the straight line A(s) = s for all s ∈ (0, t), the same picture emerges as in the two-speed case, with the McKean martingale depending only on the slope at 0 and the decoration process depending only on the slope at 1. In [28] a rather large class of functions A was considered. Here we will simplify the presentation by restricting ourselves to smooth functions. Let A : [0, 1] → [0, 1] be a non-decreasing function twice differentiable with bounded second derivative that satisfies the following two conditions: (i) For all x ∈ (0, 1): A(x) < x, A(0) = 0 and A(1) = 1. (ii) A (0) = σ2b < 1 and A (1) = σ2e > 1. Theorem 9.20 Assume that A : [0, 1] → [0, 1] satisfies (i) and (ii). Let m(t) ˜ = √ # e ) depending only on σe , and a 2t − 2 √1 2 log t. Then there is a constant C(σ random variable Yσb depending only on σb , such that (i)





) √ * − 2x # ˜ ≤ x = E e−C(σe )Yσe e lim P max xi (t) − m(t) . t↑∞

1≤i≤n(t)

(9.120)

9.3 Universality Below the Straight Line (ii) The point process



δ xk (t)−m(t) → Eσb ,σe = ˜



δ pi +σe Δ(i) , j

i, j

k≤n(t)

177

(9.121)

as t ↑ ∞, in law, where the pi are the√atoms of a Poisson point process on R # e )Yσb e− 2x dx, and the Δ(i) are the limits of the with intensity measure C(σ processes as in Theorem 7.13, but conditioned on the event {maxk x˜k (t) ≥ √ 2σe t}. √ # = 1/ 4π, and Δ(i) = δ0 , i.e. the limiting process (iii) If A (1) = ∞, then C(∞) is a Cox process. The random variable Yσb is the limit of the uniformly integrable martingale Yσb (s) =

n(s) 

e−s(1+σb )+ 2

√ 2σb x¯i (s)

,

(9.122)

i=1

where x¯i (s) is standard branching Brownian motion.

Outline of the Proof The proof of Theorem 9.20 is based on the corresponding result obtained in [27] for the case of two speeds, and on a Gaussian comparison method. We start by showing the localisation of paths, namely that the paths of all particles that reach a height of order m(t) ˜ at time t have to lie within a certain tube. Then we show tightness of the extremal process. The remainder of the work consists in proving the convergence of the finitedimensional distributions. To this end we use Laplace transforms. We introduce auxiliary two-speed BBMs whose covariance functions approximate Σ2 (s) well around 0 and t. Moreover we choose them in such a way that their covariance functions lie above and, below Σ2 (s) in a neighbourhood of 0 and t, respectively. We then use Gaussian comparison methods to compare the Laplace transforms. The Gaussian comparisons comes in three main steps. In a first step we introduce the usual interpolating process and introduce a localisation condition on its paths. In a second step we justify a certain integration by parts formula, that is adapted to our setting. Finally the resulting quantities are decomposed into a part that has a definite sign and a part that converges to zero.

Localisation of Paths An important first step is again the localisation of the ancestral paths of particles that reach extreme levels. This is essentially inherited from properties of

178

Variable Speed BBM

the standard Brownian bridge. For a given covariance function Σ2 , and a subinterval I ⊂ [0, t], define the following events on the space of paths, X : R+ → R: (( (( : ( ; (( (( (( Σ2 (s) γ 2 2 γ Tt,I,Σ2 = X (∀s : s ∈ I : (X(s) − X(t)( < (Σ (s) ∧ (t − Σ (s))) . ( ( t (9.123) Proposition 9.21 Let x denote the variable speed BBM with covariance function Σ2 . For any 12 < γ < 1 and for all d ∈ R, there exists r sufficiently large such that, for all t > 3r,    ˜ + d} ∧ xk  Tt,Iγ ,Σ2 < , (9.124) P ∃k ≤ n(t) : {xk (t) > m(t) r

where Ir ≡ {s : Σ2 (s) ∈ [r, t − r]}. To prove Proposition 9.21 we need Lemma 6.27 on Brownian bridges. Proof of Proposition 9.21 Using a first moment method, the probability in (9.124) is bounded from above by   et P BΣ2 (t) > m(t) ˜ + d, BΣ2 (·)  Tt,Iγ ,Σ2 , (9.125) r

