Topics in Occupation Times and Gaussian Free Fields 3037191090, 978-3-03719-109-5, 9783037196090, 3037196092

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Zurich Lectures in Advanced Mathematics Edited by Erwin Bolthausen (Managing Editor), Freddy Delbaen, Thomas Kappeler (Managing Editor), Christoph Schwab, Michael Struwe, Gisbert Wüstholz Mathematics in Zurich has a long and distinguished tradition, in which the writing of lecture notes volumes and research monographs plays a prominent part. The Zurich Lectures in Advanced Mathematics series aims to make some of these publications better known to a wider audience. The series has three main constituents: lecture notes on advanced topics given by internationally renowned experts, graduate text books designed for the joint graduate program in Mathematics of the ETH and the University of Zurich, as well as contributions from researchers in residence at the mathematics research institute, FIM-ETH. Moderately priced, concise and lively in style, the volumes of this series will appeal to researchers and students alike, who seek an informed introduction to important areas of current research. Previously published in this series: Yakov B. Pesin, Lectures on partial hyperbolicity and stable ergodicity Sun-Yung Alice Chang, Non-linear Elliptic Equations in Conformal Geometry Sergei B. Kuksin, Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions Pavel Etingof, Calogero-Moser systems and representation theory Guus Balkema and Paul Embrechts, High Risk Scenarios and Extremes – A geometric approach Demetrios Christodoulou, Mathematical Problems of General Relativity I Camillo De Lellis, Rectifiable Sets, Densities and Tangent Measures Paul Seidel, Fukaya Categories and Picard–Lefschetz Theory Alexander H.W. Schmitt, Geometric Invariant Theory and Decorated Principal Bundles Michael Farber, Invitation to Topological Robotics Alexander Barvinok, Integer Points in Polyhedra Christian Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis Shmuel Onn, Nonlinear Discrete Optimization – An Algorithmic Theory Kenji Nakanishi and Wilhelm Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations Erwin Faou, Geometric Numerical Integration and Schrödinger Equations Published with the support of the Huber-Kudlich-Stiftung, Zürich

Alain-Sol Sznitman

Topics in Occupation Times and Gaussian Free Fields

Author: Alain-Sol Sznitman Departement Mathematik ETH Zürich Rämistrasse 101 8092 Zürich Switzerland

2010 Mathematics Subject Classification: 60K35, 60J27, 60G15, 82B41 Key words: occupation times, Gaussian free field, Markovian loop, random interlacements

ISBN 978-3-03719-109-5 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.

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© 2012 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TE X files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Preface

The following notes grew out of the graduate course “Special topics in probability”, which I gave at ETH Zurich during the Spring term 2011. One of the objectives was to explore the links between occupation times, Gaussian free fields, Poisson gases of Markovian loops, and random interlacements. The stimulating atmosphere during the live lectures was an encouragement to write a fleshed-out version of the handwritten notes, which were handed out during the course. I am immensely grateful to Pierre-François Rodriguez, Artëm Sapozhnikov, Balázs Ráth, Alexander Drewitz, and David Belius, for their numerous comments on the successive versions of these notes. Support by the European Research Council through grant ERC-2009-AdG 245728-RWPERCRI is thankfully acknowledged.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

Generalities . . . . . . . . . . . . . . . . . . . . . . 1.1 The set-up . . . . . . . . . . . . . . . . . . . . 1.2 The Markov chain X: (with jump rate 1) . . . . 1.3 Some potential theory . . . . . . . . . . . . . . 1.4 Feynman–Kac formula . . . . . . . . . . . . . . 1.5 Local times . . . . . . . . . . . . . . . . . . . . 1.6 The Markov chain Xx: (with variable jump rate) .

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5 5 7 10 23 25 26

2

Isomorphism theorems . . . . . . . . . 2.1 The Gaussian free field . . . . . . 2.2 The measures Px;y . . . . . . . . . 2.3 Isomorphism theorems . . . . . . . 2.4 Generalized Ray–Knight theorems

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31 31 35 41 45

3

The Markovian loop . . . . . . . . . . . . . . . . . . . . . 3.1 Rooted loops and the measure r on rooted loops . . 3.2 Pointed loops and the measure p on pointed loops . 3.3 Restriction property . . . . . . . . . . . . . . . . . . 3.4 Local times . . . . . . . . . . . . . . . . . . . . . . . 3.5 Unrooted loops and the measure  on unrooted loops

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61 61 70 74 75 82

4

Poisson gas of Markovian loops . . . . . . . . . . . . . . . . . . . . 4.1 Poisson point measures on unrooted loops . . . . . . . . . . . . 4.2 Occupation field . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Symanzik’s representation formula . . . . . . . . . . . . . . . . 4.4 Some identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Some links between Markovian loops and random interlacements

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. 85 . 85 . 87 . 91 . 95 . 100

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Introduction

This set of notes explores some of the links between occupation times and Gaussian processes. Notably they bring into play certain isomorphism theorems going back to Dynkin [4], [5] as well as certain Poisson point processes of Markovian loops, which originated in physics through the work of Symanzik [26]. More recently such Poisson gases of Markovian loops have reappeared in the context of the “Brownian loop soup” of Lawler and Werner [16] and are related to the so-called “random interlacements”, see Sznitman [27]. In particular they have been extensively investigated by Le Jan [17], [18]. A convenient set-up to develop this circle of ideas consists in the consideration of a finite connected graph E endowed with positive weights and a non-degenerate killing measure. One can then associate to these data a continuous-time Markov chain Xxt , t  0, on E, with variable jump rates, which dies after a finite time due to the killing measure, as well as the Green density g.x; y/, x; y 2 E,

(0.1)

(which is positive and symmetric), x xt D the local times L

Z

t

1fXxs D xg ds; t  0, x 2 E.

(0.2)

0

In fact g.; / is a positive definite function on E  E, and one can define a centered Gaussian process 'x , x 2 E, such that cov.'x ; 'y /.D EŒ'x 'y / D g.x; y/; for x; y 2 E.

(0.3)

This is the so-called Gaussian free field. x z1 , z 2 E, have intricate relationships. For It turns out that 12 'z2 , z 2 E, and L instance Dynkin’s isomorphism theorem states in our context that for any x; y 2 E,  z  x 1 C 1 'z2 L under Px;y ˝ P G (0.4) z2E 2

has the “same law” as 1 2

.'z2 /z2E under 'x 'y P G ,

(0.5)

where Px;y stands for the (non-normalized) h-transform of our basic Markov chain, with the choice h./ D g.; y/, starting from the point x, and P G for the law of the Gaussian field 'z , z 2 E.

2

Introduction

Eisenbaum’s isomorphism theorem, which appeared in [7], does not involve htransforms and states in our context that for any x 2 E, s 6D 0,  z  x 1 C 1 .'z C s/2 L under Px ˝ P G (0.6) z2E 2

has the “same law” as 1 2

.'z C s/2





z2E

under 1 C

'x  G P . s

(0.7)

The above isomorphism theorems are also closely linked to the topic of theorems of Ray–Knight type, see Eisenbaum [6], and Chapters 2 and 8 of Marcus–Rosen [19]. Originally, see [13], [21], such theorems came as a description of the Markovian character in the space variable of Brownian local times evaluated at certain random times. More recently, the Gaussian aspects and the relation with the isomorphism theorems have gained prominence, see [8], and [19]. Interestingly, Dynkin’s isomorphism theorem has its roots in mathematical physics. It grew out of the investigation by Dynkin in [4] of a probabilistic representation formula for the moments of certain random fields in terms of a Poissonian gas of loops interacting with Markovian paths, which appeared in Brydges–Fröhlich–Spencer [2], and was based on the work of Symanzik [26]. The Poisson point gas of loops in question is a Poisson point process on the state space of loops on E modulo time-shift. Its intensity measure is a multiple ˛ of the image  of a certain measure rooted , under the canonical map for the equivalence relation identifying rooted loops  that only differ by a time-shift. This measure rooted is the -finite measure on rooted loops defined by Z 1 P dt t Qx;x .d/ rooted .d/ D ; (0.8) t x2E 0 t is the image of 1fX t D xg Px under .Xs /0st , if X: stands for the where Qx;x Markov chain on E with jump rates equal to 1 attached to the weights and killing measure we have chosen on E. The random fields on E alluded to above, are motivated by models of Euclidean quantum field theory, see [11], and are for instance of the following kind: Z .Z Q  'x2  Q  'x2  1 1 hF .'/i D F .'/ e  2 E.';'/ h e  2 E.';'/ h d'x d'x 2 2 RE RE x2E x2E (0.9) with Z 1 h.u/ D e vu d.v/, u  0, with  a probability distribution on RC ; 0

3

Introduction

and E.'; '/ the energy of the function ' corresponding to the weights and killing measure on E (the matrix E.1x ; 1y /, x; y 2 E is the inverse of the matrix g.x; y/, x; y 2 E in (0.3)). w3

w2

y1 x2

w1

x3 y2

x1

y3

Figure 0.1. The paths w1 ; : : : ; wk in E interact with the gas of loops through the random potentials.

The typical representation formula for the moments of the random field in (0.9) looks like this: for k  1, z1 ; : : : ; z2k 2 E, h'z1 : : : 'z2k i D P pairings of z1 ; : : : ; z2k

  P xx xx Px1 ;y1 ˝    ˝ Pxk ;yk ˝ Q e  x2E vx .Lx CL1 .w1 /CCL1 .wk //  P  ; Q e  x2E vx Lx

(0.10) where the sum runs over the (non-ordered) pairings (i.e. partitions) of the symbols z1 ; z2 ; : : : ; z2k into fx1 ; y1 g; : : : ; fxk ; yk g. Under Q the vx ; x 2 E, are i.i.d. distributed (random potentials), independent of the Lx , x 2 E, which are distributed as the total occupation times (properly scaled to take account of the weights and killing measure) of the gas of loops with intensity 12 , and the Pxi ;yi , 1  i  k are defined just as below (0.4), (0.5). The Poisson point process of Markovian loops has many interesting properties. We will for instance see that when ˛ D 12 (i.e. the intensity measure equals 12 ), .Lx /x2E has the same distribution as 12 .'x2 /x2E , where .'x /x2E stands for the Gaussian free field in (0.3).

(0.11)

The Poisson gas of Markovian loops is also related to the model of random interlacements [27], which loosely speaking corresponds to “loops going through infinity”. It

4

Introduction

appears as well in the recent developments concerning conformally invariant scaling limits, see Lawler–Werner [16], Sheffield–Werner [24]. As for random interlacements, interestingly, in place of (0.11), they satisfy an isomorphism theorem in the spirit of the generalized second Ray–Knight theorem, see [28].

1 Generalities

In this chapter we describe the general framework we will use for the most part of these notes. We introduce finite weighted graphs with killing and the associated continuous-type Markov chains X: , with constant jump rate equal to 1, and Xx: , with variable jump rate. We also recall various notions related to Dirichlet forms and potential theory.

1.1 The set-up We introduce in this section the general set-up, which we will use in the sequel, and recall some classical facts. We also refer to [14] and [10], where the theory is developed in a more general framework. We assume that E is a finite non-empty set

(1.1)

endowed with non-negative weights cx;y D cy;x  0, for x; y 2 E, and

cx;x D 0, for x 2 E,

(1.2)

so that E, endowed with the edge set consisting of the pairs fx; yg such that cx;y > 0, is a connected graph.

(1.3)

We also suppose that there is a killing measure on E, x  0; x 2 E;

(1.4)

x 6D 0, for at least some x 2 E.

(1.5)

cemetery state  not in E

(1.6)

and that We also consider a (we can think of x as cx; ). With these data we can define a measure on E: P cx;y C x ; x 2 E (note that x > 0, due to (1.2)–(1.5)): x D y2E

(1.7)

6

1 Generalities

We can also introduce the energy of a function on E, or Dirichlet form P P 1 E.f; f / D cx;y .f .y/  f .x//2 C x f 2 .x/; 2 x;y2E

(1.8)

x2E

for f W E ! R. Note that .cx;y /x;y2E and .x /x2E determine the Dirichlet form. Conversely, the Dirichlet form determines .cx;y /x;y2E and .x /x2E . Indeed, one defines, by polarization, for f; g W E ! R, E.f; g/ D D

1 4

ŒE.f C g; f C g/  E.f  g; f  g/

P P 1 cx;y .f .y/  f .x//.g.y/  g.x// C x f .x/g.x/; 2 x;y2E x2E

(1.9)

and one notes that E.1x ; 1y / D cx;y ; for x 6D y in E; P cx;y C x D x ; for x 2 E; E.1x ; 1x / D

(1.10)

y2E

so that the Dirichlet form uniquely determines the weights .cx;y /x;y2E and the killing measure .x /x2E . Observe also that by (1.3), (1.5), (1.8), (1.9), the Dirichlet form defines a positive definite quadratic form on the space F of functions from E to R, see also (1.39) below. We denote by .; / the scalar product in L2 .d /: P .f; g/ D f .x/g.x/ x ; for f; g W E ! R: (1.11) x2E

The weights and the killing measure induce a sub-Markovian transition probability on E, cx;y px;y D ; for x; y 2 E; (1.12) x which is -reversible: x px;y D y py;x ; for all x; y 2 E:

(1.13)

One then extends px;y ; x; y 2 E to a transition probability on E [ fg by setting x ; for x 2 E, and p; D 1; (1.14) px; D x so the corresponding discrete-time Markov chain on E [ fg is absorbed in the cemetery state  once it reaches . We denote by Zn ; n  0, the canonical discrete Markov chain on the space of discrete trajectories in E [ fg, which after finitely many steps reaches  and from then on remains at ,

(1.15)

1.2 The Markov chain X: (with jump rate 1)

7

and by Px the law of the chain starting from x 2 E [ fg.

