Probability and Partial Differential Equations in Modern Applied Mathematics (The IMA Volumes in Mathematics and its Applications, 140) 0387258795, 9780387258799

Table of contents :
Title Page
Copyright Page
FOREWORD
PREFACE
Table of Contents
NONNEGATIVE MARKOV CHAINS WITH APPLICATIONS
1. Introduction
2. Examples
3. Random dynamical systems.
4. Stationary distributions for Markov chains satisfying (1)
5. Harris irreducibility.
REFERENCES
PHASE CHANGES WITH TIME AND MULTI-SCALE HOMOGENIZATIONS OF A CLASS OF ANOMALOUS DIFFUSIONS
1. Introduction
2. A general model with two spatial scales: The first phase of asymptotics and the time scale for its breakdown.
3. The second Gaussian phase and its time scale, examples of non-Gaussian int ermediat e phases.
4. Examples of stratified media with non-Gaussian intermediate phases.
5. The non-divergence free case:
REFERENCES
SEMI-MARKOV CASCADE REPRESENTATIONS OF LOCAL SOLUTIONS TO 3-D INCOMPRESSIBLE NAVIER-STOKES
1. Introduction and preliminaries.
2. Semi-Markov cascades and local representations.
3. Time-asymptotic steady state solutions.
REFERENCES
AMPLITUDE EQUATIONS FOR SPDES: APPROXIMATE CENTRE MANIFOLDS AND INVARIANT MEASURES
1. Introduction.
2. General setting.
2.1. Assumptions.
2.2. Examples of equations.
3. Amplitude equations, main results.
3.1. Attractivity.
3.2. Approximation.
4. Applications.
4.1. Approximate centre manifold.
4.2. Dynamics of the random attractor.
5. Approximation of the invariant measure.
6. What is so special about cubic nonlinearities?
REFERENCES
ENSTROPHY AND ERGODICITY OF GRAVITY CURRENTS
1. Geophysical background.
2. Mathematical model.
3. Cocycle property.
4. Dissipativity.
5. Random dynamics: Enstrophy and ergodicity.
REFERENCES
STOCHASTIC HEAT AND BURGERS EQUATIONS AND THEIR SINGULARITIES
1. Introduction.
2. Stochastic heat and Burgers equations.
3. Stochastic general case.
4. Closeness to classical.
5. The Burgers fluid.
6. Singularities and intermittence of turbulence.
REFERENCES
A GENTLE INTRODUCTION TO CLUSTER EXPANSIONS
1. Introduction.
2. The exponential and the combinatorial exponential.
3. The equilibrium lattice gas.
3.1. The Mayer equations.
3.2. Cluster estimates.
3.3. Abstract polymer systems.
4. Polymer systems.
5. Cluster expansions.
REFERENCES
CONTINUITY OF THE ITO-MAP FOR HOLDER ROUGH PATHS WITH APPLICATIONS TO THE SUPPORT THEOREM IN HOLDER NORM
1. Introduction.
1.1. Background in Rough Path theory.
1.2 . Rough Path theory and stochastic analysis.
1.3. Rough Path theory for p E [2,3).
1.4. Definitions and outline.
2. Holder-regularity of Enhanced Brownian motion.
3. Approximations to Brownian Rough Paths.
3.1. Piecewise linear nested approximations.
3.2. Adapted dyadic approximations.
4. A primer on the Universal Limit Theorem.
5. Lipschitz regularity of Ito-rnap for HOlder Rough Paths.
6. Application to the Support Theorem.
APPENDIX
REFERENCES
DATA-DRIVEN STOCHASTIC PROCESSES IN FULLY DEVELOPED TURBULENCE
1. Introduction.
2. A careful look at data analysis.
3. Binary random multiplicative cascade process.
4. RMCP-driven data analysis.
5. Outlook: more stochastic processes.
REFERENCES
STOCHASTIC FLOWS ON THE CIRCLE
1. Introduction.
2. Flows of diffeomorphisms.
3. The Krylov Veretennikov expansion.
3.1. Lipschitz case.
3.2. Non-Lipschitz case.
3.3. A flow of infinite matrices.
4. Flows of kernels and flow of maps.
4.1. n-point motions.
4.2. Flow of maps.
4.3. Diffusive flow of kernels.
4.4. Diffusive or coalescing?
5. Classification of the solutions of the SDE.
5.1. Solutions of the SDE.
5.2. Extension of the noise and weak solutions.
REFERENCES
PATH INTEGRATION: CONNECTING PURE JUMP AND WIENER PROCESSES
1. Introduction.
2. Infinitely divisible complex distributions and complex Markov processes.
3. Regularization and the Fock space lifting.
4. Two remarks on parabolic equations in momentum representation.
REFERENCES
RANDOM DYNAMICAL SYSTEMS IN ECONOMICS
1. Introduction.
1.1. The Solow model: A dynamical system with an increasing law of motion.
1.2. The quadratic family in dynamic optimization problems.
2. Random dynamical systems.
3. Evolution.
3.1. A general theorem under splitting.
3.2. Applications of splitting.
3.2.1. Stochastic turnpike theorems.
3.2.2. Uncountable I': an example.
3.2.3. An estimation problem.
4. Iterates of quadratic maps.
REFERENCES
A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION OF THE NAVIER-STOKES EQUATIONS
1. Introduction.
2. The Navier-Stokes equations in the Galerkin approximation.
2.1. The noisy forcing term.
2.2. The Galerkin approximation.
3. The ergodic properties of the process.
4. Regularity of the transition probabilities.
5. Irreducibility and the control problem.
6. A toy model.
7. Some numerical results.
7.1. The numerical simulation.
7.2. Conclusions.
REFERENCES
INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS
1. Introduction.
2. Random dynamical systems.
3. random evolution equations.
4. The random graph transform and inertial manifolds.
6. Examples.
REFERENCES
EXISTENCE AND UNIQUENESS OF CLASSICAL, NONNEGATIVE, SMOOTH SOLUTIONS OF A CLASS OF SEMI-LINEAR SPDES
1. Introduction.
2. Linear SPDE.
3. Semi-linear SPDE.
REFERENCES
NONLINEAR PDE'S DRIVEN BY LEVY DIFFUSIONS AND RELATED STATISTICAL ISSUES
Introduction.
1. Selfsimilar solutions of evolutions equations as limits of general solutions.
2. Parabolic scaling limits for Burgers turbulence.
3. Parametric estimation in Burgers turbulence via parabolicrescaling.
4. Selfsimilar solutions and scaling limits for fractional conservationlaws.
REFERENCES
LIST OF WORKSHOP PARTICIPANTS

Citation preview

The IMA Volumes in Mathematics and its Applications Volume 140

Series Editors Douglas N. Arnold Fadil Santosa

Institute for Mathematics and its Applications (IMA) The Institute for Mathematics and its Applications was established by a grant from the National Science Foundation to the University of Minnesota in 1982. The primary mission of the IMA is to foster research of a truly interdisciplinary nature, establishing links between mathematics of the highest caliber and important scientific and technological problems from other disciplines and industry. To this end, the IMA organizes a wide variety of programs, ranging from short intense workshops in areas of exceptional interest and opportunity to extensive thematic programs lasting a year . IMA Volumes are used to communicate results of these programs that we believe are of particular value to the broader scientific community. The full list of IMA books can be found at the Web site of the Institute for Mathematics and its Applications: http:j jwww.ima.umn.edujspringerjvolumes.html. Douglas N. Arnold, Director of the IMA

********** IMA ANNUAL PROGRAMS 1982-1983 1983-1984

Statistical and Continuum Approaches to Phase Transition Mathematical Models for the Economics of Decentralized Resource Allocation 1984-1985 Continuum Physics and Partial Differential Equations 1985-1986 Stochastic Differential Equations and Their Applications 1986-1987 Scientific Computation 1987-1988 Applied Combinatorics 1988-1989 Nonlinear Waves 1989-1990 Dynamical Systems and Their Applications 1990-1991 Phase Transitions and Free Boundaries 1991-1992 Applied Linear Algebra 1992-1993 Control Theory and its Applications 1993-1994 Emerging Applications of Probability 1994-1995 Waves and Scattering 1995-1996 Mathematical Methods in Material Science 1996-1997 Mathematics of High Performance Computing 1997-1998 Emerging Applications of Dynamical Systems 1998-1999 Mathematics in Biology Continued at the back

Edward C. Waymire

Jinqiao Duan

Editors

Probability and Partial Differential Equations in Modem Applied Mathematics

With 22 Illustrations

~ Springer

Edward C. Waymire

Jinqiao Duan

Department of Mathematics Oregon State University Covallis, OR 97331 [email protected]

Department of Applied Mathematics Illinois Institute of Technology Chicago. IL 60616 [email protected]

Series Editors: Douglas N. Amold Fadil Santosa Institute for Mathematics and its Applications University of Minnesota Minneapolis, MN 55455

Mathematics Subject Classification (2000) : 35Q30 , 35Q35. 37H05 . 60Hl5, 6OG60 Library of Congress Control Number: 2005926339 ISBN-IO: 0-387-25879-5 ISBN-13 : 978-0387-25879-9

Printed on acid-free paper.

© 2005 Springer Science+Business Media. Inc. All rights reserved . This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media. Inc., 233 Spring Street. New York. NY 10013. USA), except for brief excerpts in connection with reviews or scholarly analysis . Use in connection with any form of information storage and retrieval, electronic adaptation. computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names , trademarks. service marks , and similar terms, even if they are not identified as such. is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights . Printed in the United States of America . 9 8 7 6 5 432 I springeronline.com

(MVY)

FOREWORD

This IMA Volume in Mathematics and its Applications

PROBABILITY AND PARTIAL DIFFERENTIAL EQUATIONS IN MODERN APPLIED MATHEMATICS contains a selection of articles presented at 2003 IMA Summer Program with the same title. We would like to th ank Jinqiao Duan (Department of Applied Mathematics, Illinois Institute of Technology) and Edward C. Waymire (Department of Mathematics, Oregon State University) for their excellent work as organizers of the two-week summer workshop and for editing the volume. We also take thi s opportunity to th ank th e National Science Foundation for their support of th e IMA.

Series Editors

Douglas N. Arnold, Director of the IMA Fadil Santosa, Deputy Director of the IMA

PREFACE

The IMA Summer Program on Probability and Partial Differential Equations in Modern Applied Mathematics took place July 21-August 1, 2003, a fitting segue to the IMA annual program on Probability and Statistics in Complex Systems : Genomics, Networks, and Financial Engineering which was to begin September, 2003. In addition to the outstanding resources and staff at IMA, the summer program was developed with the assistance of the following members of the organizing committee: Rabi N. Bhattacharya, Larry Chen, Jinqiao Duan, Ronald B. Guenther, Peter E. Kloeden, Salah Mohammed, Sri Namachchivaya, Mina Ossiander, Bjorn Schmalfuss, Enrique Thomann, and Ed Waymire . The program was devoted to the role of probabilistic methods in modern applied mathematics from perspectives of both a tool for analysis and as a tool in modeling. Researchers involved in contemporary problems concerning dispersion and flow , e.g. fluid flow, cash flow, genetic migration, flow of internet data packets , etc., were selected as speakers and to lead discussion groups. There is a growing recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. In organizing this program an explicit effort was made to bring together researchers with a common interest in the problems, but with diverse mathematical expertise and perspective. A probabilistic representation of solutions to partial differential equations that arise as deterministic models, e.g. variations on Black-Scholes options equations, contaminant transport, reaction-diffusion, non-linear equations of fluid flow , Schrodinger equation etc . allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. In addition such approaches can be effective in sorting out multiple scale structure and in the development of both non-Monte Carlo as well as Monte Carlo type numerical methods. There is also a growing recognition of a role for the inclusion of stochastic terms in the modeling of complex flows. The addition of such terms has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations . During the last decade, significant progress has been made towards building a comprehensive theory of random dynamical systems, statistical cascades, stochastic flows, and stochastic pde's. A few core problems in the modeling, analysis and simulation of complex flows under uncertainty are : Find appropriate ways to incorporate stochastic effects into models; Analyze and express the impact of randomness on the evolution of complex vii

viii

PREFACE

systems in ways useful to the advancement of science and engineering; Design efficient numerical algorithms to simulate random phenomena. There is also a need for new ways in which to incorporate the impact of probability, statistics, pde's and numerical analysis in the training of present and future PhD students in the mathematical sciences. The engagement of graduate students was an important feature of this summer program. Stimulating poster sessions were also included as a significant part of the program. The editors thank the IMA leadership and staff, especially Doug Arnold and Fadil Santosa, for their tremendous help in the organization of this workshop and in the subsequent editing of this volume. The editors hope this volume will be useful to researchers and graduate students who are interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in engineering and science. Jinqiao Duan Department of Applied Mathematics Illinois Institute of Technology Edward C. Waymire Department of Mathematics Oregon State University

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Nonnegative Markov chains with applications. K.Bo Athreya Phase changes with time and multi-scale homogenizations of a class of anom alous diffusions Rabi Bhattacharya 0

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Semi-Markov cascad e representations of local solut ions to 3-D incompressible Navier-Stokes Rabi Bhattacharya, Larru Chen, Ronald B . Guenther, Chris Drum, Mina Ossuuuier, En rique Thomann, and Edward C. Waymire

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Amplitude equations for SPDEs: Approxim ate cent re manifolds and invariant measures Dirk Blomker and Martin Hairer

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Enstrophy and ergodicity of gravity currents Vena Pearl Bongolan- Walsh, Jinqiao Duan, Hongjun Gao, Tamay Ozgokm en, Paul Fischer, and Traian Iliescu Stochastic heat and Burgers equations and their singularities fan M. Davies, Aubrey Truman , and Huaizhong Zhao

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A gentle introduction to cluster expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 William G. Faris

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CONTENTS

Continuity of the Ito-rnap for Holder rough paths with applications to the Support Theorem in Holder norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 117 Peter K. Friz Data-driven stochastic processes in fully developed turbulence Martin Greiner, Jochen Cleve, Jiirgen Schmiegel, and K atepalli R . Sreenivasan Stochastic flows on the circle Yves Le Jan and Olivier Raimond Path integration: connecting pure jump and Wiener processes Vassili N. Kolokoltsov Random dynamical systems in economics Mukul Majumdar A geometric cascade for the spectral approximation of the Navier-Stokes equations M. Romito

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Inertial manifolds for random differential equations. . . . . . . . . . . . . . . . .. 213 Bjorn Schmalfuss Existence and uniqueness of classical , nonnegative, smooth solutions of a class of semi-linear SPDEs Hao Wang

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Nonlinear PDE's driven by Levy diffusions and related statistical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Wojbor A . Woyczynski List of workshop participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

NONNEGATIVE MARKOV CHAINS WITH APPLICATIONS· K.S . ATHREYAt Abstract. For a class of Markov chains that arise in ecology and economics conditions are provided for the existence, uniqueness (and convergence to) of stationary probability distributions. Their Feller property and Harris irreducibility are also explored. Key words. Population mod els, stationary measures, random iteration, Harris irreducibility, Feller property.