where BΣ2 (·) is a time change of an ordinary Brownian motion. Using that Σ2 (s) is a non-decreasing function on [0, t] with Σ2 (t) = t, we bound (9.125) from above by (( ( & < =' s ( ˜ + d} ∧ ∃s ∈ [r, t − r] : ((( Bs − Bt ((( > (s ∧ (t − s))γ . (9.126) et P {Bt > m(t) t Now, ξ(s) ≡ Bs − st Bt is a Brownian bridge from 0 to 0 in time t, and it is well known that ξ(s) is independent of Bt . Therefore, it holds that (9.126) is equal to ˜ + d)P (∃s ∈ [r, t − r] : |ξ(s)| > (s ∧ (t − s))γ ) . et P(Bt > m(t) Using the standard Gaussian tail bound, ∞ 2 2 e−x /2 dx ≤ u−1 e−u /2 ,

for u > 0,

(9.127)

(9.128)

u

we have

√

t 2 ˜ /2t ˜ + d) ≤ e √ e P(Bt > m(t) e−(m(t)+d) 2π(m(t) ˜ + d) √ t = √ e− 2d ≤ M, 2π(m(t) ˜ + d) t

t

(9.129)

179

9.3 Universality Below the Straight Line

for some constant M > 0. By Lemma 6.27 we can find r large enough such that P (∃s ∈ [r, t − r] : |ξ(s)| > (s ∧ (t − s))γ ) < /M.

(9.130)

Using the bounds (9.129) and (9.129) we bound (9.127) from above by .



Proof of Theorem 9.20 We show the convergence of the extremal process  Et = δ xk (t)−m(t) (9.131) ˜ k≤n(t)

by showing the convergence of the finite-dimensional distributions and tightness. Tightness of (Et )t≥0 follows trivially from a first-order Chebyshev estimate, which shows that, for any d ∈ R and  > 0, there exists N = N(, d) such that, for all t > 0, P(Et [d, ∞) ≥ N) < .

(9.132)

To show the convergence of the finite-dimensional distributions, define, for u ∈ R, n(t)  1 xi (t)−m(t)>u , (9.133) Nu (t) = ˜ i=1

that counts the number of points that lie above u. Moreover, we define the corresponding quantity for the process Eσb ,σe (defined in (9.121)),  Nu = 1 pi +σe Λ(i) >u . (9.134) j

i, j

Observe that, in particular,   ˜ ≤ u = P (Nu (t) = 0) . P max xi (t) − m(t) 1≤k≤n(t)

(9.135)

The key step in the proof of Theorem 9.20 is the following proposition, which asserts the convergence of the finite-dimensional distributions of the process Et . Proposition 9.22 For all k ∈ N and u1 , . . . , uk ∈ R, d

{Nu1 (t), . . . , Nuk (t)} → {Nu1 , . . . , Nuk }

(9.136)

as t ↑ ∞. The proof of this proposition will be postponed to the following sections. Assuming the proposition, we can now conclude the proof of the theorem.

180

Variable Speed BBM

Equation (9.136) implies the convergence of the finite-dimensional distributions of Et . Since tightness follows from the control of the first moment, we obtain assertion (ii) of Theorem 9.20. Assertion (i) follows immediately from Eq. (9.135). To prove√Assertion (iii), we need to show that, as σ2e ↑ ∞, it holds that # e ) ↑ 1/ 4π and the processes Λ(i) converge to the trivial process δ0 . Then, C(σ  Eσb ,∞ = δ pi , (9.137) i

where (pi√, i ∈ N) are the points of a PPP with random intensity measure √1 Yσb e− 2x dx. 4π Lemma 9.23 The point process Eσb ,σe converges in law, as σe ↑ ∞, to the point process Eσb ,∞ . Proof The proof of Lemma 9.23 is based on a result concerning the cluster processes Λ(i) . We write Λσe for a single copy of these processes and add the subscript to make the dependence on the parameter σe explicit. We recall from [27] that the process Λσe is constructed as follows. Define the processes Eσe as the limits of the point processes t

Eσe ≡

n(t) 

δ xk (t)− √2σe t ,

(9.138)

k=1

√ where xt is standard BBM conditioned on the event {maxk≤n(t) xk (t) > 2σe t}. We show here that, as σe tends to infinity, the processes Eσe converge to a point process consisting of a single atom at 0. More precisely, we show that   (( √ t ( (9.139) lim lim P Eσe ([−R, ∞)) > 1 ( max xk (t) > 2σe t = 0. σe ↑∞ t↑∞

k≤n(t)