(1.16)

We will attach to the Dirichlet form (1.8) (or, equivalently, to the weights and the killing measure), two continuous-time Markov chains on E [ fg, which are time change of each other, with discrete skeleton corresponding to Zn , n  0. The first chain X: will have a unit jump rate, whereas the second chain Xx: (defined in Section 1.6) will have a variable jump rate governed by .

1.2 The Markov chain X: (with jump rate 1) We introduce in this section the continuous-time Markov chain on E [ fg (absorbed in the cemetery state ), with discrete skeleton described by Zn , n  0, and exponential holding times of parameter 1. We also bring into play some of the natural objects attached to this Markov chains. The canonical space DE for this Markov chain consists of right-continuous functions with values in E [ fg, with finitely many jumps, which after some time enter  and from then on remain equal to . We denote by X t ; t  0, the canonical process on DE ;

t ; t  0, the canonical shift on DE : t .w/./ D w. C t/, for w 2 DE ; (1.17) Px the law on DE of the Markov chain starting at x 2 E [ fg: Remark 1.1. Whenever convenient we will tacitly enlarge the canonical space DE and work with a probability space on which (under Px ) we can simultaneously consider the discrete Markov chain Zn , n  0, with starting point a.s. equal to x, and an independent sequence of positive variables Tn , n  1, the “jump times”, increasing to infinity, with increments TnC1  Tn , n  0, i.i.d. exponential with parameter 1 (with the convention T0 D 0). The continuous-time chain X t , t  0, will then be expressed as X t D Zn ; for Tn  t < TnC1 , n  0: Of course, once the discrete-time chain reaches the cemetery state , the subsequent “jump times” Tn are only fictitious “jumps” of the continuous time chain.  Examples. 1) Simple random walk on the discrete torus killed at a constant rate E D .Z=N Z/d , where N > 1, d  1,

8

1 Generalities

endowed with the graph structure, where x, y are neighbors if exactly one of their coordinates differs by ˙1, and the other coordinates are equal. We pick cx;y D 1fx; y are neighborsg ; x; y 2 E; x D  > 0: So X t , t  0, is the simple random walk on .Z=N Z/d with exponential holding times of parameter 1, killed at each step with probability 2dC , when N > 2, and  probability d C , when N D 2. 2) Simple random walk on Zd killed outside a finite connected subset of Zd , that is: E is a finite connected subset of Zd , d  1. cx;y D 1fjxyjD1g ; for x; y 2 E; P 1fjxyjD1g ; for x 2 E. x D y2Zd nE

x

x D 2

E

x0

x 0 D 0

Figure 1.1

Then X t , t  0, when starting in x 2 E, corresponds to the simple random walk in Zd with exponential holding times of parameter 1 killed at the first time it exits E.  Our next step is to introduce some natural objects attached to the Markov chain X: , such as the transition semi-group, and the Green function.

1.2 The Markov chain X: (with jump rate 1)

9

Transition semi-group and transition density Unless otherwise specified, we will tacitly view real-valued functions on E, as functions on E [ fg, which vanish at the point . The sub-Markovian transition semi-group of the chain X t , t  0, on E is defined for t  0, f W E ! R and x 2 E, by R t f .x/ D Ex Œf .X t / P t t n D e Ex Œf .Zn / nŠ n0 P t t n n D e P f .x/ D e t .P I / f .x/; nŠ n0 where I denotes the identity map on RE , and for f W E ! R, x 2 E, P (1.15) Pf .x/ D px;y f .y/ D Ex Œf .Z1 /:

(1.18)

(1.19)

y2E

As a result of (1.13) and (1.18), P and R t (for any t  0) are bounded self-adjoint operators on L2 .d /; R t Cs D R t Rs , for t; s  0 (semi-group property):

(1.20) (1.21)

We then introduce the transition density r t .x; y/ D .R t 1y /.x/

1 ; for t  0, x; y 2 E: y

(1.22)

It follows from the self-adjointness of R t , cf. (1.20), that r t .x; y/ D r t .y; x/; for t  0, x; y 2 E (symmetry)

(1.23)

and from the semi-group property, cf. (1.21), that for t; s  0, x; y 2 E, P r t .x; z/ rs .z; y/ z (Chapman–Kolmogorov equations). (1.24) r t Cs .x; y/ D z2E

Moreover due to (1.3), (1.12), (1.18), we see that r t .x; y/ > 0; for t > 0, x; y 2 E:

(1.25)

Green function We define the Green function (or Green density): Z 1 i 1 hZ 1 (1.18), (1.22) D r t .x; y/ dt Ex 1fX t D ygdt ; for x; y 2 E: g.x; y/ D Fubini y 0 0 (1.26)

10

1 Generalities

Lemma 1.2. g.x; y/ 2 .0; 1/ is a symmetric function on E  E:

(1.27)

Proof. By (1.23), (1.25) we see that g.; / is positive and symmetric. We now prove that it is finite. By (1.1), (1.3), (1.5) we see that for some N  0, and " > 0, inf Px ŒZn D , for some n  N   " > 0:

x2E

(1.28)

As a result of the simple Markov property at times, which are multiple of N , we find that .P kN 1E /.x/ D Px ŒZn 6D ; for 0  n  kN  simple Markov



.1  "/k ; for k  1:

(1.28)

It follows by a straightforward interpolation that with suitable c; c 0 > 0, 0

sup .P n 1E /.x/  c e c n ; for n  0:

(1.29)

x2E

As a result inserting this bound in the last line of (1.18) gives: sup .R t 1E /.x/  c e t x2E

P t n c 0 n 0 D c expft.1  e c /g; e nŠ n0

(1.30)

so that 1 g.x; y/  y

Z

1

.R t 1E /.x/ dt  0

whence (1.27).

c 1  c 00 < 1; y 1  e c 0

(1.31) 

1.3 Some potential theory In this section we introduce some natural objects from potential theory such as the equilibrium measure, the equilibrium potential, and the capacity of a subset of E. We also provide two variational characterizations for the capacity. We then describe the orthogonal complement under the Dirichlet form of the space of functions vanishing on a subset of K. This also naturally leads us to the notion of trace form (and network reduction).

11

1.3 Some potential theory

The Green function gives rise to the potential operators P g.x; y/ f .y/ y ; for f W E ! R (a function); Qf .x/ D

(1.32)

y2E

the potential of the function f , and P G.x/ D g.x; y/ y , for  W E ! R (a measure);

(1.33)

y2E

the potential of the measure . We also write the duality bracket (between functions and measures on E): P h; f i D x f .x/; for f W E ! R,  W E ! R: (1.34) x

In the next proposition we collect several useful properties of the Green function and the Dirichlet form. Proposition 1.3. def

E.; / D h; Gi D

P

x g.x; y/ y , for ;  W E ! R

(1.35)

x;y2E

defines a positive definite, symmetric bilinear form. Q D .I  P /1 (see (1.19), (1.32) for notation).

(1.36)

G D .L/1 ; where P cx;y f .y/  x f .x/; for f W E ! R: Lf .x/ D

(1.37)

y2E

E.G; f / D h; f i, for  W E ! R and f W E ! R:

(1.38)

2 9 > 0, such that E.f; f /  kf kL 2 .d/ , for all f W E ! R:

(1.39)

G D 1 (the killing measure  is also called an equilibrium measure of E): (1.40) Proof.  (1.35): One can give a direct proof based on (1.23)–(1.26), but we will instead derive (1.35) with the help of (1.37)–(1.39). The bilinear form in (1.35) is symmetric by (1.27). Moreover, for  W E ! R, (1.38)

0  E.G; G/ D h; Gi D E.; / (the energy of the measure ):

12

1 Generalities

By (1.39), 0 D E.G; G/ H) G D 0, and by (1.37) it follows that  D .L/ G D 0. This proves (1.35) (assuming (1.37)–(1.39)).  (1.36): By (1.29): Z 0

1

P

Z 1 P t n c 0 n tn e t dt kf k1 jP n f .x/jdt  c e nŠ 0 n0 nŠ Z 1 c 0 (1.30) D ckf k1 e t.1e / dt < 1:

e t

n0

0

By Lebesgue’s domination theorem, keeping in mind (1.18), (1.26), Z 1 (1.32) R t f .x/dt Qf .x/ D 0 Z 1 P t t n n (1.18) D e P f .x/dt nŠ 0 n0 Z 1 P P n tn D dt P n f .x/ D e t P f .x/ nŠ n0 0 n0 (1.29)

D .I  P /1 f .x/; (1 is not in the spectrum of P by (1.29)).

This proves (1.36).  (1.37): Note that in view of (1.19), L D .I  P / (composition of .I  P / and the multiplication by : , i.e. . f /.x/ D x f .x/ for f W E ! R, and x 2 E):

(1.41)

Hence L is invertible and (1.36)

(1.32)

.L/1 D .I  P /1 1 D Q 1 D G: (1.33)

This proves (1.37).  (1.38): By (1.10) we find that

P

E.f; g/ D

f .x/ g.y/ E.1x ; 1y /

x;y2E (1.10)

D

P

x f .x/ g.x/ 

x2E

P x;y2E

(1.2)

D hf; Lgi D hLf; gi:

cx;y f .x/ g.y/

(1.42)

1.3 Some potential theory

13

As a result (1.37)

E.G; f / D hLG; f i D h; f i; whence (1.38):  (1.39): Note that for x 2 E, f W E ! R (1.38)

f .x/ D h1x ; f i D E.G1x ; f /: Now E.; / is a non-negative symmetric bilinear form. We can thus apply Cauchy– Schwarz’s inequality to find that (1.38)

f .x/2  E.G1x ; G1x / E.f; f / D h1x ; G1x i E.f; f / D g.x; x/ E.f; f /: As a result we find that 2 kf kL 2 .d/ D

P

f .x/2 x 

x2E

and (1.39) follows with 1 D

P

g.x; x/ x E.f; f /;

(1.43)

x2E

P

g.x; x/ x :

x2E

 (1.40): By (1.39), E.; / is positive definite and by (1.9) (1.9)

E.1; f / D

P x

(1.38)

x f .x/ D h; f i D E.G; f /; for all f W E ! R:

It thus follows that 1 D G, whence (1.40).



Remark 1.4. Note that we have shown in (1.42) that for all f; g W E ! R, E.f; g/ D hLf; gi D hf; Lgi:

(1.44)

Since L D .I  P /, we also find, see (1.11) for notation, E.f; g/ D ..I  P /f; g/ D .f; .I  P /g/ :

(1.440 ) 

14

1 Generalities

As a next step we introduce some important random times for the continuous-time Markov chain X t , t  0. Given K  E, we define HK D infft  0I X t 2 Kg; the entrance time in K; zK D infft > 0I X t 2 K and there exists s 2 .0; t/ with Xs 6D X0 g; H the hitting time of K; TK D infft  0I X t … Kg; the exit time from K;

(1.45)

LK D supft > 0I X t 2 Kg; the time of last visit to K (with the convention sup D 0, inf D 1): zK , TK are stopping times for the canonical filtration .F t / t 0 , on DE (i.e. a HK , H def

Œ0; 1-valued map T on DE , see (1.17) above, such that fT  t g 2 F t D .Xs ; 0  s  t/, for each t  0). Of course LK is in general not a stopping time. Given U  E, the transition density killed outside U is r t;U .x; y/ D Px ŒX t D y; t < TU 

1  r t .x; y/; for t  0, x; y 2 E, y

and the Green function killed outside U is Z 1 r t;U .x; y/dt  g.x; y/; for x; y 2 E: gU .x; y/ D

(1.46)

(1.47)

0

Remark 1.5. 1) When U is a connected (non-empty) subgraph of the graph in (1.3), r t;U .x; y/, t  0, x; y 2 U , and gU .x; y/, x; y 2 U , simply correspond to the transition density and the Green function in (1.22), (1.26), when one chooses on U – the weights cx;y , x; y 2 U (i.e. restriction to U  U of the weights on E), P – the killing measure Q x D x C cx;y , x 2 U . y2E nU

2) When U is not connected the above remark applies to each connected component of U , and r t;U .x; y/ and gU .x; y/ vanish when x; y belong to different connected components of U .  Proposition 1.6 (U  E; A D EnU ). gU .x; y/ D gU .y; x/; for x; y 2 E:

(1.48)

g.x; y/ D gU .x; y/ C Ex ŒHA < 1; g.XHA ; y/; for x; y 2 E:

(1.49)

Ex ŒHA < 1; g.XHA ; y/ D Ey ŒHA < 1; g.XHA ; x/; for x; y 2 E (Hunt’s switching identity).