AMS(MOS) subject classifications. 60J05 , 92D25 , 60F05 .

1. Introduction. The evolution of many populations in ecology and that of some economies exhibit the following characteristics: a) It is random but the stochastic transition mechanism displays a time st ationary behavior, b) for small population size (and in small and fledgling economies) the growth rate is proportional to the current size with a random proportionality constant, c) for large populations the above growth rate is curt ailed by competition for resources (diminishing return in economies) . This leads to considering the following class of stochastically recursive time series model

(1) [0,00) ---+ [0,1] is continuous and decreasing , g(O) = 1, and are LLd. and independent of the initial value X o. These are called density dependent models (Vellekoop and Hognas (1997), Hassel (1974)). It is clear that {Cn}n >o defined by the above random iteration scheme is a Markov chain with stated space S = [0,00) and transition function where 9

{Cn}n~l

(2)

P(x, A)

= P(Cx g(x)

EA).

The goals of this paper are to describe some recent results on the existence of nontrivial stationary distributions, convergence to them, their uniqueness , etc .

2. Examples. a) Random logistic maps. The logistic model has been quite popular in the ecology literature to capture the density dependence as will as preypredator interaction (May (1976)). In the present context the parameter "Supported in part by Grant AFOSR IISI F49620-01-1-0076. This paper is based on the talk presented by the author at the IMA conference on Probability and P.D.E . in July-August, 2003. tSchool of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853 ([email protected]) ; and Iowa State University.

2

K.B . ATHREYA

C is allowed to vary in an LLd. fashion over time. Thus the model (1) becomes

(3)

n~ O

wit h X n E [0, 1]' Cn E [0,4] . T hus, the state space S = [0, 1] and g(x) == 1 - x has compact support . b) Random Ricker maps. Ricker (1954) proposed t he following model for th e evolution of fish population in Canada:

(4)

°

with X n E [0,00) , C n E [0, 00), < d < 00. Thus, th e st at e space S = [0,00) and g(x) == e- dx has exponential decay. c) Random Hassel maps. Hassel (1974) propos ed a model with a polynomial decay for large values. Here

(5)

°

with X n E [0, 00), C; E [0, 00), < d < 00. Here S = [0, 00), g(x) = (l+ x )-d . d) Yellekoop-Hiiqnos maps. A model that includes all th e previous cases was proposed by Vellekoop and Hognas (1997)

(6)

b>O

h : [0, 00) --+ [1,00), h(O ) = 1, h(·) is continuously differenti able and h(x ) = x~~W is nondecreasing. This family of maps exhibits behavior similar to th at of t he logistic fmaily such as pitchfork bifurcation of periodic behavior , chaotic behaivor as the parameter value is increased etc . The random logistic case was first introduced by R.N. Bhattachar ya and B.V. Rao (1993). Contribution s to it include Bhattacharya and Majumdar (2004), Bhattacharya and Waymire (1999), Athreya and Dai (2000, 2002), Athreya and Schuh (2002), Dai (2002), Athreya (2003), Athreya (2004a, b) . Deterministic interval maps have been studied a great deal in the dynamical systems literature. Random perturbations of such system have been investigated in th e book of Y. Kifer. Useful references for the deterministic case are the books by Devaney (1989), de Melo and van Strien (1993).

3. Random dynamical systems. The sto chastic recursive time series defined by (1) is an example of a random dynamical system obtained by iteration of random jointly measurable maps. This set up will be describ ed now. Let (S, s) and (K , K,) be two measur able spaces and f : K x s --+ S be jointly measurable, Le. (s x K" s) measurable. Let {Bi (W )} i ~l be a sequence of K

3

NONNEGATIVE MARKOV CHAINS WITH APPLICATIONS

valued random variables on a probability space (D, B , P). Let X o : D -. S be an S-valued r.v. Let

(7)

n ~

o.

Then for each n , X n : D -. S is a random variable and hence {X n+!(w)}n2:0 is a well defined S-valued stochastic process on (D,B,P). When {Bi };2:1 are LLd. LV. independent of X o then {X n }n2:0 is an Svalued Markov chain on (D, B, P) with transition function

P(x,A) = P{w : f(B(w), x)

(8)

f-

A} .

It turns out that if S is a polish space then for every probability tr ansition kernel P(·, .), i.e., a map from S x s -. [0,1] such that for each x, P(x , ·) is a probability measur e on (S, s) and for each A in s, P(" A) : S -. [0,1] is s measurable, there exists a random dynamical system of LLd. random maps {Ji(X,W)};2:1 from S x n -. S that is jointly measurable for each i and {Ji(·,W)}i2:1 are LLd. stoch astic processes such that the Markov chain generated by the recursive equation

(9) has transition function P(" .), i.e.

P(x , A)

=

P{w : f( x ,w) EA}.

See Kifer (1986) and Athreya and Stenflo (2000). As simple examples of this consider the following. 1. The vacillating probabilist . S=[O ,l], X _ Xn n +! 2 +

En+!

2

are LLd. Bernouilli (!) LV . Athreya (1996). 2. Sierpinski Gasket. Let S be an equilateral triangle with vertices Vl,V2,V3 and {Xn}n2:0 be define by {€n}n2:1

X

_ Xn n+! -

where

{€ n}n2:1

+ €n+! 2

are LLd. with distribution P(El

1

= Vi) = -3

i

= 1,2 ,3 .

3. Let {An , bn} n2:1 be LLd r.v. such that for each n, An is K matrix and b« is a K x 1 vector . Let

X

K real

4

K.B . ATHREYA

Suppose Elog IIAIII < 0 and E(log Ilblll)+ < 00 where "Alii is the matrix norm and Ilblll is the Euclidean norm. Then it can be shown that X n converges in distribution and the limit 1r is nonatomic (provided the distribution of (AI, bI) is not degenerate). Note that this example includes the previous two. Further, it can be shown that w.p.1 the limit point set of {Xn}n>O coincides with the support k of the limit distribution tt , This result has been used to solve the inverse problem of generating k. by running an appropriate Markov chain {Xn}n~O and looking at the limit point set of its sample path. For this the book by Barnsley (1993) may be consulted. When S is Polish and the {Jih>l are LLd. Lifschitz maps several sufficient conditions are known for the existence of a stationary distribution, its uniqueness and convergence to it . Two are given below. THEOREM 3.1. Let (S ,d) be Polish and (n,B,p) be a probability space. Let {Ji(x , w h~l be i.i.d. maps form S x n ~ S such that for each i fi is jointly measurable. Let Xn+l(w) = fn+I(Xn(w),w), n 2: 0 aJ Let Ji(·,w) be Lifschitz w.p.l and let

s(fl)

==

sup d(fl(x,w) ,fl(y,w) x #y d(x ,y)

Assume E(logs(fl)) < 0 and E(logd(fl(xo ,w),xo))+ < 00 for some Xo in S . Then, for any initial distribution, the sequence {X n} converges in distribution to a limit 1r that is unique and stationary for the Markou chain {X n } . bJ Let for some p > 0 sup E(d(fl(x,w),fl(y,w)))P < 1 x#y d(x ,y)

and for some Xo E(logd(fl(xo,w),xo))+
0, M < 00 such that i) "Ix ~ k, E(v(Xd IX o = x) - V( x) S -a. ii) "Ix E S, E(V(X l ) IX o = x) - V(x) S M . Then limf n,xo(k) 2 ",';M > O. In ecological and economic applications when S = [0, (0) , the above condition is verified for a compact set k c (0,00) so that I' is different from the delta measure at O. For proofs the above two results see Athreya (2004a, b) .

Is zr

I

Is

4. Stationary distributions for Markov chains satisfying (1) . Let {Xn}n >O be a Markov chain defined by (1). A necessary condition for the existence of a st ationary distribution 1r such that 1r(0 , (0) > 0 is provided below. THEOREM 4 .1. Let E(ln cl)+ < 00. Suppose there exists a probability distribution 1r on [0, (0) that is stationary for {Xn}n~O and 1r(0, (0) > O. Then, i) E(lncl)- < 00, ii) I Ilng(x)I1r(dx) < 00, iii) E In Cl = - I In g(x) 1r(dx) and hence strictly positive. CORO LLARY 4 .1. If E In Cl S 0 then 1r == 80 , the delta measure at 0 is the only stationary distribution for {Xn} n~O ' Furthe r, X n converges to 0 w.p.1 if E In Cl < 0 and in probability if E In Cl = O. A sufficient condition is given below. THEOREM 4.2. Let D == sup xg(x) < 00. Let i) EllnCll O, ii) Ellng(C l , D)I < 00. Then, there exists a stationary distribution 1r for { X n} such that 1r(0, (0) = 1. For the logistic case this reduces to ElnC l > 0 and Elln(4 - Cdl < 00 and for the Ricker case to E In Cl > 0 and EC l < 00. For proofs of these and more results see Athreya (2004) . The stationary distribution is not unique, in gener al. For an example in the logistic case see Athreya and Dai (2002). Under some smoothness hypothesis on the distribution of Cl uniqueness does hold as will be shown in the next section. For some convergence results see Athreya (2004a,b).

6

K.B. ATHREYA

5. Harris irreducibility. DEFINITION 5.1. A Markov chain {Xn}n>O with state space (S, s) and transition function P(·, .) is Hams irreducible with reference measure cp on

(S, s) if i) sp in a-finite and ii) cp(A) > 0 ===} P(Xn E A

for some

n ~ 11 X o = x)

is

>0

for every x in S.

(Equivalently if there exists a er-finite measure sp on (S, s) such that for each x in S, the Green's measure G(x, A) == I:~=o P(Xn E AI X o = x) dominates cp.) If S = N, the set of natural numbers and P == ((Pij)) is a transition probability matrix and if Vi, j :3 nij E Pi;ij > 0 then {Xn } is Harris irreducible with the counting measure on N as the reference measure. An important consequence of Harris irreducibility is the following THEOREM 5.1. Let {Xn}n;:::O be Hams irreducible with state space (S, s), transition function P(·,·) and reference measure cp. Suppose there exists a probability measure 1r on (S, s) that is stationary for P. Then i) n is unique. ii) For any x in S , the occupation measures r n,x(A) = ~ I:~-l P(Xj E AI X o = x) converge to 1r(') in total variation. iii) For any x in S, the empirical distribution Ln(A) ~ I:~-l lA (Xj) -+ 1r(A) w.p.l (Px ) (when X o = x) for each A in s. iv) {Xn}n>O is Hams recurrent i.e. cp(A) > 0 ::::} P(Xn) E A for some";~ 11 Xo = x) = 1 for all x in S.

The Markov chain vacillating probabilist (Example 3.1) is not Harris irreducible but will be if Ei has a distribution that has an absolutely continuous compnent. It is also known that if s is countably generated then every Harris recurrent Markov chain with state space (S, s) is regenerative in the sense its sample paths could be broken up into a sequence of LLd. cycles as in the discrete state space case. For a proof of this and Theorem 5.1 see Athreya and Ney (1978), Nummelin (1984), Meyn and Tweedie (1993). In the rest of this section conditions will be found for Harris irreducibility of {Xn}n;:::O defined by (1). Assume that {Cn}n;:::l are LLd. with values in (0, L), L :s 00 and for each c E (O,L), fc(x) == cxg(x) maps S = (O,k), k:S 00 to itself. For any function f : S -+ S the iterates of f are defined by

The first step is a local irreducibility result. THEOREM 5.2. Suppose i) :30 < a < 00, 8> 0, a Borel function lIt : J == (a - 8,a + 8) (0,00) -+ P(C1 E B) ::::: fBn] lIt(B)dB for all Borel sets B .

-+

NONNEGATIVE MARKOV CHAI NS WITH APPLICATIONS

ii) ::10 < P < 00, m f~m)(p) = p .

7

2: 1 su ch that for the fun ction f O;( x) == ax g(x ),

°-;

°

Then, ::11] > "Ix E I == (p -1] ,p + 1]) , Px(X m EA) > fo r all Bo rel sets A such that iP(A) == .\(A n 1) > where .\ is Lebesgn e m easure. COROLLARY 5. 1. Suppo se in additi on t o the hypotheses of Th eorem 5.1 , Px(Xn E I fo r some n 2: 1) is> for all x in (0, k) . Th en {X n }n2: 0 is Hams irreducible with st ate space S = (0, k). Using a deep result of Gu ckenheim er (1979) on S-unimodal m aps a sufficient condition for the hyp otheses of Corollar y 5.1 ca n be found. DEFINITION 5 .2. A ma p h : [0,1] -; [0,1] is S-unimodal if i) h( .) E C3, i. e. 3 times con ti nuous ly differentiable, ii) h(O) = h(l) = 0, iii) ::I < c < 1 ::1 h"(c) < 0, h is increasing in (0, c) an d decreasing in (c,l) an d

° °

°

' ) (S f )(X ) = - h"l(x) h' (X ) > Od zv h"(x) - '3( 2 hll(X h'(x))) 2'f Z an - 00 Z'f h' (x ) -- O

°

°

is < fo r all < x < 1. EXAMPLES. h( x) === cx (l - x ), 0 < c::; 4, h( x ) = x 2 sin JrX . DEFINITION 5 .3 . A number p in (0,1) is a stable periodic point fo r h if for som e m 2: 1: h(m )(p) = p and Ih(m)(p)1 < 1. DEFINITION 5.4 . For x in (0,1) th e orbit Ox is th e set {h (m)(x)}m 2:0 and w( x) is the limit point set of Ox. THEOREM 5.3 (Guckenheim er (1979)) . Let h be S-unimodal with a stable periodic point p . Let K = {x : < x < 1, w(x ) = w(p )} . Th en , .\(K) = 1 where .\( .) is the Lebesgu e m easure. Combining Theorem 5.2, 5.3 and Corollar y 5.1 leads to THEOREM 5.4 . Let S = [0,1] . A ssume i) "1 O < c < k , hc(x) == cx g(x ) is S -uni m odal. ii) ::I < P < 1, < a < L ::1 P is a st able peri odic point fo r hO;(x) == axg(x ). iii) ::I 8 > 0, a Bo rel fun ction Ill : J == (a - 8,a + 8) -; (0,00) ::1 P(C 1 E B) 2: f Bn J III(B)dB fo r all Borel sets B . Th en , th e Markov chain {X n} n2:0 defin ed by

°

°

°

n=0 ,1 ,2 , . .. where {Cn} n2:1 are i.i.d . is Ham s irreducibl e with state space (0,1) referenc e m easure cP(·) = .\(. n 1) where I = (p -1], p + 1]) for some approp riate 1] > 0. As a special cas e applied to random logistic map s one gets THEOREM 5 .5. Let S = [0,1]' let {Cn} n2:1 i.i.d. (0,4] valued r.v . and {X n}n2:0 be th e th e Markov chain defin ed by

n

2: 0.