Now,   (( √ t P Eσe ([−R, ∞)) > 1 (( max xk (t) > 2σe t (9.140) k≤n(t)   (( √ t t ≤ P suppEσe ∩ [0, ∞)  ∅ ∧ Eσe ([−R, ∞)) > 1 (( max xk (t) > 2σe t k≤n(t)  ∞  (( √ t t ≤ P suppEσe ∩ dy  ∅ ∧ Eσe ([−R, ∞)) > 1 (( max xk (t) > 2σe t k≤n(t) 0  ∞  (( √ t P suppEσe ∩ dy  ∅ (( max xk (t) > 2σe t = k≤n(t) 0 ' & t (( t × P Eσe ([−R, ∞)) > 1 (( suppEσe ∩ dy  ∅ .

181

9.3 Universality Below the Straight Line

Figure 9.1 The cluster process seen from infinity for σe small (left) and σe very large (right).

& (( ' t But P · (( suppEσe ∩ dy  ∅ ≡ Pt,y+ √2σe (·) is the Palm measure on BBM, i.e. the conditional law of BBM given that there is a particle at time t in dy (see Kallenberg [75, Theorem 12.8]). Chauvin, Rouault and Wakolbinger [41, Theorem 2] describe the tree under the Palm measure Pt,z as follows. Pick one particle at time t at the location z. Then pick a spine, Y, which is a Brownian bridge from 0 to z in time t. Next pick a Poisson point process π on [0, t] with intensity 2. For each point p ∈ π, start a random number ν p of independent branching Brownian motions (BY(p),i , i ≤ ν p ) starting at Y(p). The law of ν is given by the size-biased distribution, P(ν p = k − 1) ∼ kp2 k . See Figure 9.1. Now √ let z = 2σe t + y for y ≥ 0. Under the Palm measure, the point process Eσe (t) then takes the form D

Eσe (t) = δy +



nY(p),i (p)

p∈π,i γ > 1/2,   √ γ γ lim lim P ∀ s≥σ−1/2 : Y(t − s) − y + 2σ s ∈ [−(σ s) , (σ s) ] = 1, (9.142) e e e e

σe ↑∞ t↑∞

if we define the set   √ : Y(t − s) − y + 2σe s ∈ [−(σe s)γ , (σe s)γ ] , Gtσe ≡ Y : ∀t≥s≥σ−1/2 e

(9.143)

it will suffice to show that, for all R ∈ R+ ,   (t − p) ≥ y − R ∧ Y ∈ Gtσe = 0. (9.144) lim lim P ∃p ∈ π, i < ν p , j : BY(p),i j σe ↑∞ t↑∞

182

Variable Speed BBM

The probability in (9.144) is bounded by   (t − p) ≥ y − R) ∧ Y ∈ Gtσe P ∃p ∈ π, i ≤ ν p , j : BY(p),i j ⎤ ⎡ t ν p  ⎥⎥ ⎢⎢⎢ ⎢ 1 Y(p),i 1Y∈Gσ π(d p)⎥⎥⎥⎦ ≤ E ⎢⎣ Bj

0 i=1

(t−p)>y−R

e

⎡ t ⎡ ν p ⎤ (( ⎤ ⎢⎢⎢ ⎢⎢⎢ ⎥⎥⎥ ⎥⎥ ( π ( ≤ E ⎢⎢⎣ E ⎢⎢⎣ 1max j BY(p),i (t−p)≥y−R 1Y∈Gtσe ( F ⎥⎥⎦ π(d p)⎥⎥⎥⎦ ( j 0 i=1   t Y(t−s) t 2KP max B j ≥ y − R ∧ Y ∈ Gσe ds. ≤ j

0

(9.145)

Here we have used the independence of the offspring BBM and that the conditional probability given the σ–algebra Fπ generated by the Poisson process π appearing in the integral over π depends only on p. For the integral over s up to 1/σ1/2  , we just bound the integrand by 2K. For larger values, we use the localisation provided by the condition that Y ∈ Gσe , to get that the right-hand side of (9.145) is not larger than σ−1/2 t e √ 2K (9.146) ds + 2K e s P(B(s) > −R + 2σe s − (σe s)γ )ds. σ−1/2 e