(1.50)

15

1.3 Some potential theory

Proof.  (1.48): This is a direct consequence of the above remark and (1.27).  (1.49): (1.26)

g.x; y/ D Ex D Ex

hZ

hZ 0

D

i 1 1fX t D y; t < TU gdt y i 1 hZ 1 1fX t D ygdt; TU < 1 C Ex y TU Z

1

0

strong Markov

D

D

i 1 y

gU .x; y/ C Ex TU < 1;

(1.46), (1.47)

(1.26)

0 1

1fX t D ygdt

h

Fubini

TU D HA

1

h

gU .x; y/ C Ex TU < 1; EXTU

 i 1 1fX t D ygdt B TU y hZ

1

1fX t D ygdt

0

ii 1 y

gU .x; y/ C Ex ŒHA < 1; g.XHA ; y/:

This proves (1.49).  (1.50): This follows from (1.48), (1.49) and the fact that g.; / is symmetric, cf. (1.27).



Example. Consider x0 2 E. By (1.49) we find that for x 2 E (with A D fx0 g, U D Enfx0 g/ g.x; x0 / D 0 C Px ŒHx0 < 1 g.x0 ; x0 /; writing Hx0 for Hfx0 g , so that Px ŒHx0 < 1 D

g.x; x0 / ; for x 2 E: g.x0 ; x0 /

(1.51)

A second application of (1.49) now yields (with U D Enfx0 g) gU .x; y/ D g.x; y/ 

g.x; x0 /g.x0 ; y/ ; for x; y 2 E: g.x0 ; x0 /

(1.52) 

16

1 Generalities

Given A  E, we introduce the equilibrium measure of A: zA D 1 1A .x/ x ; x 2 E: eA .x/ D Px ŒH Its total mass is called the capacity of A (or the conductance of A): P zA D 1 x : Px ŒH cap.A/ D

(1.53)

(1.54)

x2A

Remark 1.7. As we will see below in the case of A D E the terminology in (1.53) is consistent with the terminology in (1.40). There is an interpretation of the weights .cx;y / and the killing measures .x / on E as an electric network grounding E at the cemetery point , which is implicit in the use of the above terms, see for instance Doyle–Snell [3].  Before turning to the next proposition, we simply recall that given A  E, by our convention in (1.45) fHA < 1g D fLA > 0g D the set of trajectories that enter A: Also given a measure  on E, we write P P D x Px and E for the P -integral (or “expectation”):

(1.55)

x2E

Proposition 1.8 (A  E). Px ŒLA > 0; XLA D y D g.x; y/ eA .y/; for x; y 2 E;

(1.56)

.XLA is the position of X: at the last visit to A, when LA > 0): def

hA .x/ D Px ŒHA < 1 D Px ŒLA > 0 D G eA .x/; for x 2 E

(1.57)

(the equilibrium potential of A). When A 6D , eA is the unique measure  supported on A such that G D 1 on A:

(1.58)

Let A  B  E then under PeB the entrance “distribution” in A and the last exit “distribution” of A coincide with eA : PeB ŒHA < 1; XHA D y D PeB ŒLA > 0; XLA D y D eA .y/; for y 2 E: (1.59) In particular when B D E, under P , the entrance distribution in A and the exit distribution of A coincide with eA .

(1.60)

17

1.3 Some potential theory

Proof.  (1.56): Both members vanish when y … A. We thus assume y 2 A. Using the discrete-time Markov chain Zn , n  0 (see (1.15)), we can write: i h S Px ŒLA > 0; XLA D y D Px fZn D y; and for all k > n; Zk … Ag n0

pairwise disjoint

D

P

Px ŒZn D y; and for all k > n; Zk … A

n0 Markov property

D

P

Px ŒZn D y Py Œ for all k > 0; Zk … A

n0 Fubini

D Ex

(1.45)

D Ex

hP

hZ

i zA D 1 1fZn D yg Py ŒH

n0 1

i zA D 1 (1.26) 1fX t D ygdt Py ŒH D g.x; y/ eA .y/: (1.53)

0

This proves (1.56).  (1.57): Summing (1.56) over y 2 A, we obtain Px ŒHA < 1 D Px ŒLA > 0 D

P

(1.33)

g.x; y/ eA .y/ D GeA .x/; whence (1.57).

y2A

 (1.58): Note that eA is supported on A and GeA D 1 on A by (1.57). If  is another such measure and  D   eA , h; Gi D 0 because G D 0 on A, and  is supported on A. By (1.35) it follows that  D 0, whence (1.58).  (1.59), (1.60): By (1.50) (Hunt’s switching identity): for y 2 E, (1.58)

EeB ŒHA < 1; g.XHA ; y/ D Ey ŒHA < 1; .GeB /.XHA / D Py ŒHA < 1: AB

18

1 Generalities

Denoting by  the entrance distribution of X in A under PeB , x D PeB ŒHA < 1; XHA D x; x 2 E; we see by the above identity and (1.57) that G.y/ D GeA .y/, for all y 2 E, and by applying L to both sides,  D eA . As for the last exit distribution of X: from A under PeB , integrating over eB in (1.56), we find PeB ŒLA > 0; XLA D y D

P

(1.57)

eB .x/ g.x; y/ eA .y/ D eA .y/; for y 2 E: AB

x2E

This completes the proof of (1.59). In the special case B D E, we know by (1.40),  (1.58) that eB D  and (1.60) follows. We now provide two variational problems for the capacity, where the equilibrium measure and the equilibrium potential appear. These characterizations are, of course, strongly flavored by the previously mentioned analogy with electric networks (we refer to Remark 1.7). Proposition 1.9 (A  E). cap.A/ D .inffE.; /I  probability supported on Ag/1 ;

(1.61)

and when A 6D , the infimum is uniquely attained at eNA D eA =cap.A/, the normalized equilibrium measure of A. cap.A/ D inffE.f; f /I f  1 on Ag;

(1.62)

and the infimum is uniquely attained at hA , the equilibrium potential of A. Proof.  (1.61): When A D , both members of (1.61) vanish and there is nothing to prove. We thus assume A 6D and consider a probability measure  supported on A. By (1.35), we have 0  E.; / D E.  eNA C eNA ;   eNA C eNA / D E.eNA ; eNA / C 2E.  eNA ; eNA / C E.  eNA ;   eNA /: The last term is non-negative, by (1.35) it only vanishes when  D eNA , and   P P  11 g.x; y/ eNA .y/ D D 0: x  eNA .x/ E.  eNA ; eNA / D cap.A/ x2E y2E ƒ‚ … „ (1.58)

D

1 cap.A/

on A

19

1.3 Some potential theory

We thus find that E.; / becomes (uniquely) minimal at E.eNA ; eNA / D

1 cap.A/2

P

(1.58)

eA .x/ g.x; y/ eA .y/ D

x;y2E

1 1 eA .A/ D : 2 cap.A/ cap.A/

This proves (1.61).  (1.62): We consider f W E ! R such that f  1A , and hA D GeA , so that hA .x/ D Px ŒHA < 1 D 1; for x 2 A. We have E.f; f / D E.f  hA C hA ; f  hA C hA / D E.hA ; hA / C 2E.f  hA ; hA / C E.f  hA ; f  hA /: Again, the last term is non-negative and only vanishes when f D hA , see (1.39). Moreover, we have (1.38)

E.f  hA ; hA / D E.f  hA ; GeA / D heA ; f  hA i  0; since hA D 1 on A, f  1 on A, and eA is supported on A. So the right-hand side of (1.62) equals (1.38)

E.hA ; hA / D E.GeA ; hA / D heA ; hA i D eA .A/ D cap.A/: This proves (1.62).



Orthogonal decomposition, trace Dirichlet form We consider U  E and set K D EnU . Our aim is to describe the orthogonal complement relative to the Dirichlet form E.; / of the space of functions supported in U : FU D f' W E ! RI '.x/ D 0; for all x 2 Kg: (1.63) To this end we introduce the space of functions harmonic in U , HU D fh W E ! RI P h.x/ D h.x/; for all x 2 U g;

(1.64)

as well as the space of potentials of (signed) measures supported on K, GK D ff W E ! RI f D G; for some  supported on Kg:

(1.65)

Recall that E.; / is a positive definite quadratic form on the space F of functions from E to R (see above (1.11)).

20

1 Generalities

Proposition 1.10 (Orthogonal decomposition). HU D GK :

(1.66)

F D FU ˚ HU , where FU and HU are orthogonal, relative to E.; /:

(1.67)

Proof.  (1.66): (1.37)

We first show that HU  GK . Indeed when h 2 HU , h D G.L/ h D G where  D Lh is supported on K by (1.64) and (1.41). Hence HU  GK . To prove the reverse inclusion we consider  supported on K. Set h D G. By (1.37) we know that Lh D LG D , so that Lh vanishes on U . It follows from (1.41) that h 2 HU , and (1.66) is proved. Incidentally note that choosing A D K in (1.49), we can multiply both sides of (1.49) by y and sum over y. The first term in the right-hand side vanishes and we then see that h D G satisfies h.x/ D Ex ŒHK < 1; h.XHK /; for x 2 E:

(1.68)

 (1.67): We first note that when ' 2 FU and  is supported on K, (1.38)

E.G; '/ D h; 'i D 0: So the spaces FU and HU are orthogonal under E.; /. In addition, given f from E ! R, we can define h.x/ D Ex ŒHK < 1; f .XHK /; for x 2 E;

(1.69)

and note that h.x/ D f .x/; when x 2 K; and that by the same argument as above, h is harmonic in U : If we now define ' D f  h, we see that ' vanishes on K, and hence f D ' C h; with ' 2 FU and h 2 HU ; is the orthogonal decomposition of f . This proves (1.67).

(1.70) 

As we now explain the restriction of the Dirichlet form to the space HU D GK , see (1.64)–(1.66), gives rise to a new Dirichlet form on the space of functions from K to R, the so-called trace form.

21

1.3 Some potential theory

Given f W K ! R, we also write f for the function on E that agrees with f on K and vanishes on U , when no confusion arises. Note that fQ.x/ D Ex ŒHK < 1; f .XHK /; for x 2 E;

(1.71)

is the unique function on E, harmonic in U , that agrees with f on K, cf. (1.67). Indeed the decomposition (1.67) applied to the case of the function equal to f on K, and to 0 on U , shows the existence of a function in HU equal to f on K. By (1.68) and (1.66), it is necessarily equal to fQ. We then define for f W K ! R, the trace form E  .f; f / D E.fQ; fQ/ (1.69), (1.70)

D

(1.72) inffE.g; g/I g W E ! R coincides with f on Kg;

where we used in the second line the fact that when g coincides with f on K, then g D ' C fQ, with ' 2 FU , and hence E.g; g/  E.fQ; fQ/ due to (1.67). We naturally extend this definition for f; g W K ! R, by setting Q E  .f; g/ D E.fQ; g/:

(1.73)

It is plain that E  is a symmetric bilinear form on the space of functions from K to R. As we now explain E  does indeed correspond to a Dirichlet form on K induced by some (uniquely defined in view of (1.10)) non-negative weights and killing measure. Proposition 1.11 (K 6D ). The quantities defined by  zK < 1; X z D y; for x 6D y in K; cx;y D x Px ŒH HK

(1.74)

D 0; for x D y in K, zK D 1; for x 2 K; x D x Px ŒH

(1.75)

zK < 1; X z D x/; for x 2 K; x D x .1  Px ŒH HK

(1.76)

  D cy;x ). The satisfy (1.2)–(1.5), (1.7), with E replaced by K (in particular cx;y  corresponding Dirichlet form coincides with E , i.e.