Suppose ::I an open interval J C (0,4) and a function Ill : J -; (0,00) -; P(C1 E B) 2: fBnJ III(B)dB fo r all Borel sets B .

8 If J

K.B . ATHREYA

n (1,4)

Cl 3 f{3(x)

= ip , assume in addition, that :3 j3 > 1 in the support of == j3x(l - x) admits a stable periodic point p in (0,1). Then

{Xn}n>O is Hams irreducible . COROLLARY 5.2 . Suppose, in addition to the hypotheses of Theorem 5.5, that:3 InCI > 0 and Elln(4-CI )1< 00 . Then,:3 a unique stationary

measure 1r for {Xn } such that i) 1r(0 , 1) = 1, ii) 1r is absolutely continuous, iii) V 0 < x < 1, Px (X n E .)

--+

1r(') in total variation.

For proofs of all the results in this section except Theorem 5.1 see the Athreya (2003) . It has been pointed out by one of the referees that the above Corollary has been obtained independently by R.N. Bhattacharya and M. Majumdar in a paper entitled "Stability in distribution of randomly perturbed quadratic maps as Markov processes" , CAE working paper 0203, Department of economics, Cornell University. REFERENCES [1] ATHREYA K.B . (1996). The vacillating mathematician, Resonance. J . Science and Education, Indian Academy of Sciences , Vo!. 1, No. 1. [2] ATHREYA KB . (2003). Harris irreducibility of iterates of LLd. random maps on R+. Tech . Report, School of ORIE, Cornell University. [3] ATHREYA K .B . (2004a) . Stationary measures for some Markov chain models in ecology and economics. Economic Theory, 23 : 107-122. [4] ATHREYA K.B . (2004b) . Markov chains on Polish spaces via LLd. random maps . Tech. Report, School of ORIE, Cornell University. [5] ATHREYA K.B. AND DAI J.J. (2000). Random logistic maps I. J . Th. Probability, 13(2): 595-608. [6] ATHREYA KB . AND DAI J .J . (2002). On the nonuniqueness of the invariant probability for LLd. random logistic maps. Ann . Prob., 30: 437-442. [7] ATHREYA K.B . AND NEYP . (1978). A new approach to the limit theory of recurrent Markov chain . Trans. Am . Math. Soc., 245: 493-501. [8] ATHREYA K.B . AND SCHUH H.J. (2003). Random logistic maps Il, the critical case . J . Th. Prob., 16(4): 813-830. [9] ATHREYA KB . AND STENFLO O . (2000). Perfect sampling for Doeblin chains . Tech . Report, School of ORIE, Cornell University. (To appear in Sankhya, 2004). [10] BHATTACHARYA R.N . AND RAo B.V . (1993). Random iteration of two quadratic maps. In Stochastic processes: A. Fetschrift in honor of G. Kallianpur, pp . 1321, Springer. [11] BHATTACHARYA R .N . AND MAJUMDAR M. (2004). Random dynamical systems: A review . Economic Theory, 23(1) : 13-38. [12J BHATTACHARYA R.N. AND WAYMIRE E .C . (1999). An approach to the existence of unique invariant probabilities for Markov processes, colloq ium for limit theorems in probability and statistics. J . Bolyai Soc. Budapest. [13] BARNSLEY M.F. (1993). Fractals everywhere. Second edition, Academic Press, New York. [14] CARLSSON N. (2004). Applications of a generalized metric in the analysis of iterated random funct ions. Economic Theory, 23(1) : 73-84. (1980) . Iterated random maps on the interval as [15] DEVANEY R.L . (1989). An introduction to chaotic dynamical systems. 2nd edition, Academic Press, New York.

NONNEGATIVE MARKOV CHAINS WITH APPLICATIONS

9

[16] DE MELO W . AND VAN STRIEN S. (1993) . One dimensional dynamics. Springer. [17] DIACONIS P . AND FREEDMAN D.A . (1999) . Iterated random function . SIAM Review, 41 : 45-76. [18] GUCKENHEIMER J . (1987) . Limit sets of S-unimodal maps with zero entropy. Comm. Math. Physics, 110: 655-659. [19] HASSEL M.P. (1974) . Density dependence in single species populations. J . Animal Ecology, 44: 283-296. [20] KIFER Y. (1986) . Ergodic theory of random transformations. Brikhauser, Boston. [21] MAY R.M . (1976). Simple mathematical models with very complicated dynamics. Nature, 261: 459-467. [22] MEYN S. AND TWEEDIE R.L . (1993) . Markov chains and stochastic stability, Springer. [23] NUMMELIN E. (1984) . General irreducible Markov chains and nonegative operators. Cambridge University Press. [24] RICKER W .E . (1954) . Stock and recruitment. Journal of Fisheries Research Board of Canada, 11:559-623. [25] VELLEKOVP M.H . AND HOGNAS G . (1997) . Stability of stochastic population model. Studia Scientiarum Hungarica, 13 : 459-476. [26J WEI BIAO Wu (2002). Iterated random functions : Stationary and central limit theorems. Tech . Report , Dept. of Statistics, University of Chicago.

PHASE CHANGES WITH TIME AND MULTI-SCALE HOMOGENIZATIONS OF A CLASS OF ANOMALOUS DIFFUSIONS* RABI BHATTACHARYA t Abstract . Composite media often exhibit multiple spatial scales of heterogeneity. When the spatial scales are widely separated, t ransport through such medi a go through distinct phase changes as time progresses. In the pres ence of two such widely separated scales, one local and one large scale , the time scale for t he appearance of the effects due to the large scale fluctuations is det ermined. In the case of t ransport in period ic media with such slowly evolving heterogeneity and divergence-fr ee velocity fields , there is a first Gaussian phase which breaks down at the above t ime scale, and a second Gaussian phase occurs at a later time scale which is also precisely determined. In between there may be non-Gaussian phases, as shown by examples. Dep ending on the structure of the large scale fluctuations , the diffusion is either super-diffusive, with the effective diffusivity increasing to infinity, or it exhibit s normal diffus ivity which increas es to a finite limit as time increases. Sub-diffusivity, with the effect ive diffusion coefficient tend ing to zero in time, is shown to arise in a cert ain class of velocity fields which are not divergence-free.

1. Introduction. Electric and thermal conduction in composite media as well as diffusion of matter through them are problems of much significance in applications (see, [5-7, 16, 21]) . Ex amples of such composite media are natural heterogeneous material such as soils, polycrystals, wood , animal and plant tissue, cell aggregates and tumors , and synthetic products such as fiber composites, cellular solids , gels, foams, colloids, concrete, etc . The evolution equation that arises in such conte xt s is gener ally a Fokker-Planck equation of the form

(1.1)

ac(t, y) 1 at = 2" \7 . (D(y)\7c) - \7 . (v(y) c),

c(O, .) = Ox

where D( ·) is a k x k positive definite matrix-valued function depending on local properties of the medium, and its eigenvalues are assumed bounded away from zero and infinity; v(·) is a vector field which arises from other sources. To fix ideas one may think of v(·) as the velocity of a fluid (say, water) in a porous medium (such as a saturated aquifer) in which c(t , y) is the concentration of a solute (e.g., a chemical pollutant) injected at a point in the medium ([12 , 16,21 ,25,31 ,36,38]) . One may also think of (1.1) as the equation of transport, or diffusion, of a substance in a turbulent fluid ([1,3,35]). One of the main aims of the study of t ransport in disordered media is to derive from the local , or microscopic, Equation (1.1) a ma croscopic equation with const ant coefficients governing c over much larger space/time *Research supported by NSF Grant DMS-OO-73865. tDepartment of Mathematics, Univ ers ity of Arizona, ([email protected]) . 11

Tu cson ,

AZ

85721

12

RABI BHATTACHARYA

scales , under appropriate assumptions. Such a derivation is known as hom ogenization in par tial differential equations. The macroscopic equation is then of the form ac(t, y) = ~ ~ D . . a _ ~ e. ac at 2 L...J t ,) ay . ay . L...J t ay. ' 2c

(1.2)

i,j=l

i= l

t)

t

where D = (Di,j) is the effective dispersion or , diffusivity. This program has been carried out in complete generality for periodic D (·), v( ·) in Bensoussan et al. (1978) (also see [1, 2, 8, 23, 30, 38]). Another popular model assumes D (·), v (·) are stationary ergodic random fields ([1, 2, 7, 23,38]). P apanicolaou and Varadhan (1980) and Kozlov (1979) independently derived homogenizations when (1.1) is in divergence form (i.e., v(·) = 0 in (1.1)) . For a class of two-dimensional problems in such random media wit h D (·) = D constant and v(.) a (divergence free) shearing motion, a der ivation of homogenization and analysis of asymptotics is carried out in Avellaneda and Majda (1990), (1992) (also see [1]). From a probabilistic point of view, homogenization of (1.1) in the form (1.2) means t hat a diffusion (Markov process ) X( ·) generated by A = ~\7. (D (.) \7) + v( ·) . \7 converges in law, under a scaling of time and space with properly large units , to a Brownian motion WO with (constant) diffusion matrix D and (constant) drift velocity vector v: (1.3)

cX(!-) - ! v --. W(t) , c;2

E

(t

~

0),

as s ]

o.

It is known t hat if the coefficients are periodic, or stationary ergodic random fields, and v(·) is divergence free, the effective diffusivity is larger than the average of the local diffusivity D(· ). We have so far considered homogenization und er a single scale of het erogeneity. Natural composite media generally exhibit multiple scales of het erogen eity, i.e., heterogeneity th at evolves with distance. It has been observed in many instances, and sometimes verified theoretic ally, that this often leads to increase in the effective dispersivity D with the spatial scale, say, L. For t he case of solute dispersion in porous media, such as saturated aquifers, one may see this by int roducing a scale parameter in v(·), or by relating D to the correlation length, and still using a single large scale ([13, 23, 38]). Our objective in the present survey is to introduce different widely separated spatial scales of heterogeneity explicitly in the model and study (i) the effective diffusivity as a funct ion of the spatial scale, and (ii) the time scales for the different (Gaussian and non-G aussian ) ph ases t he diffusion pass es through as time progresses. In the next sect ion we give a fairly complete description of this for the case of periodic coefficients and a divergence free velocity field v(·) with two widely separ at ed scales- a local scale and a large scale . The case of additional appropriately widely separated

PHASE CHANGES WITH OF A CLASS OF ANOMALOUS DIFFUSIONS

13

scales may be understood from this. Examples in Section 4 illustrate the emergence of non-Gaussian phases in between Gaussian ones. Before concluding this introduction, let us mention the classical work of Richardson (1926) who looked at already existing data on diffusion in air over 12 or so different orders of spatial scale, and conjectured that the diffusivity DL at the spatial scale L satisfies

(1.4) This was related later by Batchelor (1952) to the turbulence spectrum v ex L 1/ 3 derived by Kolmogorov (1941). The length scale L(t) and the diffusion coefficient DL(t), as functions of time t, are now related using L(t) as the root mean squared distance from the mean flow (see Ben Arous and Owhadi (2002)): L2(t) ex DL(t)t ex L4/3(t)t, leading to L(t) ex t 3/ 2 and DL(t) ex t 2. This was also derived by Obhukov (1941) by a dimensional argument similar to that of Kolmogorov (1941). In particular, DL(t) ---+ 00 as t ---+ 00 , that is, this is a case of super-diffusivity. For a precise analysis of a two-dimensional model with constant D( .) = D and a stationary ergodic v( ·), we refer to Avellaneda and Majda (1990), (1992). 2. A general model with two spatial scales: The first phase of asymptotics and the time scale for its breakdown. Consider the general model (1.1) with v(·) of the form

v(y) = b(y) + I'(~) ,

(2.1)

where a is a large parameter, b(.), and 1'( -!a) represent the local and large scale velocities, respectively. The solution to (1.1) is the fundamental solution p(t;x, y) . Consider a diffusion X(t) , t 2 0, on R k with transition probability density p, starting at x = X (0). To avoid the artificial importance of the origin, take the initial point x to be

x = axo

(2.2)

where Xo is a given point in Rk, so that the initial value of I'Ua) is I'(xo). One may represent such a diffusion as the solution to the stochastic integral equation

X(t)

axo + it {b(X(S)) + d(X(s)) + 1'( X~s)) }dS

(2.3)

+ it a(X(s))dB(s) , where a(x) JD(x), d(x) (d1(x) , .. . ,d k(x))', dj(x) L.i(fJ/fJXi) Dij(x), and BC) is a standard k-dimensional Brownian motion. Since I'(-!a) changes slowly, at the rate of s.]«, one expects that for

14

RABIBHATTACHARYA

an initial period of time the process X (.) will behave like the diffusion Y(.) governed by

Y(t) =

axo + it {b(Y(s))

+ d(Y(s)) }ds + t,8(xo)

(2.4)

+ it (1(Y(s))dB(s) . Indeed, the £i-distance between p(t;x, y) and the tr ansition density q(t;x, y) of Y (t) is negligible for t he times t « a2 / 3 . Actually, the total variation distance l!Poot - QO,tllv between the distributions Poot of the process {X (s) : 0 'S s 'S t} and the distribution QO,t of the process {Y(s) : 0 'S s 'S t} goes to zero in this range . More precisely, one has the following result obtained in [12] (also see [9]). THEOREM 2.1. Assume b(.) and its first order derivatives are bounded, as are D( ·), ,8(.) and their first and second order derivatives. Assume also that the eigenvalues of D( .) are bounded away from zero and infinity. Then

l!Poot -

(2.5)

QO,t Ilv-----. 0

as

t

a2 / 3

-----.