0

By (9.128), (9.146) is bounded from above by ∞ √ 2 γ −1/2 2Kσe + 2K e(1−σe )s+ 2σe (R+(σe s) ) ds. σ−1/2 e

(9.147)

From this it follows that (9.147) (which no longer depends on t) converges to zero, as σe ↑ ∞, for any R ∈ R. Hence we see that ' & t (( t (9.148) P Eσe ([−R, ∞)) > 1 (( suppEσe ∩ dy  ∅ ↓ 0, uniformly in y ≥ 0, as t and then σe tend to infinity. Next,  ∞  (( √ t ( P suppEσe ∩ dy  ∅ ( max xk (t) > 2σe t (9.149) k≤n(t) 0   ∞ (( √ √ P max xk (t) ≥ 2σe t + y (( max xk (t) > 2σe t . ≤ 0

k≤n(t)

k≤n(t)

But by√Proposition 7.5 in [27] the probability in the integrand converges to exp(− 2σe y), as t ↑ ∞. It follows from the proof that this convergence is uniform in y, and hence, by dominated convergence, the right-hand side of (9.149) is finite. Therefore, (9.139) holds. As a consequence, Λσe converges to δ0 . It remains to show that the intensity of the Poisson process converges as

9.3 Universality Below the Straight Line

183

# e ) defined by (9.53) claimed. Theorems 1 and 2 of [39] relate the constant C(σ √ to the intensity of the shifted BBM conditioned to exceed the level 2σe t as follows: 

E k 1 x¯k (s)> √2σe s 1  (9.150) = lim  √ √ # e ) s↑∞ P maxk x¯k (s) > 2σe s 4πC(σ ⎡ ⎤ (( √ ⎢⎢⎢ ⎥⎥ ( = lim E ⎢⎢⎣ 1 x¯k (s)−maxi x¯i (s)> √2σe s−maxi x¯i (s) (( max x¯k (s) > 2σe s⎥⎥⎥⎦ ( k s↑∞ k

= Λσe ((−E, 0]), where, by Theorem 7.5 √ in [27], E is an exponentially distributed random variable with parameter 2σe , independent of Λσe . As we have just shown that Λσe → δ0 , it follows √ that the right-hand side tends to one, as σe ↑ ∞, and # e ) ↑ 1/ 4π. Hence the intensity measure of the PPP appearing in hence C(σ √ Eσb ,σe converges to the desired intensity measure √14π Yσb e− 2x dx.  This proves assertion (iii) of Theorem 9.20.



The Finite-Dimensional Marginals We prove Proposition 9.22 via convergence of Laplace transforms. Define the Laplace transform of {Nu1 (t), . . . , Nuk (t)}, ⎛ ⎛ k ⎞⎞ ⎜⎜⎜ ⎜⎜⎜  ⎟⎟⎟⎟⎟⎟ ⎜ ⎜ Lu1 ,...,uk (t, c) = E ⎜⎜⎝exp ⎜⎜⎝− cl Nul (t)⎟⎟⎟⎠⎟⎟⎟⎠ , c = (c1 , . . . , ck )t ∈ Rk+ , (9.151) l=1

and the Laplace transform Lu1 ,...,uk (c) of {Nu1 , . . . , Nuk }. Proposition 9.22 is then a consequence of the next proposition. Proposition 9.24 For any k ∈ N, u1 , . . . , uk ∈ R and c1 , . . . , ck ∈ R+ , lim Lu1 ,...,uk (t, c) = Lu1 ,...,uk (c).

t→∞

(9.152)

The proof of Proposition 9.24 requires two main steps. First, we prove the result for the case of two-speed BBM, as in the proof of Theorem 9.12. In fact, we need a slight extension of that result in which we allow a slight dependence of the speeds on t. This will be given in the following subsection. The second step is to show that the Laplace transforms in the general case can be well approximated by those of two-speed BBM. This uses the usual Gaussian comparison argument in a slightly subtle way.