E  .f; f / D

2 P  2 P   1 c x f .x/; f .y/f .x/ C 2 x;y2K x;y x2K

for f W K ! R: (1.77)

The corresponding Green function g  .x; y/, x, y in K, satisfies g  .x; y/ D g.x; y/; for x; y 2 K:

(1.78)

22

1 Generalities

Proof. We first prove that the quantities in (1.74)–(1.76) satisfy (1.2)–(1.5), (1.7), with E replaced by K:

(1.79)

To this end we note that for x 6D y in K, (1.67) (1.44) E.z 1x ; z 1y / D E.1x ; z 1y / D Lz 1y .x/

(1.73)

E  .1x ; 1y /

D

zy .x/ D 0 1

P

D

x

D

z2E  : cx;y

(1.71) zK < 1; X z D y 1y .z/ D x Px ŒH px;z z HK Markov

(1.80) By a similar calculation we also find that for x 2 K,   P E  .1x ; 1x / D  L z 1x .z/ 1x .x/ D x 1  px;z z z2E





zK < 1; X z D x D x 1  Px ŒH HK

(1.81)

D x : We further see that P y2K

 cx;y C x

(1.74), (1.75)

D

(1.76)

zK < 1; X z D x/ x .1  Px ŒH HK

D x :

(1.82)

 . These are non-negative weights on K. From (1.80) we deduce the symmetry of cx;y Moreover when x0 ; y0 are in K, we can find a nearest neighbor path in E from x0 to y0 and looking at the successive visits of K by this path, taking (1.74) into account,  we see that K endowed with the edges fx; yg for which cx;y > 0, is a connected graph.

Further we know that for all x in E, Px -a.s., the continuous-time chain on E zK D 1 > 0, for at reaches the cemetery state  after a finite time. As a result Py ŒH least one y in K, since otherwise the chain starting from any x in K would a.s. never reach . By (1.75) we thus see that   does not vanish everywhere on K. In addition (1.7) holds by (1.82). We have thus proved (1.79).  (1.77): Expanding the square in the first sum in the right-hand side of (1.77), we see using  the symmetry of cx;y , (1.82), and the second line of (1.74), that the right-hand side

23

1.4 Feynman–Kac formula

of (1.77) equals P  2 x f .x/  x2K

P

(1.80), (1.81)

D

P x6Dy in K

 cx;y f .x/ f .y/

E  .1x ; 1x / f 2 .x/ C

x2K

P x6Dy in K

E  .1x ; 1y / f .x/ f .y/ D E  .f; f /;

and this proves (1.77).  (1.78): Consider x 2 K and x the restriction to K of g.x; /. By (1.66) we see that g.x; / D G1x ./ D Q x ./, and therefore for any y 2 K we have E .

x ; 1y /

(1.66), (1.67) (1.38) D E.G1x ; z 1y / D E.G1x ; 1y / D 1fxDyg

(1.73)

(1.38)

D E .

It follows that

x

D

 x

 x ; 1y /;

if

 x ./

D g  .x; /: 

for any x in K, and this proves (1.78).

Remark 1.12. 1) The trace form, with its expressions (1.72), (1.77), is intimately related to the notion of network reduction, or electrical network equivalence, see [1], p. 56. 2) When K  K 0  E are non-empty subsets of E, the trace form on K, of the trace form on K 0 of E, coincides with the trace form on K of E. Indeed this follows for instance by (1.78) and the fact that the Green function determines the Dirichlet form, see (1.37), (1.10). This feature is referred to as the “tower property” of traces. 

1.4 Feynman–Kac formula Given a function V W E ! R, we can also view V as a multiplication operator: .Vf /.x/ D V .x/ f .x/; for f W E ! R:

(1.83)

In this short section we recall a celebrated probabilistic representation formula for the operator e t .P I CV / , when t  0. We recall the convention stated above (1.18). Theorem 1.13 (Feynman–Kac formula). For V; f W E ! R, t  0, one has nZ t oi  h  V .Xs /ds D e t.P I CV / f .x/; for x 2 E; (1.84) Ex f .X t / exp 0

(the case V D 0 corresponds to (1.18)).

24

1 Generalities

Proof. We denote by S t f .x/ the left-hand side of (1.84). By the Markov property we see that for t; s  0, n Z tCs oi h S t Cs f .x/ D Ex f .X tCs / exp V .Xu /du 0 o nZ s o h nZ t i V .Xu /du exp V .Xu /du B t f .Xs / B t D Ex exp 0 0 o o ii h nZ t h nZ s V .Xu /du EX t exp V .Xu /du f .Xs / D Ex exp 0 0 o i h nZ t V .Xu /du Ss f .X t / D S t .Ss f /.x/ D .S t Ss /f .x/: D Ex exp 0

In other words S t , t  0, has the semi-group property S tCs D S t Ss ; for t; s  0:

(1.85)

Moreover, observe that 1 t

.S t f  f /.x/ D

1 t

D

1 t

nZ t o i h Ex f .X t / exp V .Xs /ds  f .X0 / 0

Ex Œf .X t /  f .X0 / Z t i h Rs 1 C Ex f .X t / V .Xs / e 0 V .Xu /du ds ; t

and as t ! 0, 1 t

0

Ex Œf .X t /  f .X0 / ! .P  I / f .x/; by (1.18);

whereas by dominated convergence Z t i h Rs 1 Ex f .X t / V .Xs / e 0 V .Xu /du ds ! Ex Œf .X0 / V .X0 / D Vf .x/: t

So we see that

0

1 t

.S t f  f /.x/ ! .P  I C V / f .x/: t!0

(1.86)

Then considering S tCh f .x/  S t f .x/ D .Sh  I / S t f .x/, with h > 0 small, as well as (when t > 0 and 0 < h < t ) S th f .x/  S t f .x/ D .Sh  I / S t h f .x/; one sees that (using in the second case that suput jSu f .x/j  e t kV k1 kf k1 / the function t  0 ! S t f .x/ is continuous.

25

1.5 Local times

Now dividing by h and letting h ! 0, we find that t  0 ! S t f .x/ is continuously differentiable with derivative .P  I C V / S t f .x/:

(1.87)

It now follows that the function s 2 Œ0; t ! F .s/ D e .ts/.P I CV / Ss f .x/ is continuously differentiable on Œ0; t, with derivative F 0 .s/ D e .ts/.P I CV / .P  I C V / Ss f .x/ C e .ts/.P I CV / .P  I C V / Ss f .x/ D 0: We thus find that F .0/ D F .t / so that e t.P I CV / f .x/ D S t f .x/. This proves (1.84). 

1.5 Local times In this short section we define the local time of the Markov chain X t , t  0, and discuss some of its basic properties. The local time of X: at site x 2 E, and time t  0, is defined as Z Lxt D

t

1fXs D xg ds 0

1 : x

(1.88)

Note that the normalization is different from (0.2) (we have not yet introduced Xxt , t  0). We extend (1.88) to the case x D  (cemetery point) with the convention Z  D 1; L t D

t

1fXs D gds; for t  0:

(1.89)

0

By direct inspection of (1.88) we see that for x 2 E, t 2 Œ0; 1/ ! Lxt 2 Œ0; 1/ is a continuous non-decreasing function with a finite limit Lx1 (because X t D  for t large enough). We record in the next proposition a few simple properties of the local time.

26

1 Generalities

Proposition 1.14. Ex ŒLy1  D g.x; y/; for x; y 2 E.

(1.90)

Ex ŒLyTU  D gU .x; y/; for x; y 2 E, U  E.

(1.91)

P x2E [fg

Z V .x/ Lxt

D

t

V 0 

.Xs /ds; for t  0, V W E [ fg ! R.

Lxt B s C Lxs D LxtCs ; for x 2 E, s  0 (additive function property):

(1.92) (1.93)

Proof.  (1.90): Ex ŒLy1  D Ex

hZ

1

1fX t D yg

0

dt i (1.26) D g.x; y/: y

 (1.91): Analogous argument to (1.90), cf. (1.45), (1.47), and Remark 1.5.  (1.92): P x2E [fg

V .x/ Lxt D D

P

Z

t

1fXs D xg

V .x/

x2E [fg Z t P 0 x2E [fg

0

V 

ds x

.x/ 1fXs D xg ds D

Z

t

V 0 

.Xs /ds:

 (1.93):

Rt Rs R t Cs Note that 0 V .Xu /du D . 0 V .Xu /du/B s C 0 V .Xu /du, and apply this identity with V ./ D 1x 1x ./. 

1.6 The Markov chain Xx: (with variable jump rate) Using time change, we construct in this section the Markov chain Xx: with same discrete skeleton Zn , n  0, as X: , but with variable jump rate x , x 2 E [ fg. We describe the transition semi-group attached to Xx: , relate the local times for Xx: and X: , and briefly discuss the Feynman–Kac formula for Xx: . As a last topic, we explain

1.6 The Markov chain Xx: (with variable jump rate)

27

how the trace process of Xx: on a subset K of E is related to the trace Dirichlet form introduced in Section 1.4. We define Lt D

P x2E [fg

Z Lxt D

0

t

1 X ds; t  0; s

(1.94)

so that t 2 RC ! L t 2 RC is a continuous, strictly increasing, piecewise differentiable function, tending to 1. In particular it is an increasing bijection of RC , and using the formula for the derivative of the inverse one can write for the inverse function of L: , Z u Z u

u D infft  0I L t  ug D Xv dv D Xxv dv; (1.95) 0

0

where we have introduced the time changed process (with values in E [ fg) def Xxu D Xu ; for u  0

(1.96)

(the path of Xx: thus belongs to DE , cf. above (1.17)). We also introduce the local times of Xx: (note that the normalization is different from (1.88), but in agreement with (0.2)): Z u x def x Lu D 1fXxv D xg dv; for u  0, x 2 E [ fg: (1.97) 0

Proposition 1.15. Xxu , u  0, is a Markov chain with cemetery state  and subMarkovian transition semi-group on E: xt f .x/ def R D Ex Œf .Xxt / D e tL f .x/; for t  0, x 2 E, f W E ! R

(1.98)

(i.e. Xx: has the jump rate x in x and jumps according to px;y in (1.12), (1.14)). Moreover one has the identities X t D XxL t ; for t  0 (“time t for X: is time L t for Xx: ”);

(1.99)

and x xL ; for x 2 E [ fg; t  0; Lxt D L t

x x1 ; for x 2 E: Lx1 D L

(1.100) (1.101)

28

1 Generalities

Proof.  Markov property of Xx: (sketch): Note that u , u  0, are F t D  .Xs ; 0  s  t/-stopping times and using the fact that L t , t  0, satisfies the additive functional property L tCs D Ls C L t B s ; for t; s  0; we see that Lu B v Cv D u C v, and taking inverses

uCv D u B v C v ; and

(1.102)

(1.96) XxuCv D Xu B v Cv D Xu B u D Xxu B v : (1.102)

(Note incidentally that Nu D u , for u  0, satisfies the semi-flow property NuCv D

Nu B Nv , for u; v  0, and, in this notation, the above equality reads XxuCv D Xxu B Nv .) It now follows that for u; v  0, B 2 Fv , one has for f W E [ fg ! R, Ex Œf .XxuCv / 1B  D Ex Œf .Xu B v / 1B  strong Markov for X

D

:E

x ŒEXv Œf .Xu / 1B 

D Ex ŒEXxv Œf .Xxu / 1B :

Since for v 0  v, Xxv0 D Xv0 are (see for instance Proposition 2.18, p. 9 of [12]) Fv -measurable, this proves the Markov property.  (1.98): xt , t  0, is a sub-Markovian semiFrom the Markov property one deduces that R group. Now for f W E ! R, x 2 E, u > 0, one has 1 u

xu f  f /.x/ D .R

1 u

Ex Œf .Xxu /  f .Xx0 / D

1 u

Ex Œf .Xu /  f .X0 /:

By (1.95) we see that u  cu, for u  0. We also know that the probability that X: jumps at least twice in Œ0; t is o.t/ as t ! 0. So as u ! 0, 1 u

Ex Œf .Xu /  f .X0 / D

1 u

  Ex f .Xu /  f .X0 /; X: has exactly one jump in Œ0; u  C o.1/

D Ex Œf .Z1 /  f .X0 / (1.94)

D .Pf  f /.x/

1 u

Px

1 u

Px ŒLT1  u C o0 .1/

hT

1

x

(with T1 the first jump of X: )

i (1.37)  u C o0 .1/ ! x .Pf  f /.x/ D Lf .x/: u!0

1.6 The Markov chain Xx: (with variable jump rate)

29

So we have shown that 1 u

xu f  f /.x/ ! Lf .x/: .R u!0

(1.103)

Just as below (1.86) one now shows the corresponding statement to (1.87): xu f .x/: xu f .x/ is continuously differentiable with derivative LR u0!R (1.104) xu f .x/, One then concludes in the same fashion as below (1.87), that e uL f .x/ D R and this proves (1.98).  (1.99): By (1.96), XxL t D XL t D X t , for t  0, whence (1.99).  (1.100): x xu ˇˇ d xx dL dL t  LL t D ˇ dt du uDL t dt

D 1fXxL t D xg

(1.97) (1.94)