O.

Proof By the Cameron-Martin-Girsanov Theorem (Ikeda and Watanabe (1981), pp. 176-181) ,

Z(t) (2.6)

:=

it (1-1(Y(S)){ ,8(Y~S)) ,8(Y~O)) _~ it 1(1- 1(Y(S)){ ,8(Y~s)) ,8 ( Y~O) ) }1 -

}dB(S)

_

2

ds.

Since Eexp{Z(t)} = 1, Ell - exp{Z(t)}1 = 2E(1 - exp{Z(t)})+ 'S 2[EIZ(t)1 /\ 1]. Now the expected value of the second integral in (2.6) can be shown, using Ito 's Lemma ([26]), to be bounded by [C1t2j a2 + C2t3j a2+ c3t3ja4]jA where A is the infimum of all eigenvalues of D(.), and C1, C2,C3, depend only on the upper bounds of the components of b( ·), ,8(.), D( ·) and of their first order derivatives, and also of the second order derivatives of ,8(.). Since the expected value of the square of the norm of the stochastic integral equals the expected value of the Reimann integral of the squared norm of the integrand, one has 1 1/2 l!Poot - QOotllv 'S (} + 2(}

where () = [C1t2 ja 2 + C2t3j a2 + C3t3 j a4]jA. 0 One may show by examples (see Section 4) that the large scale fluctuations (namely, fluctuations of ,8Ua)) can not be ignored in general for times t of the order a 2 / 3 or larger, i.e., the time scale in (2.5) is precise.

PHASE CHANG ES WITH OF A CLASS OF ANOMALO US DIFFUSIONS

15

Theorem 2.1 implies that a first homogenization occurs for times 1 « t « a 2/3, provided y( .) defined by (2.4) is asymptotically Gaussian. This is the case, e.g., if b(·), D( ·) are period ic, or are ergodic random fields satisfying some additional condit ions ([1- 3, 14, 34,38]). No assumption is needed on f3 (.), except the smoothness and boundedn ess conditions impos ed in Theorem 2.1. To illustrat e thi s, let b(.) and D( ·) be periodic with the same period lattice, say, and assume for simplicity that

zr.

(2.7)

divb(·)

= o.

Then, by Bensoussan et al. (1978) (or, Bhattachar ya (1985) ), and Theorem 2.1, one has

(2.8)

lim

oo, a ----> 00 , and THEOREM 3 .1 ([9, 10 , 12]) .

t

(3.15)

2 a

----> 00 ,

one has

where [... ]i denotes th e first p coordinates of the vector inside [...], and I p is the p x p iden tity m atrix. REMARK 1. Suppose t sat isfies (3.15) and t = 0(a 2 +8 ) for some 8 > O. 1

Then a = O( t (2+, < 11 >, .. .},

SEMI-MARKOV CASCADES AND NAVIER-STOKES EQUATIONS

35

v = (2 , 1 ,2, . .. ) E 8V

( 11 2)

( 2 12)

( 11 )

( 2 1)

( 1)

( 2 2 2)

(22 )

( 2)

B FIG. 1. Full binary tree with index set V and boundary 8V . The path v = (2,1,2, . . .) E 8V is indicated in bold, with vlO = B, viI = (2), vl2 = (21), and vl3 = (212).

where {I, 2}O = {O} . Also let aV = n~o{1, 2} = {I , 2}N. A stochastic model consistent with (2.1) is obtained by consideration of a multitype branching random walk of nonzero Fourier wavenumbers ~ , thought of as particle types , as follows: A particle of type ~ =1= initially at the root 0 holds for a random length of time SIJ distributed according to p(~ , dt). When this clock rings, a coin /'i,1J is tossed and either with probability ~ the event [/'i,1J = 0] occurs and the particle is terminated, or with probability ~ one has [/'i,1J = 1], the clocks are re-set and the particle is replaced by two offspring particles of types TJ , ~ - TJ selected according to the probability kernel q(~ , dTJ) . This process is repeated independently for the particle types TJ and ~ - TJ rooted at the vertices < 1 >, < 2 >, respectively. A more precise description of the stochastic model requires more notation. For v = (Vl ,V2 , ,, . ,Vk) E V, let Ivl = k, 101 = 0. For v = (Vl,V2, .,,) E aV, and j = 0,1,2 , . " let vlj = (Vl ,,,.Vj) , vlo = e. That is, for v E aV, vlO, vll , vI2, ... may be viewed as a path through vertices of the tree starting from the root vlo = e. For u , v E aV, let [u /\ vi = inf {m 2': 1 : ulm =1= vim}. The following requirements provide the defining properties of the underlying stochastic model. The model depends on the initial frequency (wave number) ~ . For fixed ~ =1= let {(~v , /'i,v) : v E V} be the tree-indexed (discrete parameter) Markov pro cess starting at (~IJ , /'i,1J) with ~IJ = ~, /'i,1J E {a, I}, taking values in the state space (R3 \{0}) x {a, I}, and defined on a probability space (0, F , Pe) by the following properties:

°

°

36

RABI BHATTACHARYA ET AL .

1. Pe (~o E B , KO= K) = !8dB ), BE

e;

K E {O, I}. 2. For any fixed v E av, (~vlo , Kvlo) , (~Vl l' Kvp), (~vI2 ' KVI2) " .. is a Markov chain wit h transit ion probabilities

(2.17)

Pdevln+l E B , Kvln+l

1

= Kla( {(eu ,Ku) : lul ::; n})) = "2 q(e, B )

for B E B(R3\{0} ), K E {0,1} . 3. For any u , v , E av , {(eulm, Ku lm )}~=o and {(~v l m , Kvlm)}~=o are conditionally independent given a( {(~w , Kw) : [w] ::; [u 1\ v i}). 4. For v E V, ~v1 + ~v2 = ~v Pe - a.s., where vj = (V1.. . vn)j := (V1.. .Vn, j), j = 1,2 , . .. is the concatenation operat ion. 5. Let {Sv : v E V} be a collection of non-negative rand om variables such that for each m 2=: 1, conditionally given eo = and et := {ev : v E v ,lvl 2=: I} , the random variables Sv,v E V are independent with respective (conditional) margin al distributions p(ev, ds). Our objective now is to use the stochastic branching model to recursively define a random funct ional related to (2.1) through its expected value. By a backward recursion one may define a (non-random) function !(z , z+, s, s+,K,K+,t ) where z E Rk\{O} , z+ E (Rk\ {O})v+,s E [0, 00), s+ E [O,oo)v+,'and K E {1,2} , K+ E {1,2} v+, where

e

V+ := {v E V : Ivl 2=: I} ,

(2.18) such t hat

! (Z, Z+, S, S+,K, K+, t ) = a(z ,t )l [t ,oo)(s) + l [o,t)(s)l{1} (K)b(z, t - s) . ! (Zl, zt , Sl , st, K1,Kt, t - s) 0 z ! (Z2, zt, S2 , st, K2, Kt ,t - s) + l [o ,t)(s )l{o}(K)c(z , S,t - s), where for x E SV we define (2.19)

(2.20)

In the event that

K v = 1 for all v the recursion may not terminate. In this case one simply defines ! (z , z", s, s+, K, K+,t ) == 0). Otherwise the backward recursion is sure to termin ate and! is well-defined. For this given functio n ! let us now define a random functional of the random fields {~v : v E V }, {Kv : v E V} , and {Sv : v E V} defined on (0, F , P ) for wE 0 by t he composition

(2.21)

X(B, t)(w)

:= ! (~o (w ) , ~t (w ) , So(w), s t

(w), KO(W), Kt(W), t ).

SEMI-MARKOV CASCADES AND NAVIER-STOKES EQUATIONS

37

A careful formulation and details of a proof is given in a PhD thesis by Orum (2004) which yields the following: THEOREM 2.1. If EIX (0,t)\ < 00 then a solution of (2.1) is given by x(~ , t) :=

Ef.o=f.X (0, t) .

We will conclude this section with an application to local existence theory for Navier-Stokes via a semi-Markov cascade representation. Taking a constant failure rate >.(~, t) = 6 > 0, this includes a special case obtained by Orum (2004) of local existence having exponentially distributed holding times. THEOREM 2.2. Assume that there is a 0 < T* :::; 00 such that for all ~ E Rk\{O}, max{la(~,t)l, Ib(~,t)l,suPo< s 0,

one obtains a local solution to (FNS) in

F h o,r ,O,T •.

SEMI-MARKOV CASCADES AND NAVIER-STOKES EQUATIONS

39

3. Time-asymptotic steady state solutions. In this section we compute time-asymptotics under supplemental conditions for the existence of a unique global solution to 3-d incompressible Navier-Stokes equations. For this we begin with the following theorem quoted from Bhattacharya et al. (2003) for ease of reference. THEOREM 3 .1. Let h (~ ) be a standard maj orizing kern el with exponent B = 1. Fix 0 < T ~ +00. Suppos e that luol h .o . T ~ (/2ii) 311 / 2 and I(- ~) -1 gl.1"h .O . T ~ (/2ii)3 112 /4. Th en there is a un ique solution u

in the ball Bo(O ,R) cen tered at 0 of radius R = (/2ii )311/2 in the space Moreover the Fouri er transform of the solution is given by u(~ , t) =

Fh ,O ,T .

h(~)Ef.X (T(~ , t)) , ~ E W~3) and t ~ T . As an immediate consequ ence one readily obtains st eady-st at es as follows. COROLLARY 3 .1. Under the conditions of Th eorem 3.1 with T = 00, suppose further that limt->oo 9(~ , t) = 900(~) exists for each ~ -1= O. Then Uoo(~) :=

lim u(~ , t)

t-s- co

exists and satisfies the st eady state Na vier-Stokes (FNS)oo defin ed by

Uoo(O

= t OO e-VIf.12S { I~I (27r) -~ ( uoo (1])

lo

lR3

® f.

uoo (~ -1] )d1] + 900(0 }ds.

Proof. Note that the und erlying discrete par am et er binary branching is critical. Thus limt->oo X(B, t) exists a.s. as a finit e random product. Moreover, under the conditions of Theorem 3.1 one has for each t ~ 0, with probability one IX(B, t)1 ~ 1. Thus, by Lebesgue's Dominated Convergence Theorem Xoo (~ ) := lim X(~ , t)

t-+oo

exists for each Z. Now again apply Dominated Convergence to (FNS)h to obtain

Multiplication through by h(~) proves the assertion for uoo(~) 2g oo (f.) ( C) -- vlf.12h(f.)· an d IIL = sup { 14>(x)1 , 14>(x) - 4>(y)I} Ilx-yll ' x ,y E'H

Le. one has

(5.3) This will give us convergence for prob abilities and , as we have uniform bounds on arbitrary moments (cf. Theorem 3.2), convergence of moments and other statistical quantities. The second norm is the total variation norm 1 ·IITv, which is defined as the dual norm to th e Loo-norm. Since 11 4>IIL 2 11 4>1100 , the total variation norm is stronger th an th e Wasserstein norm. Note that the Wasserstein norm depends strongly on the metric that equips the underlying spa ce, whereas the total variation norm is indep endent of that metric. For exa mple, the Wasserstein norm between two Dirac measures 8x and 8y is given by min{l , Ilx - YII}, whereas 118x - 8yll T V is given by 1 if x -=1= y and 0 otherwise. (Actually, one can show that the total variation norm is equal to

54

DIRK BLOMKER AND MARTIN HAIRER

the supremum over all possible metrics of the corresponding Wasserstein norms. See [38] for a beautiful discussion of the relationship and properties between various metrics on probability measures .) Before we state our results, we introduce one more notation. Similar to the proof of Theorem 4.2, we will rescale the solutions of (2.1) by e such that they are concentrated on a set of order 1 instead of a set of order c. Furthermore, we will rescale the equation to the slow time-scale T = tc 2 . We denote by J.L; the invariant measure of the rescaled version of (2.1). We furthermore denote by the invariant measure for the pair of processes (a, c'ljJ). Note that v; depends on e by the rescaling of 'ljJ and by the fact that equations (2.4) and (2.5) are coupled through the noise, but do not live on the same time scale. However, the marginal of on N is independent of e and its marginal on S depends on e only through the trivial scaling of c'ljJ. We denote these two marginals by and With these notations, our result in the Wasserstein distance is the following: THEOREM 5.1. Let Assumptions 1, 3, 4, and 6 hold. Then, for every K > 0, one has

v;

v;

vz

v:.

(5.4) 5.2. Actually, one also has IIJ.L; - v;IIL = O(c 2- K ) , but the above formulation is mo re interesting, since vZ and v: can be characterised explicitly, whereas v; can not, unless the covarian ce operator Q is blockdiagonal with respect to the splitting 1-£ = N EB S . Idea of proof Denote by Qt the Markov transition semigroup (acting on measures) associated to the rescaled version of (2.1), and by Pt the transition semigroup associated to the evolution of (a(t) ,c'ljJ(c-2t)) . Then, the main ingredient for the proof of Theorem 5.1 is that there exists a time T such that, for every pair (J.L, v) of probability measures with finite first moment, one has REMARK

In order to prove (5.5), one uses the strong contr action properties of the linear dynamic in S and that the strong mixing properties of the nondegenerate noise in N . Once (5.5) is established, the proof of Theorem 5.1 follows in a rather straightforward way. One first obtains from Theorem 3.4 that IIJ.L~

- v;IIL :::;

+ IIPTJ.L~ - PTv;liL - 1/; IlL + O(c 2 ) ,

IIQTJ.L~ - PTJ.L~IIL

:::; O(c 2 -

K )

1 + 211J.L~

and therefore IIJ.L; -v;IIL = O(c 2 - K ) . The bound 111/; -vZ0v:liL = O(c 2 - K ) is then obtained by using the smoothness of the density of vZ with respect to

AMPLITUDE EQUATIONS FOR SPDE

55

the Lebesgue measure, combined with the separation of time scales between 0 the dynamics on N and on S. The first result in the total variation norm only considers the marginals of the invariant measures on N. THEOREM 5.2 . Let Assumptions 1, 3, 4, and 6 hold. Then, for every r: > 0, one has

(5.6) Idea of proof We combine the smoothing properties of PcPtPc with the result previously obtained in Theorem 5.1 to show that

(5.7)

v:

C

2-1
0 such that, for all , E [0, 'a], :F : (11.,)3 ~ 11.,-0: and A : 11.' ~ 'H'-O: are continuous. Furth ermore, the operator Q-1 is con tinuous from 11.'0-0: to 11. and for som e a E [O,~) we have 11(1- L)'o- ciQIIHs(x) < 00 . THEOREM 5 .3 . Let Assumptions 1, 3, 4, and 7 hold. Th en, for every r: > 0, one has

'A

(5.10)

56

DIRK BLOMKER AND MARTIN HAIRER

Idea of proof We denote by to the linear system (5.11)

i\

the transition probabilities associated

du = c- 2Ludt + QdW(t) .