184

Variable Speed BBM

Approximating Two-Speed BBM As we will see later, it is enough to compare the process with covariance function A with processes whose covariance function is piecewise linear with single change in slope. We will produce approximate upper and lower bounds by choosing these in such a way that the covariances near zero and near one are below, resp. above that of the original process. In fact, it is not hard to convince oneself that the following holds. Lemma 9.25 There exist families of functions At and At that are piecewise linear, continuous functions with a single change point for the derivative, such that A(0) = A(0) = 0 and A(1) = A(1) = 1. Moreover, 

lim At (0) = lim At (0), t↑∞

t↑∞



and lim At (1) = lim At (1). t↑∞

(9.153)

t↑∞

Moreover, (i) for all s with Σ2 (s) ∈ [0, t1/3 ] and Σ2 (s) ∈ [t − t1/3 , t], 2

Σ (s) ≥ Σ2 (s) ≥ Σ2 (s);

(9.154)

(ii) if A(x) = 0 on some finite interval [0, δ], then Eq. (9.154) only holds for all s with Σ2 (s) ∈ [t − t1/3 , t] while, for s ∈ [0, (δ ∧ b)t), it holds that 2

Σ (s) = Σ2 (s) = Σ2 (s) = 0.

(9.155) 2

Let {yi , i ≤ n(t)} be the particles of a BBM with speed function Σ and let {y , i ≤ n(t)} be particles of a BBM with speed function Σ2 . We want to show i that the limiting extremal processes of these processes coincide. Set N u (t) ≡

n(t) 

1yi (t)−m(t)>u , ˜

(9.156)

i=1

N u (t) ≡

n(t) 

1yi (t)−m(t)>u . ˜

(9.157)

i=1

Lemma 9.26

For all u1 , . . . , uk and all c1 , . . . , ck ∈ R+ , the limits ⎛ ⎛ k ⎞⎞ ⎜⎜⎜ ⎜⎜⎜  ⎟⎟⎟⎟⎟⎟ ck N ul (t)⎟⎟⎟⎠⎟⎟⎟⎠ lim E ⎜⎜⎝⎜exp ⎜⎜⎝⎜− t↑∞

(9.158)

l=1

and

⎛ ⎛ k ⎞⎞ ⎜⎜⎜ ⎜⎜⎜  ⎟⎟⎟⎟⎟⎟ ck N ul (t)⎟⎟⎟⎠⎟⎟⎟⎠ lim E ⎜⎜⎜⎝exp ⎜⎜⎜⎝− t↑∞ l=1

exist. The two limits coincide with Lu1 ,...,uk (c).

(9.159)

9.3 Universality Below the Straight Line

185

Proof We first consider the case when A (1) < ∞. To prove Lemma 9.26, we show that the extremal processes Et =

n(t) 

δyi −m(t) ˜

and Et =

i=1

n(t) 

δy −m(t) ˜ i

(9.160)

i=1

both converge to Eσb ,σe , defined in (9.121). Note that this implies first convergence of Laplace functionals. The assertion in the case when σe = ∞ follows directly from Lemma 9.23. 

Gaussian Comparison We now come to the heart of the proof, namely the application of Gaussian comparison to control the process with covariance function A in terms of those with the piecewise linear covariances. From now on we distinguish the expectation with respect to the underlying tree structure and the one with respect to the Brownian movement of the particles. (i) En : expectation with respect to Galton–Watson process. (ii) EB : expectation with respect to the Gaussian process conditioned on the σ–algebra Ftree t generated by the Galton–Watson process. The proof of Proposition 9.24 is based on the following Lemma that compares the Laplace transform Lu1 ,...,uk (t, c) with the corresponding Laplace transform for the comparison processes. Lemma 9.27

For any k ∈ N, u1 , . . . , uk ∈ R and c1 , . . . , ck ∈ R+ , we have ⎛ ⎛ k ⎞⎞ ⎜⎜⎜ ⎜⎜⎜  ⎟⎟⎟⎟⎟⎟ ⎜ ⎜ Lu1 ,...,uk (t, c) ≤ E ⎜⎝⎜exp ⎜⎝⎜− cl N ul (t)⎟⎟⎟⎠⎟⎟⎟⎠ + o(1), (9.161) l=1 ⎛ ⎛ k ⎞⎞ ⎜⎜⎜ ⎜⎜⎜  ⎟⎟⎟⎟⎟⎟ ⎜ ⎜ cl N ul (t)⎟⎟⎟⎠⎟⎟⎟⎠ + o(1). (9.162) Lu1 ,...,uk (t, c) ≥ E ⎜⎜⎝exp ⎜⎜⎝− l=1