1 (1.96) 1 dLxt D 1fX t D xg D ; X t x dt

except when t is a jump time of X: , and integrating we find (1.100).  (1.101): x x1 D Lx1 , that is (1.101). Letting t ! 1 in (1.100) yields L



One then has the Feynman–Kac formula for Xx: . Theorem 1.16 (Feynman–Kac formula for Xx: ). For V; f W E ! R, u  0, one has nZ u oi x V .Xxv /dv D e u.LCV / f .x/; for x 2 E: (1.105) Ex Œf .Xu / exp 0

Proof. The proof is similar to that of (1.84). One simply uses (1.98) in place of (1.18).  Remark 1.17. As a closing remark for Chapter 1, we briefly sketch a link between the Markov chain obtained as the trace of Xx: on a non-empty subset K of E and the trace form E  , cf. (1.72) and Proposition 1.11. To this end we introduce Z u ˚  P x xK x L L D D 1 Xxv 2 K [ fg dv; for u  0; (1.106) u u x2K[fg

0

30

1 Generalities

which is a continuous non-decreasing function of u tending to infinity, and its rightcontinuous inverse, xK

N K v D inffu  0I L u > vg; for v  0:

(1.107)

The trace process of Xx: on K is defined as x ; for v  0 Xx K v D XN K v

(1.108)

(intuitively at time v, Xx K : is at the location where Xx: sits once Lx K: accumulates v C " units of time, with " ! 0). With similar arguments as in the case of Xx: (see the proof of Proposition 1.15, in particular, using the strong Markov property of Xx: and the x K N ), one can show that fact that, in the notation from below (1.102), Xx K uCv D X u B N K v under Px , x 2 K [ fg, Xx K , v  0, is a Markov chain on K with cemetery state . v One can further show that its corresponding sub-Markovian transition semi-group on K has the form t L xK xK R f .x/; for x 2 K, f W K ! R, t f .x/ D Ex Œf .X t / D e

where in the notation of (1.74), (1.76), P  L f .x/ D cx;y f .y/  x f .x/; for x 2 K, f W K ! R:

(1.109)

(1.110)

y2K

To see this last point one notes that if L stands for the generator of Xx K : , the inverse of L has the K  K matrix: i i hZ 1 hZ 1 x Ex 1fXx K D yg dv D E 1f X D yg du x u v 0 0 (1.111) (1.101), (1.90)

D

(1.78)

g.x; y/ D g  .x; y/; for x; y 2 K:

A similar identity holds for the continuous-time chain on K with variable jump rate  and the killing measure : . Its generator is L (by : attached to the weights cx;y Proposition 1.15), and we thus find that L D L . 

2 Isomorphism theorems

In this chapter we will discuss the isomorphism theorems of Dynkin and Eisenbaum mentioned in the introduction, cf. (0.4)–(0.7), as well as some of the so-called generalized Ray–Knight theorems, in the terminology of Marcus–Rosen [19]. We still need to introduce some objects such as the Gaussian free field and the measures on paths entering the Dynkin isomorphism theorem. We keep the same set-up and notation as in Chapter 1.

2.1 The Gaussian free field In this section we define the Gaussian free field. We also describe the conditional law of the field given its values on a given subset K of E, as well as the law of its restriction to K. Interestingly this brings into play the orthogonal decomposition under the Dirichlet form and the notion of trace form discussed in Section 1.4. As we now see we can use g.x; y/, x; y 2 E, as the covariance function of a centered Gaussian field indexed by E. An important step to this effect is (1.35). We endow the canonical space RE of functions on E with the canonical product  -algebra and with the canonical coordinates 'x W f 2 RE ! 'x .f / D f .x/; x 2 E:

(2.1)

Proposition 2.1. There exists a unique probability P G on RE , under which .'x /x2E is a centered Gaussian field with covariance E G Œ'x 'y  D g.x; y/; for x; y 2 E.

(2.2)

Proof. Uniqueness.

P Under such a P G , for any  W E ! R, h; 'i D x2E x 'x is a centered Gaussian variable with variance P P x y g.x; y/ D E.; /: x y E G Œ'x 'y  D x;y

As a result

x;y2E

1

E G Œe ih;'i  D e  2 E.;/ ; for any  W E ! R.

This specifies the characteristic function of P G , and hence P G is unique.

(2.3)

32

2 Isomorphism theorems

Existence. We give both an abstract and a concrete construction of the law P G . Abstract construction: We choose ` , 1  `  jEj an orthonormal basis for E.; /, cf. (1.35), of the space of measures, and consider the dual basis fi , 1  i  jEj, of functions so: h` ; fi i D ı`;i , for 1  i, `  jEj. If i , i  1 are i.i.d. N.0; 1/ variables on some auxiliary space .; A; P /, we define the random function P i .!/ fi ./: (2.4) .; !/ D 1ijE j

For any x 2 E, 1x D

PjE j `D1

E.1x ; ` / ` , so that P

.x; !/ D h1x ; .; !/i D

1`;ijE j

E.1x ; ` / i .!/h` ; fi i D

jE Pj

E.1x ; ` / ` :

`D1

It now follows that for x; y 2 E, P

E P Π.x; !/ .y; !/ D

1`;`0 jE j

P

D

1`jE j Parseval

D

So the law of

E.1x ; ` / E.1y ; `0 / E P Œ` `0 

E.1x ; ` / E.1y ; ` /

E.1x ; 1y / D g.x; y/:

.; !/ on RE satisfies (2.2).

Concrete construction: The matrix g.x; y/, x; y 2 E, has inverse hL1x ; 1y i, x; y 2 E, cf. (1.37), and hence under the probability PG D

˚ 1 exp  p .2/ det G jEj 2

1 2

E.'; '/

 Q

d'x

(2.5)

x2E

P (1.44) (using that x;y2E 'x 'y hL1x ; 1y i D hL'; 'i D E.'; '/), .'x /x2E is a centered Gaussian vector with covariance g.x; y/, x; y 2 E, i.e. (2.2) holds.  Remark 2.2. In the above abstract construction the dual basis fi , 1  i  jEj, of ` , 1  `  jEj, is simply given by fi D Gi ; 1  i  jEj:

(2.6)

Indeed h` ; fi i D h` ; Gi i D E.` ; i / D ı`;i . Note that fi , 1  i  jEj, is an orthonormal basis under E.; /: (1.38)

E.fi ; fj / D E.Gi ; Gj / D hi ; Gj i D E.i ; j / D ıi;j :



33

2.1 The Gaussian free field

Conditional expectations We consider K  E, U D EnK, and want to describe the conditional law under P G of .'x /x2U given .'x /x2K , as well as the law of .'x /x2K . The orthogonal decomposition in Proposition 1.10 together with the description of the trace form in Proposition 1.11 will be useful for this purpose. We write P G;U for the law on RE of the centered Gaussian field with covariance E G;U Œ'x 'y  D gU .x; y/; for x; y 2 E;

(2.7)

(so 'x D 0, P G;U -a.s., when x 2 K). Proposition 2.3 (K 6D ). For x 2 E, define on RE the .'y ; y 2 K/-measurable hx D Ex ŒHK < 1; 'XHK  P D Px ŒHK < 1; XHK D y 'y (so hx D 'x , for x 2 K):

(2.8)

y2K

Then we can write 'x D

x

C hx ; for x 2 E (so

x

D 0, for x 2 K):

(2.9)

Under P G , .

x /x2E

is independent from .'y ; y 2 K/;

(2.10)

.

x /x2E

is distributed as .'x /x2E under P G;U :

(2.11)

and

In addition, in the notation of (1.77), .'x /x2K has the law .2/

jKj 2

˚ 1 exp  p detKK G

1 2

E  .'; '/

 Q

d'x ; (2.12)

x2K

where detKK G denotes the determinant of the K  K-matrix obtained by restricting g.; / to K  K. Proof.  (2.10): For any x 2 E, x belongs to the linear space generated by the centered jointly Gaussian collection 'z , z 2 E. In addition when x 2 E and y 2 K, we find that by

34

2 Isomorphism theorems

(2.8), (2.9) and (2.2), EGŒ

x

'y  D E G Œ'x 'y   E G Œhx 'y  P Px ŒHK < 1; XHK D z g.z; y/ D g.x; y/  z2K (1.49)

D g.x; y/  Ex ŒHK < 1; g.XHK ; y/ D 0: The claim (2.10) now follows.  (2.11): Since x D 0, for x 2 K, and P G;U -a.s., 'x D 0, for x 2 K, we only need to focus on the law of . x /x2U under P G . When F is a bounded measurable function on RU , jEj p we find that by (2.5), setting c as the inverse of .2/ 2 det G, we have Z   Q     ˚ 1 E G F . x /x2U D c F .'x  hx /x2U exp  E.'; '/ d'x : 2

RE

x2E

Using Proposition 1.10 and (1.71), (1.72), we see that for ' in RE , E.'; '/ D E.'  h; '  h/ C E  .'jK ; 'jK /; where 'jK denotes the restriction to K of ' 2 RE . We then make a change of variables in the above integral. We set 'x0 D 'x  hx , for x 2 U , and 'x0 D 'Q x , for x 2 K, 0 and Q note that the Jacobian of this transformation equals 1, so that x2E d'x D x2E d'x . We thus see that for all F as above that   E G ŒF . x /x2U  Z (2.13)  Q  ˚ 1 1 F .'x /x2U exp  E.'U ; 'U /  E  .'jK ; 'jK / d'x ; Dc 2

RE

2

x2E

where we have set 'U .x/ D 1U .x/'x . Integrating over the variables 'x , x 2 K, we find that for a suitable constant c 0 , Z  ˚ 1  Q 0 Dc F .'x /x2U exp  E.'U ; 'U / d'x RU

2

x2U

and using Remark 1.5 and (2.5)    D E G;U F .'x /x2U : This proves (2.11).  (2.12):

    A simple modification of the last calculation replacing F . x /x2U by H .'x /x2K , with H a bounded measurable functions on RK , yields (2.12). 

35

2.2 The measures Px;y

Remark 2.4. As a result of Proposition 2.3, when x is a given point of U , under P G , conditionally on the variables 'y , y 2 K, 'x is distributed as a Gaussian variable with mean Ex ŒHK < 1; 'XHK , and variance gU .x; x/:

(2.14)

Note that by Proposition 1.9 (and Remark 1.5), for x 2 U , gU .x; x/ is the inverse of the minimum energy of a function taking the value 1 on x and 0 on K

(2.15)

(this provides an interpretation of the conditional variance as an effective resistance between x and K [ fg). 

2.2 The measures Px;y In this section we introduce a further ingredient of the Dynkin isomorphism theorem, namely the kind of measures on paths that appear in (0.4), which live on paths in E with finite duration that go from x to y. We provide several descriptions of these measures, and derive an identity for the Laplace transform of the local time of the path, which prepares the ground for the proof of the Dynkin isomorphism theorem in the next section. We introduce the space of E-valued trajectories with duration t  0:  t D the space of right-continuous functions Œ0; t ! E; with finitely many jumps, left-continuous at t .

(2.16)

We still denote by Xs , 0  s  t , the canonical coordinates, and by convention we set Xs D  (cemetery point) if s > t . We then define the space of E-valued trajectories with finite duration as [ D t :

(2.17)

t>0

For  2 , we denote the duration of  by ./ D the unique t > 0 such that  2  t .