It is then possible to show as above by Girsanov theorem that (5.12) Furthermore, the fast relaxation of the S-component of the solutions to (5.11) toward its equilibrium measure, combined with the fact that the N marginals of J..t; and of IJ; are close by Theorem 5.2, allows to show that II'PTJ..t; - IJ~ Q9 1J: IITV :S Cc, provided that T» c2 • The result then follows 0 by choosing T of the order c 2 - o for some small value of 8. 6. What is so special about cubic nonlinearities? Cubic nonlinearities are not special, we can extend the method to a lot of different types of nonlinearities. Suppose we have a multi-linear nonlinearity, which is homogeneous of degree n. Then the noise strength in the SPDE (2.1) should be changed to c(n+l) /(n-l) instead of c 2. Now with the ansatz

and a similar formal calculation as in section 2, we derive the amplitude equation

which now contains also a nonlinearity that is homogeneous of degree n. We can verify this result rigorously. After minor changes the local theorems immediately carry over to these kinds of equations. For example for stable odd nonlinearities at least the order 1 approximation (local and global) is completely analogous . The local approximation results also carry over to even nonlinearities, but one problem for global results is that we do not have nonlinear stability of the equations. In some cases, we can however get global results for even nonlinearities, if we already have good a-priori bounds for the solutions. But the main problem with quadratic nonlinearities B (u) = B (u, u) is that in many examples PcB(a) == for a E N. In this case, the previously mentioned result will give us only the linearisation, meaning that we still look at solutions that are too small to capture the nonlinear features of the equation. To illustrate this problem , we will briefly discuss Burgers equation, which is given by OtU = o;u + J..tU - uOxu + (j€~ . For periodic boundary conditions and J..t = O(c 2 ) we get N = span{l} but now already B(l) = 0. If we consider Dirichlet boundary conditions on [0,11"], for example, then the linear instability arises for J..t = 1 + O(c 2 ) .

°

57

AMPLITUDE EQUATIONS FOR SPDE

Furthermore, N = span{sin} and Be(sin) = 0, where we used the shorthand notation B; = PeB and B, = PsB. There are numerous examples in the physics literature of equations with quadratic nonlinearities and the same property, as described above. One example is the growth of rough amorphous surfaces. See for example [6] and the references therein. Another example is the celebrated KuramotoSivashinsky equation, but the probably most important example is the Rayleigh-Benard problem (see e.g. [23] or [15]) which is the paradigm of pattern formation in convection problems . If we want to take into account nonlinear effects, we then have to look at the coupling of the slow dominant modes to the fast modes. This was done in [8] for the local result. Let us now briefly comment on these results. Consider an equation of the type

with Be(a,a) = 0 for a EN, where B is symmetric and bilinear. We make the ansatz

with a E N (£) and 'l/J E PsX. This yields in lowest order in e the following system of formal approximations. First of order 0(£2) on the fast timescale t in PsX.

Secondly of order (6.2)

£3

Bra(T)

=

in N (£) on the slow time-scale T = £2t

Aea(T) + 2Be(a(T ),'l/J(e- 2T» + Pe€(T),

where f.(T) = £-1~(£-2T) is a rescaled version of the noise. These equations are on one hand a dominating equation (6.2) on a slow time-scale coupled to an equation (6.1) on the fast time-scale. Equations with a similar structure are treated in [3] for stochastic ODEs, or in [16, 17] where tracers in a fast moving velocity field are considered . The aim is now to get an effective equation for the slow component completely independent of the fast modes. First rescale (6.1) to the slow time-scale T by 'l/J(t) = (£2t). Hence,

As L; is invertible on PsX, we get in lowest order of e that (T) = -£;lB s(a(T),a(T)) . This together with (6.2) establishes a single approximation equation.

(6.3)

fJra(T) = Aea(T) - 2Be (a(T), £-;1e, (a(T), a(T)) )

+ pi(T) ,

58

DIRK BLOMKER AND MARTIN HAIRER

Surprisingly, this equation involves a cubic nonlinearity, although the nonlinearity in the original equation was quadratic. The main results of [8] show that these formal calculations can be made rigorous in the sense of Theorems 3.1 and 3.3.

REFERENCES [1] L. ARNOLD, Random dynamical systems, Springer Monographs in Mathematics. Springer, Berlin, 1998. [2] B . AULBACH, Continuous and discrete dynamics near manifolds of equilibria, Lecture Notes in Math., 1058, Springer, Berlin, 1984. [3] N. BERGLUND AND B . GENTZ, Geometric singular perturbation theory for stochastic differential equations, J . Differential Equations 191(1): 1-54 (2003) . [4] D . BLOMKER, Amplitude equations for locally cubic non-autonomous nonlinearities, SIAM J. Appl. Dyn. Sys., 2(2) : 464-486 (2003). [5J D . BLOMKER, S . MAIER-PAAPE, AND G . SCHNEIDER, The stochastic Landau equation as an amplitude equation , Discrete and Continuous Dynamical Systems, Series B , 1(4) : 527-541 (2001). [6] D. BLOMKER, C .GUGG, AND M . RAIBLE, Thin-mm-growth models: roughness and correlation functions, European J . Appl. Math., 13(4): 385-402 (2002). [71 D . BLOMKER AND M. HAIRER, Multiscale expansion of invariant measures for SPDEs, to appear in Commun. Math. Phys., 2004. [8] D . BLOMKER, Approximation of the stochastic Rayleigh-Benard problem near the onset of instability and related problems, Submitted for publication, 2003. [9] D . BLOMKER, M . HAIRER, AND G . PAVLIOTIS, Stochastic amplitude equations in large domains, In Preparation, 2004. [lOJ Z. BRZEZNIAK AND S . PESZAT, Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process , Studia Math. 137(3): 261-299 (1999) . [l1J Z. BRZEZNIAK AND S . P ESZAT, Strong local and global solutions for stochastic Navier-Stokes equations . Infinite dimensional stochastic analysis (Amsterdam, 1999), pp. 85-98, Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. , 52, R . Neth. Acad. Arts ScL , Amsterdam (2000). [12] P . COLLET AND J .P . E CKMANN, Instabilities and fronts in ext end ed systems, Princeton Univ. Press, Princeton, NJ, 1990. [13] P . COLLET AND J .P. ECKMANN, Th e time dependent amplitude equation for the Swiit-Hohenoetg problem , Comm. Math. Phys., 132(1): 139-153 (1990) . [14] H . CRAUEL, A . DEBUSSCHE, AND F . FLANDOLI, Random attractors, J . Dynam. Differential Equations, 9(2): 307-341 (1997). [15] M .C . CROSS AND P .C . HOHENBERG, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 : 851-1112 (1993). [16J G . PAVLIOTIS AND A . STUART, White noise limits for inertial particles in a random field. Preprint (2003). [17] G . PAVLIOTIS AND A . STUART, Ito versus Stratonovich white noise limits. Preprint

(2003). [18] G . DA PRATO AND J . ZABCZYK, Stochastic Equat ions in Infinite Dimen sions, Cambridge University Press, 1992. [19] G . DA PRATO AND J . ZABCZYK, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Note Series, 229, Cambridge University Press (1996). [20J J . Du AN AND V .J. ERVIN, On nonlinear amplitude evolution under sto chastic forcing, Appl. Math. Comput. 109(1): 59-65 (2000). [21J J. DUAN, K. Lu , AND B . SCHMALFUSS , Invariant manifolds for stochastic partial differential equations , The Annals of Probability, to appear.

AMPLITUDE EQUATIONS FOR SPDE

59

[22] W. E AND D. LIU, Gibbsian dynamics and invariant measures for stochastic dissipative PDEs, Journal of Statistical Physics, 108: 4773-4785 (2002) . [23] A.V . GETLlNG, Reyleigh-Betierd Convection - Structures and Dynamics, World Scientific Press, 1998. [24] M. HAIRER AND J .C . MATTINGLY, Ergod icity of the 2D Navier- Stokes Equations with Degenerate Sto chastic Forcing. Preprint, 2004. [25] H. HAKEN, Synergetics. An introduction. Nonequilibrium phase transitions and self- organization in physics, chemistry, and biology, Springer Series in Syn ergetics, Vol. 1. Berlin et c.: Springer, 1983. [26] D. HENRY, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840. Berlin etc .: Springer, 1981. [27] P .C . HOHENllERG AND J .B . SWIFT, Effect s of additive noise at the onset of Rayleigh-Benard convection, Physical Review A, 46 : 4773-4785 (1992). [28] B.B . KING , O. STEIN , AND M. WINKLER, A fourth-order parabolic equation modeling epitaxial thin JJlm growth, J . Math. Anal. Appl. , 286(2) : 459-490 (2003) . [29] R. KUSKE, Multi-scale analysi s of noise-sensitivity near a bifurcation, Proceedings of the IUTAM Symposium held in Monticello, IL, USA, 26-30, August 2002 (eds . N. Sri Namachchivaya and Y.K. Lin) , Kluwer , 2002. [30J S.B . KUKSIN AND A. SHlRIKYAN , A Coupling Approach to Randomly Forced Nonlinear PDE's. I, Commun. Math . Phy s., 221 : 351- 366 (2001) . [311 A. LUNARDI, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and th eir Applications, 16 , Birkhauser Verlag, Basel , 1995. [32] J .C . MATTINGLY, Exponential convergence for the stochastically forced NavierStokes equations and other partially dissipative dyn amics, Commun. Math . Phys. , 230 : 421-462 (2002). [33] S.P . MEYN AND R.L. TWEEDlE, Markov Chains and Stochastic St ability, Springer , New York , 1994. [34] A. MIELKE, G. SCHNEIDER, AND A. ZIEGRA , Comparison of inertial man ifolds and application to modulated systems, Math. Nachr., 214 : 53-69 (2000). [35] A. MIELKE AND G . SCHNEIDER, Attractors for modulation equations on unbounded domains - exist ence and compari son, Nonlinearity, 8 : 743-768 (1995). [361 S. MOHAMMED , T. ZHANG, AND H. ZHAO, The Stable Manifold Theorem for Semilinear Sto chastic Evolution Equations and Stochastic Partial Differential Equations . Part I: The Stoch astic Semiflow , preprint (2003). [37] A . PAZY, Sem igroups of Lin ear Operators and Application to Partial Differential Equations, Springer, 1983. [38] S. RACHEV, Probability metrics and the stability of stochastic models, John WHey & Sons Ltd., Chichester, 1991. [39] M. SCHEUTZOW, Comparison of various concepts of a random at tractor: a case study, Arch . Math. (Basel) ,18(3) : 233-240 (2002). [40] G . SCHNEIDER, Bifurcation theory for dissipative systems on unbounded cylindrical domains-an introduction to the mathematical th eory of modulation equations, ZAMM (Z. Angew. Math. Mech.) 81(8) : 507-522 (2001). [41J B. SCHMALFUSS, Measure attractors and random attractors for stochastic partial differential equations, Stochastic Anal. Appl. , 11(6): 1075-1101 (1999).

ENSTROPHY AND ERGODICITY OF GRAVITY CURRENTS VENA PEARL BONCOLAN-WALSH', JINQIAO DUAN*, HONCJUN

cxot,

TAMAY OZCOKMENt , PAUL FISCHER§, AND TRAIAN ILIESCU'

Abstract. We study a coupled deterministic system of vorticity evolution and salinity transport equations, with spatially correlated white noise on the boundary. This system may be considered as a model for gravity currents in oceanic fluids . The noise is due to uncertainty in salinity flux on fluid boundary. After transforming this system into a random dynamical system, we first obtain an asymptotic estimate of enstrophy evolution, and then show that the system is ergodic under suitable conditions on mean salinity input flux on the boundary, Prandtl number and covariance of the noise . Key words. Random dynamical system, stochastic geophysical flows, enstrophy, climate dynamics, ergodicity. AMS(MOS) subject classifications. Primary 60H15 ; Secondary 86A05 , 34D35 .

1. Geophysical background. A gravity current is the flow of one fluid within another driven by the gravitational force acting on the density difference between th e fluids. Gravity currents occur in a wide variety of geophysical fluids. Oceanic gravity currents are of particular importance, as they are intimately related to the ocean's role in climate dynamics. The thermohaline circulation in the ocean is strongly influenced by dense-water formation that takes place mainly in polar seas by cooling and in marginal seas by evaporation. Such dense water masses are released into the large-scale ocean circulation in the form of overflows, which are bottom gravity currents .

We consider a two-dimensional model for oceanic gravity currents, in terms of the Navier-Stokes equation in vorticity form and the transport equation for salinity. The Neumann boundary conditions for this model involve a spatially correlated white noise due to uncertain salinity flux at the inlet boundary of the gravity currents. In the next section, we present the model and reformulate it as a random dynamical system, and then discuss the eocycle property and dissipativity of this model in §3 and §4, respectively. Main results on random 'Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA (bongvenat i t edu: duanlUiit.edu) . tDepartment of Mathematics, Nanjing Normal University, Nanjing 210097, China (gaohjenjnu . edu. en) . tRSMAS /MPO, University of Miami, Miami, Florida, USA (tamaylUrsmas .miami. edu) . §Argonne National Laboratory, Argonne, Illinois, USA (fiseherlUmes.anl.gov) . 'Mathematics Department, Virginia Tech, Blacksburg, VA 24061, USA (ilieseulUealvin .math . vt . edu) . i

61

62

VENA PEARL BONGOLAN-WALSH ET AL.

attractors, enstrophy and ergodicity are in §5. Enstrophy is one half of the mean-square spatial integral of vorticity. Ergodicity implies that the time average for observables of the dynamical system approximates the statistical ensemble average , as long as the time interval is sufficiently long.