Proof The proofs of (9.161) and (9.162) are very similar. Hence we focus on proving (9.161). As we go along, however, we indicate what has to be changed when proving the lower bound. For simplicity all overlined names depend on 2 2 Σ . Corresponding quantities where Σ is replaced by Σ2 are underlined. Set ⎞ ⎛ n(t) k ⎟⎟⎟ ⎜⎜⎜   ⎟⎟⎟ . ⎜ (9.163) cl 1 xi (t)−m(t)>u f (x(t)) ≡ f (x1 (t), . . . , xn(t) (t)) ≡ exp ⎜⎝⎜− ˜ l⎠ i=1 l=1

186

Variable Speed BBM

We want to control ⎛ ⎛ k ⎛ ⎛ k ⎞⎞ ⎞⎞ ⎜⎜⎜ ⎜⎜⎜  ⎜⎜⎜ ⎜⎜⎜  ⎟⎟⎟⎟⎟⎟ ⎟⎟⎟⎟⎟⎟ cl Nul (t)⎟⎟⎟⎠⎟⎟⎟⎠ − EB ⎜⎜⎜⎝exp ⎜⎜⎜⎝− cl N ul (t)⎟⎟⎟⎠⎟⎟⎟⎠ EB ⎜⎜⎝⎜exp ⎜⎜⎝⎜− l=1 l=1   

= EB f (x1 (t), . . . , xn(t) (t)) − EB f (y1 (t), . . . , yn(t) (t) .

(9.164)

A straightforward application of the interpolation formula from Lemma 3.3 will produce a term that is very similar to what appears in a second moment estimate. As always, this will not give a useful estimate unless we introduce an appropriate truncation that reflects the typical behaviour of ancestral paths of extrema particles. This has to be done at the right place. Define the interpolating processes √ √ xih = hxi + 1 − hyi . (9.165) Remark The interpolating process {xih , i ≤ n(t)} is a new Gaussian process with the same underlying branching structure and speed function 2

Σ2h (s) = hΣ2 (s) + (1 − h)Σ (s). The right-hand side of Eq. (9.164) is equal to  1  d h EB f (x (t))dh , 0 dh

(9.166)

(9.167)

where $ % n(t) 1 1 d f (xh (t)) 1  ∂ h = f (x1h (t), . . . , xn(t) (t)) √ xi (t) − √ yi (t) (9.168) dh 2 i=1 ∂x j h 1−h and

⎛ k ⎞ ⎜⎜⎜ ⎟⎟⎟ ∂ h h f (x1h (t), . . . , xn(t) (t)) = − ⎜⎜⎜⎝ cl δ(xih (t) − m(t) − ul )⎟⎟⎟⎠ f (x1h (t), . . . , xn(t) (t)), ∂x j l=1 (9.169) where δ denotes the Dirac delta-function. Recall that we learned in Chapter 4 that working with the delta-function in this context can be justified rigorously by using smooth approximations of indicator functions. We must introduce the condition on the path of xih into (9.168) at this stage. To do so, we insert into the right-hand side of (9.168) a one in the form 1 = 1 xih ∈T γ

¯ 2 t,I,Σ h

with

+ 1 xih T γ 2 , ¯ t,I,Σ

(9.170)

h

I¯ ≡ t(δ , respectively. The next lemma shows that S > does not contribute to the expectation in (9.168), as t → ∞. Lemma 9.28

With the notation above, we have   1 EB (|S > |)dh = 0. lim En t→∞

(9.172)

0

The proof of this lemma will be postponed. We continue with the proof of Lemma 9.27. We are left with controlling ⎛ n(t) %⎞ $ ⎜⎜⎜ 1  ∂ f (xh (t)) yi (t) ⎟⎟⎟⎟ xi (t) ⎜ γ ⎟⎟⎠ . (9.173) 1 xih ∈T ¯ 2 √ − √ EB (S > ) = EB ⎜⎜⎝ t,I,Σ 2 i=1 ∂x j h 1−h h By the definition of Tt,γI,Σ ¯ 2, h