(2.18)

The  -algebra we choose on  is simply obtained by “transport”. We use the bijection ˆ W 1  .0; 1/ ! :   ˆ  2 ; .w; t/ 2 1  .0; 1/ ! ./ D w t

36

2 Isomorphism theorems

where we endow 1 .0; 1/ with the canonical product -algebra (and 1 is endowed with the -algebra generated by the maps Xs , 0  s  1, from 1 into E). We thus take the image by ˆ of the -algebra on 1  .0; 1/, and obtain the -algebra on . We define for x; y 2 E, t > 0, the measure Px , y from DE \ fX t D yg into  t . /.

t the image on  t of 1fX t D yg Px;y

under the map .Xs /0st

(2.19)

t is Note that the total mass of Px;y t Px;y Πt  D

1 (1.22) Px ŒX t D y D r t .x; y/: y

We then define the finite measure Px;y on  via Z 1 t Px;y ŒB D Px;y ŒB dt;

(2.20)

(2.21)

0

t defines a finite measure for any measurable subset B of  (noting that t > 0 ! Px;y kernel from .0; 1/ to ). The total mass of Px;y is Z 1 Z 1 (2.20) (1.26) t Px;y ΠD Px;y Πdt D r t .x; y/ dt D g.x; y/: (2.22) 0

0

The next proposition describes some relations between the measures Px;y and Px . In particular it provides an interpretation of Px;y as a non-normalized h-transform of Px , with h./ D g.; y/, see for instance Section 3.9 of [19]. The Remark 2.6 below gives yet an other description of Px;y . Proposition 2.5 (x; y 2 E). For 0 < t1 <    < tn , x1 ; : : : ; xn 2 E, one has Px;y ŒX t1 D x1 ; : : : ; X tn D xn  D Px ŒX t1 D x1 ; : : : ; X tn D xn  g.xn ; y/

(2.23)

D r t1 .x; x1 / r t2 t1 .x1 ; x2 / : : : r tn tn1 .xn1 ; xn / g.xn ; y/ x1 : : : xn : If K  E and HK is defined as in (1.45), for B 2 .XHK ^s ; s  0/ and  as in (2.18), Px;y ŒB; HK   D Ex ŒB \ fHK < 1g; g.XHK ; y/: (2.24)

37

2.2 The measures Px;y

Proof.  (2.23): Z 1 (2.21) t Px;y ŒX t1 D x1 ; : : : ; X tn D xn ; dt Px;y ŒX t1 D x1 ; : : : ; X tn D xn  D 0 Z 1 dt Px ŒX t1 D x1 ; : : : ; X tn D xn ; X t D y D y tn Markov property at time t

Z

1

D

Ex ŒX t1 D x1 ; : : : ; X tn D xn ; r t tn .xn ; y/ dt tn

(1.26)

D Px ŒX t1 D x1 ; : : : ; X tn D xn  g.xn ; y/;

and the second equality of (2.23) follows from the Markov property.  (2.24): Z

1

Px;y ŒB; HK   D 0 (2.18), (2.19)

t Px;y ŒB; HK   dt

Z

1

D

Ex ŒB; HK  t; X t D y 0

strong Markov property

Z

dt y

1

D

Ex ŒB; HK  t; r t HK .XHK ; y/ dt 0

Fubini

D Ex ŒB; HK < 1; g.XHK ; y/:

(1.26)



Remark 2.6. Given  2 , we can introduce N. /  0, the number of jumps of  strictly before ./, the duration of , and when N. / D n  1, we can consider 0 < T1 ./ <    < Tn ./ < ./ the successive jump times of Xs , 0  s  ./. As we now explain, for n  1, ti > 0, 1  i  n, t > 0, and x1 ; : : : ; xn 2 E, one has the following formula complementing Proposition 2.5: Px;y ŒN D n; XT1 D x1 ; : : : ; XTn D xn ; T1 2 t1 C dt1 ; : : : ; Tn 2 tn C dtn ;  2 t C dt cx;x1 cx1 ;x2 : : : cxn1 ;y ıxn ;y 1f0 < t1 < t2 <    < tn < tg e t dt1 : : : dtn dt; D x x1 : : : xn1 y (2.25)

38

2 Isomorphism theorems

where the precise meaning of (2.25) is obtained by considering some subsets A1 ; : : : ; An , A of .0; 1/, replacing “T1 2 t1 C dt1 ”,: : : ;“Tn 2 tn C dtn ”, “ 2 t C dt” by fT1 2 A1 g; : : : ; fTn 2 An g, f 2 Ag, in the left-hand side of (2.25), and in the righthand side multiplying by 1A1 .t1 / : : : 1An .tn / 1A .t/, and integrating the expression over the variables t1 ; : : : ; tn ; t. To find (2.25), we note that for t > 0, t ŒN D n; XT1 D x1 ; : : : ; XTn D xn ; T1 2 t1 C dt1 ; : : : ; Tn 2 tn C dtn  Px;y (2.19)

D Px ŒX: has n jumps in Œ0; t, XT1 D x1 ; : : : ; XTn D xn , T1 2 t1 C dt1 ; : : : ; Tn 2 tn C dtn  ıxn ;y y1 D Px ŒZ1 D x1 ; Z2 D x2 ; : : : ; Zn D xn  Px ŒTn < t < TnC1 ; T1 2 t1 C dt1 ; : : : ; Tn 2 tn C dtn  ıxn ;y y1 D

cx;x1 cx1 ;x2 : : : cxn1 ;y 1f0 < t1 <    < tn < tg e t dt1 : : : dtn ıxn ;y ; x x1 : : : xn1 y

where we made use of (1.12), (1.15), and Remark 1.1. Multiplying both sides by dt readily yields (2.25), in view of (2.18), (2.21). Similarly, we see that for n D 0, we have Px;y ŒN D 0;  2 t C dt D ıx;y e t dt:

(2.26)

We record an interesting consequence of (2.25), (2.26). We denote by { 2  the “time reversal” of  2 , i.e. { is the element of  such that .{  / D ./, {.0/ D ./, {./ D .0/, and {.s/ D lim"#0 .  s  "/, for 0 < s < ./. Note that on fN D ng one can reconstruct  from T1 ; : : : ; Tn ; , and XT1 ; : : : ; XTn . It is then a straightforward consequence of (2.25), (2.26) that Py;x is the image of Px;y under the map  ! {:

(2.27)

We will later see some analogous formulas to (2.25), (2.26) for rooted loops in Proposition 3.1, see also (3.40).  R1 We will now provide some formulas for moments of 0 V .Xs /ds and Lz1 under the measure Px;y . Proposition 2.7 (x; y 2 E). For V W E ! R and n  0, one has, in the notation of (1.32) and (1.83), n i h Z 1   Ex;y V .Xs /ds D nŠ .QV /n g y .x/; with g y ./ D .G1y /./ D g.; y/: 0

(2.28)

39

2.2 The measures Px;y

For x1 ; x2 ; : : : ; xn 2 E, one has i hQ n P Lx1i D g.x; x.1/ / g.x.1/ ; x.2/ / : : : g.x.n/ ; y/; Ex;y i D1

(2.29)

2n

with n the set of permutations of f1; : : : ; ng. When k GjV j k1 < 1, one has h n P oi     Ex;y exp V .z/ Lz1 D .I  GV /1 g y .x/ D .I  GV /1 G1y .x/: z2E

(2.30) Proof. We begin with a slightly more general calculation and consider V1 ; : : : ; Vn W E ! R and Ex;y

Z hQ n i D1

1

i hZ Vi .Xs /ds D Ex;y

Rn C

0

i V1 .Xs1 / : : : Vn .Xsn / ds1 : : : dsn

and decomposing over the various orthants Z P Ex;y ŒV1 .Xs1 / : : : Vn .Xsn / ds1 : : : dsn D  2n

(2.23)

D

0 0 and  2 R, see Chapter 17 of [11]. The assumption (4.29) however rules out such a choice since it implies that log h is convex (by Hölder’s inequality). Nevertheless, it simplifies the presentation made below. We write h  ih for the expectation relative to Px G;h , i.e. R Q '2 1 2 E.';'/ h. 2x / d' RE F .'/ e x2E (4.32) hF ih D R Q 2 1  2 E.';'/ h. '2x / d' RE e x2E

when, for instance, F W RE ! R is a bounded measurable function. Note that when  D ı0 , then h D 1E , and one recovers the free field: Px G;hD1E D P G ; in the notation of (2.2).

(4.33)

Symanzik’s formula will provide a representation of the moments of the random field governed by Px G;h in terms of the occupation field of a Poisson gas of loops and of the local times of walks interacting via random potentials. A variant of the formula is, for instance, used in Section 3 of [2], to obtain bounds on critical temperatures. z Pz / enz A; As a last ingredient, we introduce an auxiliary probability space .; dowed with a collection V .x; !/, z x 2 E, of non-negative random variables (the random potentials) such that z the variables V .x; !/, under P, z x 2 E, are (non-negative) i.i.d. -distributed. (4.34) Let us point out that the probability Q in (0.10) coincides with Pz ˝ P˛D 1 . We are 2 now ready to state and prove Symanzik’s formula.

94

4 Poisson gas of Markovian loops

w3

w2 y1

x3

x2

w1

y2 x1

y3

Figure 4.2. The paths w1 ; w2 ; w3 in E, and the gas of loops interact through the random potentials (k D 3, and the z1 ; : : : ; z6 are distinct.

Theorem 4.8 (Symanzik’s representation formula). For any k  1, z1 ; : : : ; z2k 2 E, one has h'z1 : : : 'z2k ih D

P

Ex1 ;y1 ˝    ˝ Exk ;yk

P

x x V .x;!/.L z   x .!/CL1 .w1 /CCL1 .wk //  z ˝ E 1 e x2E ˝E 2



z ˝ E1 e E

pairings of f1;:::;2kg

P



x2E

V .x;!/L z x .!/ 

;

2

(4.35)

where fxi ; yi g D fz` I ` 2 Di g; 1  i  k, and D1 ; : : : ; Dk stands for the (unordered) pairing of f1; : : : ; 2kg, and wi denotes the integration variable under Pxi ;yi . z x; y 2 E, Proof. By (3.60), we know that for ! z 2 , 



Ex;y e

P z2E

V .z;!/L z z1 

D gV . ;!/ z .x; y/;

(4.36)

which is a symmetric function of x, y, so that the expression under the sum in the right-hand side of (4.35) only depends on the pairing and is therefore well defined. Let F be a bounded measurable function on RE . Denote by .F /h the numerator /h . We have the identity of (4.32) so that hF ih D .F .1/h .F /h

z D E

(4.34)

hZ F .'/ e

1 2 ŒE.';'/C

P x2E

2 V .x;!/' z x

i d'

RE

 jEj 1 z z ˝ E G; V . ;!/ F .'/.2/ 2 .det GV .;!/ DE z /2 z ˝ E1 ˝ E D E

(4.11)

2

G;V . ;!/ z





F .'/ e

P x2E

V .x;!/L z x .!/ 

.2/

jEj 2

1

.det G/ 2 ; (4.37)

95

4.4 Some identities 1 where the second line is a consequence of (2.5), (2.36), and .V .; !/L/ z GV . ;!/ z . As a result of (4.28), we thus find that

below (3.45)

D

.'z1 : : : 'z2k /h P  V .x;!/L z x .!/   P z ˝ E 1 GV . ;!/ x2E E D z .x1 ; y1 / : : : GV . ;!/ z .xk ; yk / e 2

pairings of f1;:::;2kg

(4.36)

D

 .2/

P

jEj 2

1

.det G/ 2

z Ex1 ;y1 ˝    ˝ Exk ;yk ˝ E

pairings of f1;:::;2kg



˝ E1 e

P



x2E

x x V .x;!/.L z x .!/CL1 .w1 /CCL1 .wk // 

2

 .2/

jEj 2

1

.det G/ 2 : (4.38)

In the same way, we find by (4.37) that P

V .x;!/L z x .!/    jEj 1 z ˝ E 1 e x2E .1/h D E .2/ 2 .det G/ 2 :

(4.39)

2



Taking the ratio of (4.38) and (4.39) precisely yields (4.35). Remark 4.9. When k D 1, Symanzik’s representation formula (4.35) becomes P

h'x 'y ih D

Ex;y

z V .z;!/.L z   z .!/CL1 .w//  z ˝ E 1 e z2E ˝E 2

P

V .z;!/L z z .!/    z ˝ E 1 e z2E E

:

2

We explain below another way to obtain this identity. Recall that (4.27) combines Dynkin’s isomorphism theorem and the identity in law of .Lx /x2E under P 1 , with 2

. 12 'x2 /x2E under P G , stated in Theorem 4.5. P z z2 z ˝ E G Œ'x 'y e  12 z2E V .z;!/' By (4.27) the numerator equals E , whereas by P G 1 V .z;!/' z z2 z z2E 2 (4.24) the denominator equals E ˝ E Œe . Keeping in mind (4.29) and (4.34), one easily recovers the left-hand side of the above quality. 

4.4 Some identities In this section we discuss some further formulas concerning the Poisson gas of Markovian loops. In particular given two disjoint subsets of E, we derive a formula for the probability that no loop of the Poisson gas visits both subsets. In the next section, as an application, we will link the so-called random interlacements with various notions of “loops going through infinity” for the Poisson cloud of Markovian loops.