2. Mathematical model. Oceanic gravity currents are usually down a slope of small angle (order of a few degrees) . We model the gravity currents in the downstream-vertical plane , and ignore the variability in the cross-stream direction. This is an appropriate approximate model for, e.g., the Red Sea overflow that flows along a long narrow channel that naturally restricts motion in the lateral planar plane [11] . In fact, we will ignore small slope angle and the rotation, both affect the following estimates nonessentially, i.e., our results below still hold with non-essential modification of constants in the estimates and in the conditions for the ergodicity. Thus we consider the gravity currents in the downstream-vertical (x, z )-plane. It is composed of the Boussinesq equations for ocean fluid dynamics in terms of vorticity q(x, z, t), and the transport equation for oceanic salinity S(x,z,t) on the domain D = {(x ,z): 0::; x,z::; I} :

qt + J(q ,'IjJ) =tJ.q - RaoxS, (1)

1

s, + J(S, 'IjJ) =Pr tJ.S,

where

q(x, z, t) = -tJ.'IjJ is the vorticity in terms of stream function 'IjJ , Pr is the Prandtl number and Ra is the Rayleigh number. Moreover, J(g, h) = gxhz-gzhx is the Jacobian operator and ~ = oxx + ozz is the Laplacian operator. All these equations are in non-dimensionalized forms. For the simplicity, we let Pr = 1. Note that the Laplacian operator tJ. in the temperature and salinity transport equations is presumably oxx+~152ozz with 8 being the aspect ratio, and KH, KV the horizontal and vertic~ diffusivities of salt, respectively. However, our energy-type estimates and the results below will not be essentially affected by taking a homogenized Laplacian operator tJ. = oxx + ozz. All our results would be true for this modified Laplacian. The effect of the rotation is parameterized in the magnitude of the viscosity and diffusivity terms as discussed in [19]. The fluid boundary condition is no normal flow and free-slip on the whole boundary

'IjJ = 0, q = O. The flux boundary conditions are assumed for the ocean salinity S . At the inlet boundary {x = 0, 0 < z < I} the flux is specified as:

(2)

oxS = F(z)

+ w(z, t),

ENSTROPHY AND ERGODI CITY OF GRAVIT Y CURRENTS

63

with F (z) being the mean freshwater flux , and the fluctuating par t w(z, t ) is usu ally of a shorte r t ime scale t han t he response t ime scale of t he oceanic mean salinity. So we neglect t he a utocorr ela t ion t ime of this fluctuating process and t hus ass ume that the noise is white in t ime. The spatially correlat ed white-in-time noise w(z , t) is desc ribed as the generalized t ime derivative of a Wi en er pro cess w(z, t) defined in a pr ob ability space (D, IF, JPl) , with mean vect or zer o and covar ia nce operator Q. On t he outlet boundary {x = 1, 0< z < I}:

At the t op boundar y z

= 1, and at t he

bottom bo undary z

= 0:

This is a system of deterministic partial differ ential equat ions with a stochastic boundary condit ion.

3. Cocycle property. In this section we will show that (1) has a unique solution, and by reformulating t he model, we see it defines a eo cycle or a random dynamical system. For t he followin g we need some tools from t he t heory of par ti al different ial equations . Let Wi (D) be t he Sobolev space of fun ct ions on D wit h first generalized der ivative in L 2 (D ), t he function space of square int egr able functions on D with norm and inn er produ ct 1

IIullL2 = ( l

lu (x )12dD)

2,

(u, V)L 2

=

1

u(x)v(x)dD,

u, v E L 2(D ).

The space Wi (D ) is equipped wit h t he norm

1;Jotivated by the zero-boundary condit ions of q we also introduce the space W~(D) which contains ro ughly spe aking func tions wh ich are zero on the boundary oD of D. This space ca n be equipped with the norm

(3) Simil arl y, we can define fun cti on spaces on t he int er val (0, 1) denoted by L 2 (0, 1). Another Sob olev space is given by Wi( D) which is a subs pace of Wi( D) consisting of fun cti ons h such t hat hdD = 0. A norm equivalent to t he Wi-norm on Wi (D ) is given by t he right han d side of (3) . For t he subs pace of functions in L 2(D ) having t his prop erty we will de note as £ 2(D ).

ID

64

V ENA PEARL BONGOLAN-WALSH ET AL.

We reformulate th e above stoc hastic initi al-boundary value problem into a random dynamical syste m [1] . For convenience, we introduce vector not ati on for unknown geophysical quantities u = (q, S).

(4)

Let w be a white noise in L 2 (0, 1) with finite trace of th e covariance operator Q, and t he Wiener process w(t) be defined on a probability space

(n, IF, JP). Now we choose an appro priate phase spa ce H for t his syste m. We assume th at t he mean salinity flux F E L 2(0 , 1). Not e that

:t

l

1 1

SdD

=

[F (z) + w(z, t) ]dz = constant .

It is reasonable (see [5]) to assume that

1 1

(5)

[F( z) + w(z, t)]dz = 0,

and thus JD SdD is constant in time and , without loss of generality, we may assume it is zero (otherwise, we subt ract thi s constant from S):

l.

SdD=O.

Thus S E L2 (D ), and we have t he usual Poincare inequality for S. Define the phas e space

We rewrite th e coupled syste m (1) as:

(6)

du dt + Au = F 1 (u) + F2 (u),

u(O ) = Uo E H ,

where

Au =

F1 (u)[x , z] =

-~q (

)

1

- - ~S

,

Pr

- J( q, 'I/J) ) [x , z], ( - J( S, 'I/J)

and

F2 (u)[x, z] = (

°

Ra(-oxS ) )

[x, z].

ENSTROPHY AND ERGODICITY OF GRAVITY CURRENTS

65

The boundary and initial conditions are: q = O,onoD ,

oS on (7)

=

0, on oD\{x = 0, 0< z < I} ,

oS

ox = F(z) +w(z,t), on {x = 0, 0 < z < I},

u(O) = uo = (

t ),

where n is the out unit normal vector of Bl), The system (6) consists of deterministic partial differential equations with sto chastic Neumann boundary conditions. We now transform the above system (6) into a system of random partial differential equations (Le., evolution equations with random coefficients) with homogeneous boundary conditions, whose solution map can be easily seen as a cocycle. Thus we can investigate this dynamics in the framework of random dynamical systems [1] . Note that we have a nonhomogenous stochastic boundary condition for salinity S , so the first step is to homogenize this boundary condition. To this end, we need an Ornstein-Uhlenbeck stochastic process solving the linear differential equation

(8) with the following same boundary conditions as for S, and zero initial condition

OxTJ1(t,0,Z ,W) = F(z)

+ w(z , t) ,

OxTJ1(t ,I,z,w) = 0, OzTJ1(t, x ,0,w) = 0, (9)

OzTJ1(t ,x,I ,w) =0, TJl(O ,X,Z,w) = 0,

1

TJ1dD = 0.

LEMMA 3.1. Suppose that the covariance Q has finite trace : tr L2 Q < Then the Ornstein-Uhlenbeck problem (8)-(9) has a unique stationary solution in L 2(D) generated by

00 .

66

VENA PEARL BONGOLAN-WALSH ET AL.

In fact , we can write down the solution Zabczyck [13, 12] as

(10)

T}l(t , x , z,w) = (-Ll)

l

T}l

following Da Prato and

t

S(t - s)N(X)ds

where I is the identity operator in L2(D) , and N is the solution operator to the elliptic boundary value problem Llh - )"h = with the boundary conditions for h the same as T}l, that is oh/an = X on aD with hdD = 0, where n is the unit outer normal vector to aD and X is

°

ID

x= ( F(ZlTZ,Sl ) .

Here X is chosen so that this elliptic boundary value problem has a unique solution. Since hdD = 0, we can choose X = 0. Moreover, S(t) is a strongly continuous semigroup , symbolically, eAt, that is, the generator of S(t) is Ll. Now we are ready to transform (6) into a random dynamical system in Hilbert space H. Define

ID

(11)

~l

T}(t ,x,z,w) = (

)

and recall

Let (12)

v:= U

-

T} .

Then we obtain a random partial differential equation

v(o) = Vo E H ,

(13)

where vx(t,O,z,w), vx(t ,I , z,w), vz(t , x ,O,w) and v z(t ,x,l,w) are now all zero vectors.i.e., homogenous boundary conditions, and the initial condition is still the same as for u

v(O ,x,z,w) = (

t: ).

However, because of the Jacobian, we will have to do nonlinear analysis on (13) to resolve v.

ENSTROPHY AND ERGODICITY OF GRAVITY CURRENTS

67

We introduce another space

For sufficiently smooth functions v = (ij, S) , we can calculate via integration by parts

(Au, V ) H =

(14)

1

\lq. \lijdD 1

r

-

+ Pr ) D \lS . \lS dD since now, we have only homogenous boundary conditions. Hence on the space V we can introduce a bilinear form a(.,.) which is continuous, symmetric and positive

a(u, v) =

1

\lq. \lijdD +

1vs vs.u:

This bilinear form defines a unique linear continuous operator A : V -; V' such that (Au, v) = a(u, v). Recall

Ft(u)[x, z] = (

=;~~,~~

)

[x,z]

and

) F2 (u)[x, z] = ( Ra(-oxS) 0 [x, z]. LEMMA 3 .2. The operator

Fi : V -; H is continuous. In particular,

we have

(Fl(U),U ) = O. Proof We have a constant

Cl

> 0 such that

(15) for any q E Wi (D) which follows st raight forwardly by regularity properties of a linear elliptic boundary problem. Note th at wt is a Sobolev space with respect to the third derivatives. Hence we get :

IIJ(S,1/')11£2::; sup (lox1/'(x,z)1 + loz1/'(x,z)l) x (x ,z )ED

X

(l'OxS(X, z)1

+ lozS(x,z)ldD) .

68

VENA PEARL BONGOLAN-WALSH ET AL.

The second factor on the right hand side is bounded by

On account of the Sobolev embedding Lemma, we have some positive constants C2, C3 such that sup (Iax 'l/J(x, z)/ + laz'l/J(x, z)l) :Sc211'V'l/J 1 1wi (D ) :Sc31Iqllwi(D) :Sc31Iullv. (x ,z)ED Hence we have a positive constant

C4

such that

for u E V . We now show that

(J(S, 'l/J ),S) =

o.

We obtain via integration by parts

LaxSaz'l/J SdD - LazSax'l/J SdD = - La;zS'l/J SdD + La;xS'l/JS dD -LaxS'l/JazS dD

+

razS'l/JaxSdD + r

}D

J(O ,I)

axS'l/JSI~~5dx -

r

J(O,I)

azS'l/JSI~~5dz = 0

because 'l/J is zero on the boundary Bl), This relation iso true for a set of sufficiently smooth functions 'l/J , S which are dense in W~(D) x Wi(D) . By the continuity of F I , as just shown in Lemma 3.2, we can extend this o . property to W~(D) x Wi(D). 0 LEMMA 3.3 . The following estimate holds

for some positive constant cs . Proof By simple calculation, the proof is obtained.

0 We have obtained a differential equation without white noise but with random coefficients. Such a differential equation can be treated samplewise for any sample w . We are looking for solutions in v E C([O, T]; H) n L 2(0, T; V) ,

for all T > O. If we can solve this equation then u := v + 7J defines a solution version of (6). For the well posedness of the problem we now have the following result.

ENSTROPHY AND ERGODICITY OF GRAVI TY CURRENT S

69

3.4 (Well-Posedness) . For any time r > 0, there exists a uni que solution of (13) in C( [O, r ]; H ) n £ 2(0, r ; V). In particular, the solution mapping THEOREM

IR+ x

nx

H 3 (t ,w , vo) ---+ v(t) E H

is measurable in its arguments and the solution mapping H 3 Vo ---+ v(t ) E H is continu ous. Proof By the properties of A and Ft (see Lemm a 3.2), t he random differenti al equation (13) is essentially similar to t he 2 dim ensional Navier Stokes equation. Note th at F2 is only an affine mapping. Hence we have 0 existence and uniqueness and the above regulari ty assertions. On account of t he transform ation (12), we find t hat (6) also ha s a uniqu e solution. Since the solution mapping IR+ x n x H 3 (t ,w, vo)

---+

v(t ,w , vo) = : 4?(t,w,vo ) E H

is well defined, we can introdu ce a random dyn amical system . On n we can define a shift operator ()t on th e paths of the Wiener process t hat pushes our noise:

w( ·,{hw) = w( · + t ,w) - w(t,w)

for t E IR

which is called the Wiener shift. Then {()t} tE IR forms a flow which is ergodic for the probability meas ure JP>. The properties of t he solution mapp ing cause the following rela tions

4?(t + r ,w ,u) = 4?(t , ().,w, 4?(r ,w ,u))

for

t, r 2: 0

4?(O,w, u) = u for any wE n and u E H. This prop erty is called t he eocycle proper ty of 4? which is import ant to study the dynamics of random syste ms. It is a generalization of t he semigroup property. The eocycle 4? toget her with the flow () forms a random dynamical system. 4. Dissipativity. In t his section we show t hat the random dynamical syste m (13) for gravity curre nts is dissip ative, in the sense t hat it has an absorbing (random) set . This means that the solut ion v is cont ained in a particular region of t he phas e space H after a sufficiently long tim e. This dissip ativity will help us to obtain asymptoti c est ima tes of the enst rophy and salinity evolution. Dynamical properties that follow from this dissipat ivity will be considered in the next sect ion. In par ticular , we will show that the system has a ra ndom at t rac tor , and is ergodic und er suit able conditions. We introduce the spaces

fI = £2(D ) V = Wi(D).

70

VENA PEARL BONGOLAN-WALSH ET AL.

We also choose a subset of dynamical variables of our system (1)

v=

(16)

S-

TJl .