1 xih ∈T γ¯ 2 = 1∀s∈I¯:|ξi (Σ2h (s))|≤(Σ2h (s)∧(t−Σ2h ))γ ,

(9.174)

t,I,Σ

h

where ξi (·) is a Brownian bridge from 0 to 0 in time t, that is independent of xih (t). We want to apply a Gaussian integration by parts formula to (9.173). However, we need to take care of the fact that each summand in (9.173) depends on the whole path of ξi through the term in (9.174). However, this is not really a problem, because the condition (9.174) only depends on the properties of the Brownian bridge, which is independent of the endpoint. By the Gaussian integration by parts formula from Lemma 3.4, we get      n(t) ∂2 f (xh (t)) ∂ f (xh (t)) EB xi (t) . 1 xih ∈T γ¯ 2 = EB (xi (t)xhj (t))EB 1 xih ∈T γ¯ 2 t,I,Σ t,I,Σ ∂x j ∂x j ∂xi h h j=1 (9.175) The same formula holds, of course, with xi replaced by yi . Hence EB (S > ) =

 EB (xi (t)x j (t)) − EB (yi (t)y j (t)) EB 1 xih ∈T γ

n(t)  i, j=1

 ∂2 f (xh (t)) , ¯ 2 t,I,Σ ∂xi ∂x j h

i j

(9.176) where k ∂2 f (xh (t))  h = cl cl δ(xih − m(t) ˜ − ul )δ(xih − m(t) ˜ − ul ) f (x1h (t), . . . , xn(t) (t)). ∂xi ∂x j  l,l =1 (9.177)

188

Variable Speed BBM

Introducing 1 = 1d(xih (t),xhj (t))∈I¯ + 1d(xih (t),xhj (t))I¯

(9.178)

into (9.176), we rewrite (9.176) as (T 1) + (T 2), where (T 1) =

n(t) 

EB (xi (t)x j (t)) − EB (yi (t)y j (t))

i j=1



 ∂2 f (xh (t)) , t,I,Σ2 ∂xi ∂x j h

1d(xih (t),xhj (t))∈I¯1 xih ∈T γ¯

× EB

(T 2) =

n(t) 

EB (xi (t)x j (t)) − EB (yi (t)y j (t))

i j=1

 × EB

(9.179)

1

d(xih (t),xhj (t))I¯

1

γ xih ∈T ¯ 2 t,I,Σ h

(9.180)

 ∂2 f (xh (t)) . ∂xi ∂x j

The term (T 1) is controlled by the following lemma. Lemma 9.29 With the notation above, there exists a constant C¯ < ∞ such that ((  1 (( ( (( 2 (( # (( −s+Σ2 (s)+O(sγ ) (( −s+Σ (s)+O(sγ ) (( ( (9.181) (T 1)dh ( ≤ C (e −e ((En ( ds. ( 0 I¯ Moreover, we have: Lemma 9.30 If Σ2 satisfies the two conditions (i) and (ii) stated at the start 2 of Section 9.3, and if Σ is as defined in Lemma 9.25, then (( ( 2 2 γ γ ( (9.182) lim (((e−s+Σ (s)+O(s ) − e−s+Σ (s)+O(s ) ((( ds = 0. t→∞ ¯ I

The proofs of these lemmas are technical but fairly straightforward and will not be given here. Details can be found in [28]. Up to this point the proof of (9.162) works exactly as the proof of (9.161) 2 when Σ is replaced by Σ2 . For (T 2) and (T 2) we have: Lemma 9.31 For almost all realisations of the Galton–Watson process, the following statements hold: lim(T 2) ≤ 0,

(9.183)

lim(T 2) ≥ 0.

(9.184)

t↑∞

and t↑∞

9.4 Bibliographic Notes

189

The proof of this lemma is technical and will be skipped. From Lemmaa 9.29, 9.30 and 9.31, together with (9.173), the bound (9.161) follows. As pointed out, using Lemma 9.31, the bound (9.162) also follows. Thus, Lemma 9.27 is proved.  We can now conclude the proof of Proposition 9.24. Proof of Proposition 9.24 Taking the limit as t ↑ ∞ in (9.161) and (9.162) and using Lemma 9.26 gives, in the case A (1) < ∞, lim sup Lu1 ,...,uk (t, c) ≤ Lu1 ,...,uk (c), t↑∞

lim inf Lu1 ,...,uk (t, c) ≥ Lu1 ,...,uk (c). t↑∞

(9.185)

Hence limt↑∞ Lu1 ,...,uk (t, c) exists and is equal to Lu1 ,...,uk (c). In the case A (1) = ∞, the same result follows if in addition we take ρ ↑ ∞ after taking t ↑ ∞. This concludes the proof of Proposition 9.24. 