96

4 Poisson gas of Markovian loops

Given U  E, we can consider the field of occupation times of loops contained in U ,  LU x .!/ D h1f  U g !; Lx i P P 1fi  U g Lx .i /; if ! D ı i 2 ; x 2 E: D i2I

(4.40)

i2I

Here we used the slightly informal notation 1fi  U g in place of 1fi 2 Lr;U g, where i 2 Lr is any rooted loop such that   .i / D i , and Lr;U has been defined in (3.37). Proposition 4.10 (˛ > 0). Given K  E, U D EnK, and V W E ! RC , 

E˛ e



P x2E

V .x/.Lx LU x /

D D

 det G

V

det G

det GU ˛ det GU;V

 det

GV  ˛ detKK G

(4.41)

KK

def

(where detKK A D det.AjKK /, for A an E  E-matrix, and we write det GU , resp. det GU;V , in place of detU U GU , resp. detU U GU;V , with the notation from below (3.58)). If K1 \ K2 D , and Ui D EnKi , for i D 1; 2, then P˛ Œ no loop intersects both K1 and K2  D D

 det G det G  U1 \U2 ˛ ; det GU1 det GU2  K1 K1 GU2 ˛ detK1 K1 G

 det

(4.42)

(and we used a similar convention as above for det GU ). Proof.  (4.41): Observe that Lx  LU x , x 2 E, is the field of occupation times of loops which are not contained in U , whereas LU x , x 2 U , is the field of occupation times of loops which are contained in U : Lx  LU x D h1f  U gc !; Lx i; x 2 E; LU x D h1f  U g !; Lx i; x 2 E:

97

4.4 Some identities

By (4.5) we see that they constitute independent collections of random variables. So we find that for V W E ! RC ,  det G ˛ V

det G



(4.11)

D E˛ e

independence

D

(3.58)

D

(4.7)

P



x2E









E˛ e

E˛ e

V .x/Lx  P

x2E

P x2E

V .x/.Lx LU x / V .x/.Lx LU x /



 E˛ e

P



x2E

 det G

U;V

det GU

V .x/LU x 

˛ :

The equality in the first line of (4.41) follows. To prove the second equality in (4.41) we will use the next lemma: Lemma 4.11 (Jacobi’s determinant identity). Assume that A is a .k C `/  .k C `/matrix which is invertible and that



W X B C 1 ; A D AD ; Y Z D E where B and W are k  k matrices. Then det Z D det A det B: Proof. We know that

W B C I D Y D E

BW C C Y X D DW C EY Z

(4.43)

BX C C Z ; DX C EZ

and hence BX C C Z D 0 .k  `-matrix) and DX C EZ D I (`  `-matrix). As a result, we find that





B 0 B BX C C Z I X B C : D D D I D DX C EZ 0 Z D E Taking determinants, we find that det A1 det Z D det B; and the claim (4.43) follows.



98

4 Poisson gas of Markovian loops

We choose A D L, A1 D G, B D GjKK , Z D LjU U , so that det Z D det.LjU U / D .det GU /1 ;

det A D .det G/1 ;

and (4.43) yields   notation det G D det GjKK D detKK G : det GU

(4.44)

In the same way, with A D V  L, A1 D GV , B D GV jKK , Z D .V  L/U U , we find that det GV D det GV jKK : (4.45) det GU;V Coming back to the expression after the first equality in (4.41), we see that this det G expression equals . det GV jKK /˛ , and this completes the proof of (4.41). jKK

 (4.42): We first note that the left-hand side of (4.42) equals hD  P  P E i P˛ !; Lx Lx D 0 D

x2K1

x2K2

D P˛ 1fN > 1g !; (4.4)

n

 P

Lx

x2K1

 

P

D exp  ˛   W

 P

E Lx

x2K2

Lx .  / > 0 and

x2K1

D0

P

i

o Lx .  / > 0; and N.  / > 1 :

x2K2

(4.46) Now we have  ŒN.  / > 1 < 1 (it equals  log det.I  P / by (3.19) and (3.67)), so we can write:   P P  N > 1; Lx > 0 and Lx > 0 x2K1





D  N > 1;  

x2K2

   P Lx > 0 C  N > 1; Lx > 0

P x2K1

  N > 1;

P

x2K1 [K2

 Lx > 0 :

x2K2

(4.47)

The next lemma will be useful to evaluate the above terms. Lemma 4.12 (˛ > 0, K  E, U D EnK). With the notation of (4.15), one has ˛  det G Q ˛    P Q P˛ Lyx D 0 D det GjKK x D x ; (4.48) det GU x2K x2K x2K     Q P  N > 1; Lx > 0 D log det GjKK x : (4.49) x2K

x2K

99

4.4 Some identities

Proof. Note that (4.49) is a direct consequence of (4.48) and the identity n  (4.4) o  P  P P˛ Lyx D 0 D exp  ˛ N > 1; Lx > 0 : x2K

x2K

We hence only need to prove (4.48). We use (4.20) with the choice V D 1K where  " 1. We then find that P  det I  P V D 1K ˛ yx  L   P   (4.20) y x2K P˛ Lx D 0 D lim E˛ e D lim : det I  P !1 !1 x2K (4.50) V D 1K V D 1K Observe that P ! 1U P (i.e. lim !1 P f .x/ D 1U .x/.Pf /.x/, !1

for all x 2 E), by the definition of P V below (4.19). The matrix for I  1U P is block diagonal,

I 0 ;  .I  P /jU U and, therefore,   det.I  P V D1K / ˛  det.I  P / jU U / ˛ D : det.I  P / det.I  P / !1 Q Now, det GU D .det.L/jU U /1 D . x2U x /1 .det.I  P /jU U /1 , and similarly,  Q 1  1 det G D x det.I  P / : lim

x2E

Coming back to (4.50), we have shown that Q 1 0 det G x ˛  i h P ˛ Q x2E A D det G Q P˛ Lyx D 0 D @ x ; det GU x det GU x2K x2K x2U

and the proof of (4.48) is completed with (4.44).



We now return to (4.46), (4.47), and find that (recall K1 \ K2 D ) P˛ Œ no loop intersects both K1 and K2   det G det G det GU1 \U2 ˛  det G det GU1 \U2 ˛ D D det GU1 det GU2 det G det GU1 det GU2 ˛  det (4.44) K1 K1 G D ; detK1 K1 GU2 since K1 D U2 n.U1 \U2 /. This concludes the proof of (4.42) and of Proposition 4.10. 

100

4 Poisson gas of Markovian loops

Special case: loops going through a point We specialize the above formula (4.42) to find the probability that loops in the Poisson cloud going through a base point x all avoid some K not containing x:

E x

K

Figure 4.3

Corollary 4.13 (˛ > 0). Consider x 2 E, and K  E not containing x, then P˛ Œ all loops going through x do not intersect K  Ex ŒHK < 1; g.XHK ; x/ ˛ D 1 : g.x; x/

(4.51)

Proof. In the notation of (4.42) we pick K1 D fxg, K2 D K. Setting U D EnK, (4.42) yields that the left-hand side of (4.51) equals  g .x; x/ ˛ (1.49)  g.x; x/  E ŒH < 1; g.X ; x/ ˛ U x K HK D ; g.x; x/ g.x; x/ and (4.51) follows.



4.5 Some links between Markovian loops and random interlacements In this section we discuss various limiting procedures making sense of the notion of “loops going through infinity”, and see random interlacements appear as a limit object. We begin with the case of Zd , d  3. Random interlacements have been introduced in [27], and we refer to [27] for a more detailed discussion of the Poisson point process of random interlacements. We will recover random interlacements on Zd ,

4.5 Some links between Markovian loops and random interlacements

101

d  3, by the consideration of “loops going through infinity”. More precisely, we consider d  3, and Un , n  1, a non-decreasing sequence of finite connected S U n D Zd , subsets of Zd , with

(4.52)

n

as well as x 2 Zd ; a “base point”:

(4.53)

For fixed n  1, we endow the connected subset Un , playing the role of E in (1.1), with the weights n D cx;y

1 2d

and with the killing measure P xn D

y2Zd nUn

1fjx  yj D 1g; for x; y 2 Un ;

1 2d

1fjx  yj D 1g; for x 2 Un ;

very much in the spirit of what is done in Example 2) above (1.18) (except for the fact P 1 n we now replace 1 by 2d ). Note that nx D y2Un cx;y C xn D 1, for all x 2 Un . We write n for the space corresponding to (4.2), of pure point measures on the set of unrooted loops contained in Un , and P˛n for the corresponding Poisson gas of Markovian loops at level ˛, see (4.6).

Un x

Figure 4.4. An unrooted loop contained in Un and going through x ; first the limit n ! 1, then the limit x ! 1.

We want to successively take the limit n ! 1, and then x ! 1. The first limit corresponds to the construction of a Poisson gas of unrooted loops on Zd . We will not really discuss this Poisson measure, which can be defined in a rather similar fashion to what we have done at the beginning of this chapter, but of course escapes the set-up of a finite state space E with weights and killing measure satisfying (1.1)–(1.5). For the second limit (i.e. x ! 1), we will also adjust the level ˛ as a function of x . The fashion in which we tune ˛ to x is dictated by the Green function of

102

4 Poisson gas of Markovian loops

simple random walk on Zd : gZd .x; y/ D

d ExZ

hZ

1

i 1fX t D ygdt ; for x; y 2 Zd ,

(4.54)

0

d

where PxZ denotes the canonical law of continuous-time simple random walk with jump rate 1 on Zd starting at x, and X t , t  0, the canonical process. Taking advantage of translation invariance we introduce the function def

g.x/ D gZd .0; x/; for x 2 Zd (so gZd .x; y/ D g.y  x/):

(4.54’)

The function g./ is known to be positive, finite (recall d  3), symmetric, that is, g.x/ D g.x/, and has the asymptotic behavior g.x/  cd jxj.d 2/ ; as x ! 1;   2 d d 1 D  where cd D 2 2 .d  2/ jB.0; 1/j

1

d 2

;

(4.55)

where jxj stands for the Euclidean norm of x, and jB.0; 1/j for the volume of the unit ball of Rd (see for instance [15], p. 31). We will choose ˛ according to the formula ˛Du

g.0/ jx j2.d 2/ ; with u  0: cd2

(4.56)

We introduce for ! 2 n , the subset of Un of points visited by the unrooted loops in the support of the pure point measure !, which pass through the base point x : ˚ Jn;x .!/ D z 2 Un I there is a   in the support of!, (4.57) which goes through x and z , for ! 2 n . Note that Jn;x .!/ D , when x … Un , and Jn;x .!/ 3 x , when at least one   in the support of ! goes through x . For the next result we will use the fact that (1.57) and (1.58) in the case of continuous-time simple random walk with jump rate 1 on Zd take the following form: when K is a finite subset of Zd , P d gZd .x; y/ eK .y/; for x 2 Zd PxZ ŒHK < 1 D (4.58) y2K

zK as in (1.45)); (with H where the equilibrium measure

zK D 1 1K .y/; y 2 Zd eK .y/ D PyZ ŒH zK as in (1.45)); (with H d

103

4.5 Some links between Markovian loops and random interlacements

is the unique measure supported on K such that the equality in (4.58) holds for all x 2 K. Its mass capZd .K/ is the capacity of K. The next theorem relates the so-called “random interlacement at level u” to the set Jn;x when n ! 1, and then x ! 1, under the measure P˛n , with ˛ as in (4.56). In this set of notes we will not introduce the full formalism of the Poisson point process of random interlacements but only content ourselves with the description of the random interlacement at level u, see Remark 4.15 below. Theorem 4.14 (d  3). For u  0 and K  Zd finite, one has lim P n

lim

x !1

2.d 2/ ˛Du g.0/ 2 jx j

n!1

ŒJn;x \ K D  D e u capZd .K/ :

(4.59)

c

d

Proof. By (4.51) we have, as soon as x 2 Un and x … K, P˛n ŒJn;x \ K D  D P˛n Œall loops going through x do not meet K 

d

ExZ ŒHK < TUn ; gUn .XHK ; x / D 1 gUn .x ; x /

˛

;

with gUn .; / the Green function of simple random walk on Zd killed when exiting Un . Clearly, by monotone convergence, gUn .x; y/ " gZd .x; y/; for x; y 2 Zd , when n ! 1: So we see that when x … K, then 

lim n

P˛n ŒJn;x

d

ExZ ŒHK < 1; gZd .XHK ; x / \ K D  D 1  gZd .x ; x /

˛

(4.60)

(the formula holds also when x 2 K). Now gZd .x ; x / D g.0/, and, as x ! 1, we have by (4.55) (4.55)

d

d

ExZ ŒHK < 1; gZd .XHK ; x /  PxZ ŒHK < 1 cd jx j.d 2/ (4.58)

 .cd jx j.d 2/ /2 capZd .K/;

(4.55)

and, in particular, with ˛ as in (4.56), lim

˛

x !1 g.0/

d

ExZ ŒHK < 1; gZd .XHK ; x / D u capZd .K/:

Coming back to (4.60) we readily obtain (4.59).



104

4 Poisson gas of Markovian loops

Remark 4.15. One can define a translation invariant random subset of Zd denoted by  u , the so-called random interlacement at level u, see [27], with distribution characterized by the identity: PΠu \ K D  D e u capZd .K/ ; for all K  Zd finite.