To calculate the energy inequality for V, we apply the chain rule to We obtain by Lemma 3.2

Ilvllk.

~ Il vlll + 211V' v11L

(17)

=2(J(TJl, 1jJ), v). The expression V'v is defined by (V' x,zv). We now can estimate the term on the right hand side. By the Cauchy inequality, integration by parts and Poincare ine9uality AlllqllL~ "V'qIIL~ for q E W~(D) and A211iillL2 < IIV'iiIIL2 for v E Wi(D), we have

s

(18) For q, we have the following estimate

(19) From (18) and (19), we have

(20)

:t (2 11v111 + Ra~A~ Ilqll2) + IIV'vIIL +

(Ra;A~ -

2 2,xi11TJl1 }1V'qI12

$

~~ IITJlI1 2.

DEFINITION 4 .1. A random set B = {B(W)}wEO consisting of closed bounded sets B(w) is called absorbing for a random dynamical system ep if we have for any random set D = {D(W)}WEO, D(w) E H bounded, such that t ---+ SUPyED(O,w) IlyllH has a subexponential growth for t ---+ ±oo

(21)

ep(t,w,D(w)) C B(Btw) ep(t, B_tw, D(B_tw)) C B(w)

for for

t2to(D,w) t 2 to(D, w).

B is called forward invariant if ep(t,w,uo)EB(Btw)

ifuoEB(w)

fort20.

Although v is not a random dynamical system in the strong sense we can also show dissipativity in the sense of the above definition. LEMMA 4.2. Let ep(t , w, vo) E H for Vo E H be defined in (6), and 1

Ra2A~ -

2

2

2A 1lEIITJlI > O.

ENSTROPHY AND ERGODICITY OF GRAVITY CU RRENTS

71

Then the closed ball B(O,RI(w)) with radius RI(w) =

2[°00 e"'T:~ 1I1']1112dT

is forward invariant and absorbing. The proof of this lemma follows by int egration of (20) . For the appli cations in th e next section we need that the elements which are contained in th e absorbing set satisfy a particular regularity. To this end we introduce th e functi on space

:= {u EH : Ilull; := IIA~ ull~ < oo}

'W

where s E R The operator As is th e s-th power of th e positive and symmetric operator A. Note that these spaces are embedded in the Slobodeckij spaces HS, s > 0. The norm of thes e spaces is denoted by I1 . 1I n-. This norm can be found in Egorov and Shubin [7]' P age 118. But we do not need this norm explicitly. We only mention that on re th e norm I . lisof HS is equivalent to the norm of re for < s, see [8] . Our goal is it to show that v(l, w , D) is a bounded set in 'H" for some s > 0. This property causes th e complete continuity of th e mapping v(l, w, .). We now derive a differential inequ alit y for tllv(t )II;. By the chain rule we have d d

°

dt (t [[ v(t)II; ) =

Il v(t)ll; + t dt Il v(t)II ;·

Note that for the emb edding const ant

Cs

between H' and V

1Ilvll;ds::; 1Il vll~ds t

t

c;

for s ::; 1

such th at the left hand side is bound ed if the initial condit ions Vo are contained in a bounded set in H . The second term in the above formula can be expressed as followed:

d ..

(d

)

t dt(A 2V,A 2V)H =2t dt v,Asv H

= - 2t(Au, ASV)H + 2t(FI( v + l'](B t w)), ASV)H

+ 2t(F2(v + rJ(Btw)) ,ASV)H. We have

(Av , ASV)H

= IIA~+ ~vIIH = Ilvlli+s'

Similar to the argument of [21] and th e estim ate for the existence of absorbing, and applying some embedding theorems, see Temam [18] Page 12 we have got

Ilvll; ::; C(t , IlvollH , sup tElO,I]

111']1IlD(A')) ' for t

E [0,1] .

72

VENA PEARL BONGOLAN-WALSH ET AL .

By the results of [12] and [21], we know sup 11711 IlD(AO) ::; C(trL2Q)
O

This may be viewed as a statement of the principle that a ratio of partition functions is the exponential of a difference of thermodynamic potentials. Consider an abstract polymer system. The set of particle locations is a graph, where two locations are connected if they are incompatible. It may be shown that c(M) =J 0 implies that the support of M is connected. This is one sense in which the coefficients of the expansion may be thought of as being associated with connected clusters of locations. For an abstract polymer system the cluster estimate is (3.35)

L

Iw(q)1 exp(A(q)) :S A(p).

qE[(p)

The stability bound for the probability of a particle at q is (3.36)

1 LM N(q) MI1c(M)/IwAIM :S w(q)exp(A(q)). .

The stability bound for the expected number of particles at locations that are incompatible with location p is

(3.37) 4. Polymer systems. A polymer system is a realization of an abstract polymer system. There is a given set T of sites. The set P of locations consists of all finite non-empty subsets of T. If Y is a finite nonempty subset of T, the presence or absence of a particle at location Y is identified with the presence or absence of a polymer occupying the sites in Y . The exclusion interaction between locations is such that if the subsets

GENTLE INTRODUCTION TO CLUSTER EXPANSIONS

109

Y and Y' overlap, then there cannot be a polymer occupying the set Y and also another polymer occupying the set Y'. The set combinatorial exponential may be expressed as a polymer system. The object of interest is (4.1 )

K(X)

L

=

C(Yd .. · C(Yk ) .

f={Y1 ,...,Ykl

The sum is over partitions I' = {Y1 , . . . , Yk } . Each}j has at least one point, the Yj are disjoint, and the union of the }j is X. Then K is the combinatorial exponential of C. The interpretation is that K (X) is the partition function, which is a function of the variables C(Y) . Each r is a polymer configuration. The interaction is the condition that no pair of sets Y in the partition can overlap. This gives rise to a problem: The union constraint is not a two-location interaction. However, this problem has a solution. Suppose Y having one point implies that C(Y) = 1. The sum is now over sets r = {Y1 , ... , Yr } . Each }j has at least two points, and the }j are disjoint. However there is no longer a requirement on the union of the }j. This is now precisely a polymer system, where the weights are the C(Y), and the two-body interaction is the disjointness condition. In this combinatorial setting the cluster estimate is expressed in terms of positive quantities A(Y) that depend on the set Y. A typical choice is A(Y) = a\YI, where a > 0 is a constant and IYI is the number of points in Y . The cluster estimate is a condition on the cluster coefficients of the form

L

(4.2)

IC(Z)I exp(A(Z)) ::; A(Y) .

ZnYji0

Typically there is some kind of graph structure on T, and C(Z) = 0 except subsets Z that are connected. This limits the number of terms in the sum for a given size IZ I. Furthermore, the analysis is restricted to a regime where C(Z) approaches zero very rapidly as the size IZI gets large. Thus there are situations where such a cluster estimate holds . THEOREM 4 .1. Suppose that the interaction coefficients K(X) are the set combinatorial exponential of the cluster coefficients C(Z). Suppose that IZI = 1 implies C(Z) = 1. Assume that the cluster estimate holds. Then the probability that a polymer occupies the set Y is

(4.3)

C(Y) K(X \ Y) = K(X)

L M(Y) c(~) IT C(Z)M(Z) . M

M.

ZcX

Here the c(Y) are the cluster coefficients for the polymer system. These satisfy the estimate (4.4)

L M(Y) 1c(~)1 IT IC(Z)IM(Z) ::; exp(A(Y)). M

M.

zs:x

110

WILLIAM FARIS

Thus the ratio is expanded in a series whose radius of convergence depends on Y but not on X. Proof The proof amounts to a translation from the polymer language to the combinatorial language . The set of locations A is the set of nonempty subsets of X . Each location p in A is a subset Ye X. The weight w(p) is the coefficient C(Y). Similarly, the partition function Z/\(w) is the coefficient K(X). The interaction is an exclusion interaction: t(p, q) = 1 translates to Z n Y =I- 0. The translation of the polymer system identity

(4.5)

Z/\(w) = exp(L

c~) IT w(p)M(p))

M

pE/\

into the combinatorial language is

(4.6)

K(X) = exp(L M

c(~)

IT C(Z)M(Z)).

M. z-:x

The ratio Z/\\I(p)(w)/Z/\(w) of partition functions becomes the combinatorial ratio K(X \ Y)/ K(X). Now use the exponential representation (4.7)

The cluster estimate (4.8)

L

Iw(q)1 exp(A(q)) ::; A(p)

qEI(p)

implies the stability bound and convergence.

o

5. Cluster expansions. This section is a brief sketch of the general problem of constructing a probability measure from a density on a high dimensional space using cluster expansions. The book of Malyshev and Minlos [6] gives a much more complete account. Let E be a countable infinite set. Consider a probability measure J-L defined on the Borel subsets of the space RC . In the following the same notation J-L will be used for the expectation associated with this probability measure. Thus, if f is a bounded Borel measurable function on RC, then the expected value J-L(J) = Jf dJ-L is the integral of f with respect to the probability measure J-L. More than one measure may be considered; typically the expectation will be identified with the corresponding measure. If A is a finite subset of E, and f is a function on the finite dimensional space R /\, then f defines a corresponding function fA on RC that depends only on the coordinates in A. The value of f/\ on w in RC is f/\(w) = f(w/\),

GE NTLE INTRODU CTIO N TO CLUSTER EXPA NSIONS

111

where WA in RA is the restriction of w to A. Every function that depends only on th e coordin at es in A arises in this way. Fur thermore, the prob ability measure J.l is determined by the corresponding expect at ions for bounded measurable functions th at depend on only finitely many coordinates, for all such choices of finite subsets A c 1:-. It is assumed th at there is an initial measure , denoted simply by u, and that one can already do calculations or at least estimates with this measur e. For instance, it could be a product measure or a Gaussian measure. For each finite subset A c I:- let PA > 0 be a positive function on R.c that depends only on the coordinates in A. The task is to calculate with a new measure J.l~ with expectation given by this density : (5.1)

I

J.lA

(1)

=

J.lUPA) () . J.l PA

The hope is that even if there is no limiting densit y as A approaches 1:-, there will be a limiting measure J.l' . To show this , the main task is to get estimates that are independent of A. The idea is to express the expectation with J.l~ in a series in terms of certain expectations with u. The problem is that one needs to exhibit a cancellation between the numerator and denominator. This will be difficult unless there is a factorization of th e density with some approxim at e independ ence prop erty. In t his case, it someti mes possible to express the denominator (partition function)

(5.2) as a combinatorial exponential

(5.3)

K(A) =

L IT C(Y) , r

YEr

where r ranges over partitions of A. The C(Y) are called cluster coefficients . The idea is that if the factorization has good independence properties, then the domin ant contribution will come from the partition into one point subsets. There will be a similar expression for the numerator. There is often a formula for the numerator expressing it in terms of computable quantities together with the K(A \ Z) , where Z ranges over subsets of A. So the problem reduces to an estimation of th e rations K(A \ Z) / K (A) that is uniform in A. If the cluster coefficients corresponding to subsets with more than one point are small, then there is some hope of obtaining an estimate of these ratios. This is done by an expansion in terms of the cluster coefficients. In each case, the analys is is in stages: find the cluster representation, estimate the cluster coefficients, and then carr y out the analysis of convergence. Here we only indicate the first stage.

112

WILLIAM FARIS

As a first example consider a perturbation of a Gaussian measure by a density PA . Suppose that PA factors as

(5.4)

PA =

IT A

p

pEA

with a factor for each point in A. The factor Ap is a function that only depends on the p coordinate. There are two procedures that are used in such a situation. The first procedure is the usual perturbation expansion in terms of potential energy. The second is an expansion in terms of the density factors . The first method is expansion in the potential. The idea is to write Ap = exp( -(3pUp) in terms of a potential Up. The parameter {3p > plays a role similar to that of an inverse temperature. It is convenient for the expansion to allow (3p to depend on p. Such a representation in terms of potential energy is natural in physics. Then the relation between the combinatorical exponential and the exponential gives an explicit cancellation. This leads to an elegant representation for the expectation in terms of cumulants of the Up . PROPOSITION 5.1. Let the measure /lA be expressed in terms of the reference measure /l and a density PA that is the product of the Ap = exp( -(3pUp) for p in A. Let fj be functions that each depend on only finitely many coordinates corresponding to some subset of A. Th en the /lA

°

cumulants of the and Up by (5.5)

Ii

are expressed in terms of the /l cumulants of the

C'(L) =

Ii

LM M1,C(L,M)(-{3)M . .

for L #- 0, where the sum is over all multi-indices M supported in A. Proof Write tf = LjE} tj/j and {3U = LpEA {3pUp. Then the /l'

moment generating function for the

(5.6)

Ii

has the representation

'( ( f)) - /l(exp(tf - (3U)) /l exp t - /l(exp(-{3U))

as a quotient of /l moment generating functions for the fj and the Up. This may be written as a relation for exponential generating functions :

(5.7) Write each of these exponential generating functions for moments in terms of a thermodynamic potential that is an exponential generating function for cumulants. The quotient then becomes a difference. This gives the relation for the potentials:

(5.8)

GENTLE INTRODUCTION TO CLUSTER EXPANSIONS

113

When we write this out, we see a spectacular cancellation: the troublesome division becomes a simple subtraction. Th e explicit expression is

(5.9)

",1,

",11

L

L

M

L.JL!C(L)t =L.JL!M!C(L,M)t (-{3) , L

L ,M

where the sum is over all L =I 0 and all M . Equating the coefficients of t L gives the result stated in the proposition. 0 The second method is expansion in the interaction factor. The idea is to use the combinatorial exponential with cumulants of the Ap , which express dependence. The price one pays is that the result involves an ugly quotient. However this problem is solved by use of the polymer expansion explained in the previous section . PROPOSITION 5.2 . Let the measure fLA be expressed in terms of the Gaussian reference measure fL and a density PA that is the product of the Ap for p in A. Let K(X) = fL(I1 PE x Ap ) be the moment corresponding to the subset X. Let Ji be function s that depend only on the j coordinate. Let fw = I1 j Ew fj· Then the expectation has a representation (5 .10)

, fLA(fW)

=

L

G(Z)

K(A \ Z) K(A) .