9.4 Bibliographic Notes 1. Variable speed BBM was introduced by Derrida and Spohn [47]. In the recent surge of interest in BBM, it has been studied by Fang and Zeitouni [52, 51], who obtained the order of the maximum and the logarithmic corrections in the case of piecewise constant speed, both for BBM and for branching random walks. Maillard and Zeitouni [88] obtained the precise order of the maximum for the case of strictly concave speed functions. 2. Mallein [89] considered a case of a branching Brownian motion which has a √ random offspring law. Here the corrections to the linear behaviour are n plus a logarithmic term that is identified in terms of the probability of two Brownian bridges to remain ordered. 3. Nolen et al. [96] consider a F-KPP equation with time-inhomogeneous diffusion constant from the PDE point of view. 4. The results discussed in this chapter are based on work with Lisa Hartung [27, 28]. 5. One can construct speed functions that stay close to the straight line in such a way that the logarithmic term in the function m(t) interpolates between the 1 of the REM and the 3 in BBM. An example with a piecewise constant function is given by Kistler and Schmidt in [79].

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Index

ancestral path, 61 asymptotics of upper tails, 124 auxilliary process, 132 Berman condition, 35 boundary of Galton–Watson process, 56 branching Brownian motion definition, 60 variable speed, 153 branching process, 55 Brownian bridge, 61, 77, 98, 118 basic estimates, 98 cluster extremes, 134 clusters in extremal process, 138 complete Poisson convergence, 32 conditionally compact, 24 convergence almost sure, 23 vague, 23 weak, 5, 23 convergence theorem, 69 covariance, 34, 61 Cox process, 59 derivative martingale, 70, 72, 130 Dynkin’s theorem, 17 embedding in R of Galton–Watson tree, 145 entropic repulsion, 63, 80, 102 ergodic theorem, 75 exceedances point process of, 30 extremal distributions, 2 extremal process, 15 extremal types theorem, 11

F-KPP equation, 65 existence of solutions, 80 linear, 95 Feynman–Kac formula, 89, 106 representation, 76 finite mass condition, 112 first moment estimate, 48, 62 FKG inequality, 118 Fr´echet distribution, 11 Galton–Watson process, 55 tree, 55, 145 Galton–Watson tree embedding in R, 145 Gaussian comparison, 35, 186 integration by parts, 36 process, 35, 60 sequence stationary, 35 tail bounds, 3 generalised random energy model, 46, 49 Gibbs measure, 47 Girsanov formula, 103 GREM, 46, 49, 153 Gumbel distribution, 4, 49, 69 random shift, 58 Hamiltonian, 45 Hamming distance, 45 integration by parts Gaussian, 36 intensity measure, 18 interpolating process, 186 interpolation, 36

199

200 inverse left continuous, 6 Kahane’s theorem, 37 Kallenberg’s theorem, 26 Khintchine’s theorem, 7 Kolmogorov’s theorem, 84 labelling, 56 Ulam-Neveu-Harris, 57 Laplace functional, 18, 52, 66 convergence, 127 transform, 18 law of maximum, 74 REM, 49 lexicographic distance, 46 linear F-KPP equation, 76, 80, 95 localisation of paths, 62 martingale, 64, 156 max-stability, 5 max-stable distribution, 8 maxima of iid sequences, 1 maximum principle, 81 mild formulation, 66 solution, 66 more stretched, 86 multi-index, 56 labelling, 56 multi-indices, 145 normal comparison, 35 distribution, 40 process, 35 random variable, 3 sequence, 34

Index order statistics, 12 Paley–Szygmund inequality, 48 particle number, 64 partition function, 73 Picard iteration, 80 point measure, 15 simple, 16 point process, 15, 16, 66 of BBM, 128 Poisson, 19 Poisson point process, 19 random energy model, 47 relatively compact, 22 REM, 47 second moment method, 48, 50 truncated, 51 Skorokhod’s theorem, 23 Slepian’s lemma, 39 spin glass, 45 standard conditions, 67, 76 stationary Gaussian sequence, 35 stretched more, 86 supercritical, 57 tightness, 24, 128, 179 travelling wave, 67 Ulam-Neveu-Harris labelling, 57 ultrametric, 46, 56 uniqueness of solutions, 80 vague convergence, 21 variable speed, 153 weak convergence, 5 of point processes, 21 Weibull distribution, 11