(4.61)

Coming back to (4.59), note that for any disjoint finite subsets K; K 0 of Zd one has by an inclusion-exclusion argument: P˛n ŒJn;x \ K D and Jn;x K 0  hY Y  i c c D En˛ 1Jn;x .x/ .x/ 1  1 Jn;x  x2K 0

x2K

D

P AK 0

.1/jAj P˛n ŒJn;x \ .K [ A/ D ;

where we expanded the last product in the second line to find the last line. In the same fashion, we see that for disjoint finite subsets K; K 0 of Zd we have PΠu \ K D and  u K 0  P P D .1/jAj PΠu \ .K [ A/ D  D .1/jAj e u capZd .K[A/ : AK 0

AK 0

As a result, Theorem 4.14 can be seen to imply that under the measure Pn

2.d 2/ ˛Du g.0/ 2 jx j

;

c

d

the law of Jn;x converges in an appropriate sense (i.e. convergence of all finite dimensional marginal distributions) to the law of  u , as n ! 1, and then x ! 1.  We continue with the discussion of links between random interlacements and “loops going through infinity” in the Poisson cloud of Markovian loops. We begin with a variation on (4.59) in the context of Zd , d  3, where we will give a different meaning to the informal notion of “loops going through infinity”. We consider a sequence Un , n  1, as in (4.52) of finite connected subsets of Zd , d  3, which increases (in the wide sense) to Zd . The role of the base point x , cf. (4.53), is now replaced by the complement of the Euclidean ball: def

BR D fx 2 Zd I jxj  Rg; with R > 0.

(4.62)

By analogy with (4.57), we introduce for ! 2 n , Kn;R .!/ D fz 2 Un I there is a   in the support of !, c and zg. which goes through BR

(4.63)

105

4.5 Some links between Markovian loops and random interlacements

Zd R K

Un

0 BR

first the limit n ! 1, then the limit R ! 1

an unrooted loop contained in Un c and touching BR Figure 4.5

We now choose ˛ according to ˛Du

Rd 2 , with u  0, and cd as in (4.55). cd

(4.64)

The corresponding statement to (4.59) is now the following. By the argument of Remark 4.15, it can be interpreted as a convergence of the law of Kn;R to the law of  u , as n ! 1 and then R ! 1. Theorem 4.16 (d  3). For u  0 and K  Zd finite, one has lim

R!1

lim P n

n!1

˛Du

Rd 2 cd

ŒKn;R \ K D  D e u capZd .K/ :

(4.65)

Proof. We assume that R is large enough so that K  BR and n sufficiently large so that BR   Un . In the notation of (4.42), we chose K1 D K and K2 D Un nBR , so that K1 \ K2 D . Then (4.42) yields that 

P˛n ŒKn;R

detKK GBR \ K D  D detKK GUn

By (1.49), we write detKK GBR D det.An  Bn.R/ /;

˛

:

106

4 Poisson gas of Markovian loops

where An is the K  K-matrix An .x; y/ D gUn .x; y/; for x; y 2 K; and Bn.R/ , the K  K-matrix d

Bn.R/ .x; y/ D ExZ ŒHBRc < TUn ; gUn .XHB c ; y/; for x; y 2 K: R

Likewise, by the above definitions, we find that detKK GUn D det.An /: When n ! 1, lim An D A;

where A.x; y/ D gZd .x; y/, for x; y 2 K

n

(4.66)

d

lim Bn.R/ D B .R/ ; where B .R/ .x; y/ D ExZ ŒHBRc < 1; gZd .XHB c ; y/; (4.67) n R for x; y 2 K. The matrix A is known to be invertible (one can base this on a similar calculation as in the proof of (1.35), see also [25], P2, p. 292). So we find that 

lim P n ŒKn;R \ K n!1 ˛

det.A  B .R/ / D  D det A

˛

 ˛ D det.I  A1 B .R/ / : (4.68)

d

For x 2 K, PxZ -a.s., HBRc < 1, and XHB c 2 @BR , so that R

Zd

B .R/ .x; y/ D Ex ŒgZd .XHB c ; y/ R

cd  d 2 ; for x; y 2 K, as R ! 1: R

(4.55)

It follows that det.I  A1 B .R/ / D 1 





cd 1 Tr.A1 1KK / C o d 2 ; as R ! 1; (4.69) d 2 R R

where 1KK denotes the K  K matrix with all coefficients equal to 1. C is the K K-matrix Coming back to (4.58), we see that A1 1KK D C , where P with coefficients C.x; y/ D eK .x/, for x; y 2 K. Since x2K eK .x/ D capZd .K/, we have found that det.I  A

1



B

.R/



cd 1 / D 1  d 2 capZd .K/ C o d 2 ; as R ! 1: R R

(4.70)

Inserting this formula into (4.68), with ˛ as in (4.64), immediately yields (4.65).



4.5 Some links between Markovian loops and random interlacements

107

Complement: random interlacements and Poisson gas of loops coming from infinity on a transient weighted graph. So far we only discussed links between random interlacements and a Poisson cloud of “loops going though infinity”, in the case of Zd , d  3. We now discuss another construction, which applies to the general set-up of an (infinite) transient weighted graph with no killing. We consider a countable (in particular infinite) set  endowed with non-negative weights cx;y , x; y 2  (i.e. satisfying (1.2) with  in place of E), so that  endowed with the set of edges consisting of fx; yg such that cx;y > 0, is connected, locally finite, (i.e. each x 2  has a finite number of neighbors),

(4.71)

and the simple random walk with jump rate 1 on  induced by these weights cx;y , x; y 2 , is transient.

(4.72)

This is what we mean by a transient weighted graph (i.e. with no killing). We consider, as in (4.52), Un , n  1, a non-decreasing S sequence of finite connected subsets of  increasing to  (i.e. Un D  and Un  UnC1 ),

(4.73)

n1

as well as a point x not in  (which will play the role of the point “at infinity for each Un ”). (4.74) We consider the finite graph with vertex set En D Un [ fx g, endowed with the weights obtained by collapsing UNc on x : n D cx;y ; when x; y 2 Un , cx;y P n n cx ;y D cy;x D cx;y ; when y 2 Un : 

(4.75)

x2GnUn

In addition we choose on En the killing measure xn , x 2 En , concentrated on x , so that xn D n > 0; with lim n D 1; xn D 0; for x 2 Un D En nfx g:

(4.76)

For the continuous-time walk on Pwith jump rate 1, one can show that when K is a finite subset of G, setting 0x D y2 cx;y , for x 2 , and g .; / for the Green

108

4 Poisson gas of Markovian loops

Un

Un " G

G

Figure 4.6

function (i.e. g .x; y/ D

1 0 y

R1

Ex Œ

Px ŒHK < 1 D

0

1fX t D ygdt, for x; y 2 ),

P

g .x; y/ eK .y/; for x 2 ;

(4.77)

y2K

where eK is the equilibrium measure of K: zK D 1 1K .y/ y0 ; for y 2  eK .y/ D Py ŒH

(4.78)

(for instance one approximates the left-hand side of (4.77) by Px ŒHK < TUn , with n ! 1, and applies (1.57), (1.53) to the walk killed when exiting Un ). The total mass of eK is the capacity of K: P eK .y/: cap .K/ D

(4.79)

y2K

We write n for the space of unrooted loops on En and P˛n for the Poisson gas of Markovian loops at level ˛ > 0, on the above finite set En endowed with the weights c n in (4.75) and the killing measure  n in (4.76). We also introduce the random subset of Un : Jn .!/ D fz 2 Un I there is a   in the support of ! which goes through x and zg:

(4.80)

4.5 Some links between Markovian loops and random interlacements

109

We now specify ˛ via the formula, see (4.76), ˛ D u n ; with u  0:

(4.81)

The statement corresponding to (4.59) and (4.65), which in the present context links the Poisson gas of loops on En going through “the point x at infinity”, with the interlacement at level u on G is coming next. We refer to Remark 1.4 of [27] and [29] for a more detailed description of the Poisson point process of random interlacements in this context. Theorem 4.17. For u  0 and K  G finite, one has n ŒJn \ K D  D e u capG .K/ : lim P˛Du n

(4.82)

n!1

Proof. For large n, K  Un , and by (4.51) we can write 

P˛n ŒJn \ K D  D 1 

Exn ŒHK < 1; gn .XHK ; x / gn .x ; x /

˛

;

where Pxn stands for the law of the walk on En with unit jump rate, starting at x 2 En , attached to the weights and killing measure in (4.75), (4.76) and gn .; / for the corresponding Green function. By (2.71), we know that gn .x ; z/ D 1 n ; for all z 2 En ;

(4.83)

and, as a result,  ˛ P˛n ŒJn \ K D  D 1  Pxn ŒHK < 1 (1.57)

D

(4.83)



1

1 n

˛ capn .K/ ;

(4.84)

where capn .K/ stands for the capacity of K in En . By (1.53) and the fact that  n vanishes on Un , we know that P n zK D 1 0x : Px ŒH capn .K/ D x2K

In addition, we know that Pxn -a.s., 1 zK g \ H zK D 1g D fHx < H .fHK D 1g/; fH x

(4.85)

110

4 Poisson gas of Markovian loops

because Un is finite and the walk is only killed at x . So applying the strong Markov property at time Hx we find that zK   Pxn ŒHK D 1 zK D 1 D Pxn ŒHx < H Pxn ŒH  zK   .1  Pxn ŒHK < 1/; D Px ŒTUn < H  using the fact that the walk on G and on En “agree up to time TUn ”. Note, in addition, that (1.57) P P 0 (4.76) (4.83) 1 Pxn ŒHK < 1  gn .x ; y/ y0 D y ! 0; n y2K

(1.53) y2K

and that

n!1

zK  # Px ŒH zK D 1; as n ! 1: Px ŒTUn < H

Coming back to (4.85), we have shown that lim capn .K/ D n

P x2K

zK D 1 0x D cap .K/: Px ŒH (4.78)

(4.79)

(4.86)

If we now insert this identity in (4.84) and keep in mind that ˛ D u n , we readily find (4.82).  Remark 4.18. 1) By a similar argument as described in Remark 4.15, the above n theorem can be seen to imply that under P˛Du , the law of Jn converges to the law n u of  in the sense of finite dimensional marginal distributions, as n goes to infinity. 2) A variation on the approximation scheme, which we employed to approximate random interlacements on a transient weighted graph, can be used to prove an isomorphism theorem for random interlacements, see [28]. One can define the random field .Lx;u /x2 of occupation times of continuous-time random interlacements at level u (this random field is governed by a probability denoted by P). One can also define the canonical law P G on R of the Gaussian free field attached to the transient weighted graph under consideration: under P G the canonical field .'x /x2 is a centered Gaussian field with covariance E G Œ'x 'y  D g .x; y/, for x; y 2 , with g .; / the Green function. The isomorphism theorem from [28] states that  'x2 x2 under P ˝ P G has the same law as p  1 .'x C 2u/2 x2 under P G :



Lx;u C

1 2

(4.87)

2

The above identity in law is intimately related to the generalized second Ray–Knight  theorem, see Theorem 2.17, and characterizes the law of .Lx;u /x2 .

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Index

capacity, 16, 18 variational problems, 18 conductance, 16 continuous-time loop, 65 stationarity property, 65 time-reversal invariance, 68 Dirichlet form, 6 orthogonal decomposition, 19 trace form, 19, 21, 29 tower property, 23 discrete loop, 65 stationarity property, 65 time-reversal invariance, 68 energy, 6, 11 entrance time, 14 equilibrium measure, 16, 18 equilibrium potential, 16, 18 exit time, 14 Feynman diagrams, 92 Feynman–Kac formula, 23, 29 Gaussian free field, 31, 90 conditional expectations, 33 generalized Ray–Knight theorems first Ray–Knight theorem, 45 second Ray–Knight theorem, 52, 56, 110 Green function, 9 killed, 14

jump rate 1, 7 killing measure, 5 local time, 25, 75 loops going through a point, 100 loops going through infinity, 101, 104, 107 Markov chain Xx: , 26 Markov chain X: , 7 Markovian loops, 90 measure Px;y , 35, 81 occupation field, 87 occupation field of Markovian loop, 87 occupation field of non-trivial loops, 89 occupation time, 90 pointed loops, 62 measure p on pointed loops, 70 Poisson gas of Markovian loops, 86 Poisson point measure, 86 potential operators, 11 random interlacements, 56, 95, 100, 104, 107 at level u, 103, 104, 109 Ray–Knight theorem, 45 restriction property, 74, 80 rooted loops, 61 measure r on rooted loops, 63 rooted Markovian loop, 61

hitting time, 14 Symanzik’s representation formula, 94 Isomorphism theorem time of last visit, 14 Dynkin isomorphism theorem, 41 Eisenbaum isomorphism theorem, 43 trace Dirichlet form, 19, 20, 29 for random interlacements, 110 trace process, 30

114 transient weighted graph, 107 transition density, 9 killed transition density, 14

Index

unrooted loops, 81 measure r on unrooted loops, 81 variable jump rate, 26

unit weight, 82 weights, 5