WCZCA

Furthermore, the moments

(5.11)

K(X)

=L r

IT C(Y) YEr

have a representation as the combinatorial exponential of the cumulants o] the Ap . Proof The representation given above of the denominator K(A) = fL(PA) = fL(I1 PE A Ap ) is just the representation of a moment as a sum of

products of cumulants. For the numerator the factors Ap for p in Ware replaced by factors JpA p. This changes the moments and cumulants to (5.12)

K1(X) =

L IT C1(Y) r

YEr

However for each Y that does not intersect W th e cumulant Cl (Y) C(Y) . This is because the combinatorial logarithm formula shows that it is determined by moments K 1 (Z) = K (Z) for Z c Y . For each partition

I', there are certain Y in I' that have a non-zero intersection with W . Let the union of these Y be denoted Z. Then the numerator is (5 .13)

K1(A) =

L WcZcA

G1(Z)K(A \ Z) .

114

WILLIAM FARIS

Here (5.14)

G1(Z) =

L IT C1(Y), ~

YE~

where t1 ranges over partitions of Z with the property that every element of the partition has a non-zero intersectiori'with W. 0 As a second example, consider the perturbation of a product measure by a density PII. . PROPOSITION 5.3. Let the measure tL~ be expressed in terms of the product reference measure tL and a density PII. . Let fw be a functions on R w, where W c A. The tL~ expectation of fw has a representation (5.15) where

(5.16)

L IT C(Y)

K(X) =

fCII. YEf

is the sum over partitions. Proof. The idea is to use the combinatorial exponential to write

(5.17)

LIT py .

PX =

f

YEf

By the combinatorial logarithm formula py depends only on the oz for Z c Y . Since az depends only on the coordinates in Z, it follows that py depends only on the coordinates in Y . Since tL is a product measure, if we set K(X) = JL(px) and C(Y) = tL(Py), then we get the representation (5.18)

tL(PII.)

IT C(Y)

= K(A) = L f

where

r

ranges over partitions of A. For the numerator we can write

Iw o«

(5.19)

L

=

gz

WCZCII.

where (5.20)

r

YEf

L IT py, f

YEr

is summed over partitions of A \ Z. The factor gz is given by

gz =

Lfw ~

IT py, YE~

where t1 is summed over partitions of Z with the property that each element Y of t1 has a non-empty intersection with W. Let G(Z) = JL(gz) . Then we get (5.21)

~(fwplI.)

=

L G(Z)K(A \ Z). z

o

GENTLE INTRODUCTION TO CLUSTER EXPANSIONS

115

Acknowledgements. The author is grateful for the hospitality of the Courant Institute of Mathematical Sciences, New York University, where this work was begun. He also thanks Daniel Ueltschi for helpful comments.

REFERENCES [1] A. BOVIER AND M. ZAHRADNiK , A simple inductive approach to the problem of convergence of cluster expansions of polymers, J . Stat. Phys. 100 (2000), pp. 765-778. [2] D.C . BRYDGES, A short course on cluster expansions, in K. Osterwalder and K. Stora, eds ., Critical Phenomena, Random Systems, Gauge Theories, Les Houches, Session XLIII, 1984, Elsevier, Amsterdam, 1986, pp . 129-183. [3] R.L. DOBRUSHIN, Perturbation methods of the theory of Gibbsian fields, in Lectures on Probability Theory and Statistics (Lectures Notes in Math. #1648), Springer-Veriag, Berlin, 1996, pp . 1-66 . [4] W .G . FARIS AND R.A . MINLOS, A quantum crystal with multidimensional harmonic oscillators, J. Stat. Physics 94 (1999), pp. 365-387. [5] R . KOTECKY AND D . PREISS, Cluster expansions for abstract polymer models, Commun. Math. Phys. 103 (1986), pp . 491-498. [6] V.A . MALYSHEV AND R .A. MINLOS, Gibbs Random Fields: Cluster Expansions, Kluwer, Dordrecht, 1991. [7] S. MIRACLE-SOLE, On the convergence of cluster expansions , Physica A 279 (2000) , pp . 244-249. [8] D. UELTSCHI, Cluster expansions and correlation functions , Moscow Math. J ., 4 (2004) , pp. 509-520.

CONTINUITY OF THE ITO-MAP FOR HOLDER ROUGH PATHS WITH APPLICATIONS TO THE SUPPORT THEOREM IN HOLDER NORM PETER K. FRIZ' Abstract . Rough Path theory is currently formulated in p-variation topology. We show that in the context of Brownian motion, enhanced to a Rough Path, a more natural Holder metric p can be used . Based on fine-estimates in Lyons ' celebrated Universal Limit Theorem we obtain Lipschitz-continuity of the Ito-rnap (between Rough Path spaces equipped with p). We then consider a number of approximations to Brownian Rough Paths and establish their convergence w.r.t . p. In combination with our Holder ULT this allows sharper results than the p-variation theory. Also, our formulation avoids the so-called control functions and may be easier to use for non Rough Path specialists. As concrete application, we combine our results with ideas from [MS] and [LQZ] and obtain the Stroock-Varadhan Support Theorem in Holder topology as immediate corollary. Key words. Rough Path theory, Ito-rn ap, Universal Limit Theorem, p-variation vs. Holder regularity, Support Theorem. AMS(MOS) subject classifications. 60Gxx .

-

®etoiDmet oem 2!nDeneen an 13rof. Dtto 13regL -

1. Introduction. 1.1. Background in Rough Path theory. Over the last years T . Lyons and co-authors developed a deterministic theory of differential equations capable of dealing with "rough" driving signals such as typical realizations of Brownian Motion . To explain what is now known as Rough Path Theory we consider controlled ordinary differential equations. To fix ideas , set V = ]Rd and W = ]RN and assume (Xt)tE[O,lj is a V-valued path of (piecewise) C 1_ regularity,

(1.1) Let I = (h, ..., Id) be a collection of d vector-fields on W, identified with a map

(1.2)

I :W

--+

L (V, W) ,

'Courant Institute, NYU , New York, NY 10012. The author acknowledges financial support by the Austrian Academy of Science. 117

118

PETER K. FRIZ

and consider the controlled ODE d

dYt = f(Yt)dxt = f(YdXt dt =

(1.3)

L fi(Ydx~dt . i=l

Of course, under standard conditions on f and for fixed initial point Yo, there is a unique W-valued solution path on our chosen time-horizon

[0,1]. Assume that one can find a metric d on Cl ([0,1], V) resp. Cl ([0,1], W) such that the Ita-map C l ([0,1] , V)

(1.4)

{

X .......

Cl ([0,1], W)

Y

is uniformly continuous (at least on bounded sets). Then the meaning of (1:"3) can be extended to driving signals in the closure of Cl-paths, x E Cl ([0,1] , V),

(1.5)

the closure being taken in the topology induced by d. EXAMPLE 1. Let V = ]R2, W =]R3 and consider

with

h = (1,0, _y 2 / 2), 12 = (0,1, y 1 /2) . With Yo =

(1.6) (1.7)

°

integration yields y 3(t) =

t r (x 2 lo

~

ldx 2 _

x 2dx l )

== At(x) .

Note that At(x) has a simple geometric interpretation as area. particular,

leads to At(x(m)) = nt independent of m. On the other hand xt(m) ---uniformly as m tends to infinity.

In

°

The above example shows that the uniform topology is not suited to carry out the extension (1.5). It was realized by T. Lyons that stronger path-space topologies, such as p-variation topology, are suited. We recall

ROUGH PATHS AND SUPPORT THEOREM IN HOLDER NORM

119

DEFINITION 1.1. The p-variation semi-norm of a path x with values in a normed vector-space is defined as

''tJ' (~ lx"

(1.8)

- x" _'IP )

I Jp

where sup D runs over' all dissections D = {a = to < tl < ... < tlDI = I} of [0, 1]. For fixed starting point XQ this provides a genuine norm on pathspace. EXAMPLE 2 . Almost every Brownian path has finite p-variation for p > 2, an immediate consequence of its lip-Holder regularity. On the other hand, its p-variation with pE [1,2] is known to be almost surely infinity. Actually, in the case p E [1,2) one does not have to carry out the closing procedure (1.5) . The differential equation (1.3) may be interpreted directly as Young-integral equation. Recall from [Y] the classical THEOREM 1.1 (L .C . Young) . Letv ,w E C([O ,I],lR) offinitep- resp. q-variation such that

1 p

1 q

-+->1. Then the Riemann-sum below converges and defines the Young Integral :

:J

lim

LVti (Wti - Wti_l ) ==

m"h( D )-.O i

lot v dw,

where D are dissections of [0, t]. We cite from the recent survey paper [Le] the following THEOREM 1.2 . Let f : W -+ L (V, W) be C2 with bounded derivatives up to order 2. Then for any x E C ([0,1], V) of finite p-variation with p E [1,2) and fixed starting point Yo , there exists a unique y E C ([0, 1] , W) , also of finite p-variation, such that (1.3) makes sense as Young integral equation. Moreover, the Ito-mop x f--4 y is continuous in p-variation norm. It is clear from example 2 that this result does not cover differential equations driven by Brownian motion. A new idea is needed and we give some motivation: • From (1.6) we see that iterated integrals are related to the issue of Ita-map continuityIdiscontinuity. • Consider the case of a linear ODE, e.g . f = A, a constant coefficient tensor in W 0 ~V* 0 V * so that (1.3) reads N

dy = Aydx

{:=}

dy~

d

L A~,kyt dx~. j=lk=l

= L

At least when x E Cl ([0,1], V) we may expand u

Yt = Ys + i t A (YS

+i

AYsdx v

+ ...)

dx.,

120

PETER K. FRIZ

and see that, keeping A fixed, the evolution from Ys to Yt is fully determined by iterated integrals of form

X:,t :=

1

S

jm (t)'1J jm

jm

(x) ,

of the velocity field, once inserted into (1.1), motivates the dynamical toymodel amplitude equations

(5.3)

OtVl

= ik 1

L c(l; 2, 3)V2V

3 -

lJk;Vl + it

,

2 ,3

known as the hierarchical shell model [20]. The last two terms only act at the small and large scales. Here it would be very interesting to find out which, if any, structure of the coupling c(1=jlml ; 2=12m2, 3=J3m3) between wavelet modes is required to reproduce the spatial statistics of the random multiplicative branching processes. This then would serve as a first link between the empirical random multiplicative cascade processes and the Navier-Stokes equation. More dreams in this direction are always at work! REFERENCES (1) A .S . MONIN AND A .M . YAGLOM, Statistical Fluid Mechanics, Vols . 1 and 2, MIT Press, 1971. [2J U . FRISCH, Thrbulence, Cambridge University Press, 1995. [3J R. BENZI, S . CILIBERTO , R . TRIPICCIONE, C . BAUDET, F . MASSAIOLOI, AND S . SUCCI, Extended self-similarity in turbulent flows, Phys. Rev. E, 48 (1993) , pp . R29-R32. [4) B .R . PEARSON , P .A . KROGSTAD , AND W . VAN DE WATER, Mea.surements of the turbulent energy dissipation rate, Phys. Fluids, 14 (2002) , pp . 1288-1290 . [5) B . DHRUVA, An Experimental Study of High Reynolds Number Thrbulence in the Atmosphere, PhD thesis , Yale University, 2000 . [6] J. CLEVE, M. GREINER , AND K .R . SREENIVASAN , On the surrogacy of the energy dissipation field in fully developed turbulence, Europhys. Lett ., 61 (2003) , pp . 756-761. [7) M. GREINER, P . LIPA, AND P . CARRUTHERS, Wavelet-correlations in the p-model, Phys.Rev. E, 51 (1995) 1948-1960. [8) M. GREINER, J. GIESEMANN, P. LIPA, AND P. CARRUTHERS, Wavelet-correlations in hierarchical branching processes, Z. Phys. C, 69 (1996), pp. 305-321. [9) M . GREINER, H . EGGERS, AND P . LIPA, Analytic multivariate generating function for random multiplicative cascade processes, Phys. Rev . Lett., 80 (1998) , pp . 5333-5336.

150

MARTIN GREINER ET AL.

[101 M . GREINER, J . SCHMIEGEL, F . EICKEMEYER, P . LIPA, AND H. EGGERS, Spatial [11] [12]

[13] [14] [15] [16J [17]

[18] [19J [20]

correla tions of singularity strengths in multifractal branching processes , Phys. Rev. E, 58 (1998), pp. 554-564. H. EGGERS, T . DZIEKAN , AND M . GREINER, Translat ionally invariant cumulants in energy cascade models of turbulence, Phys. Lett. A, 281 (2001), pp. 249- 255. J . CLEVE, T . DZIEKAN , J . SCHMIEGEL, O .E . BARNDORFF-NIELSEN, B.R. PEARSON , K.R. SREENIVASAN, AND M. GREINER, Data-driven derivation of the turbulent energy cascade generator, arXiv:physics/0312113 . J . CLEVE, M . GREINER, B .R. PEARSON, AND K .R. SREENIVASAN, On the intermittency exponent of the turbulent energy cascade, arXiv:physics/0402015. B . JOUAULT , P . LIPA, AND M. GREINER, Multiplier phenomenology in random multiplicative cascade processes, Phys. Rev . E , 59 (1999) , pp. 2451-2454. B. JOUAULT, M . GREINER, AND P. LIPA, Fix-point multiplier distr ibutions in discrete turbulent cascade models , Physica D, 136 (2000) , pp. 125-144. M. GREINER AND B . JOUAULT , An experimentalists view of discrete and con ti n uous cascade models in fully developed turbulence, Fractals, 10 (2002) , pp . 321-327. J . SCHMIEGEL, J . CLEVE, H. EGGERS , B . PEARSON , AND M . GREINER, Sto chastic energy-cascade model for 1+1 dimensional fully developed turbulence, Phys. Lett. A, 320 (2004), pp. 247-253. J . GIESEMANN, M . GREINER, AND P . LIPA, Wavelet cascades, Physica A, 247 (1997), pp. 41-58. M . GREINER AND J . GIESEMANN, Correlation structure of wavelet cascades, in Wavelets in Physics, L .Z. Fang and R.L. Thews (ed.), (1998) , pp. 89-131. R . BENZI, L . BIFERALE, R. TRIPICCIONE, AND E. TROVATORE, (1+1)-dimensional turbulence, Phys. Fluids, 9 (1997) , pp. 2355-2363.

STOCHASTIC FLOWS ON THE CIRCLE YVES LE JAW AND OLIVIER RAIMOND* Abstract. Brown ian flows on th e circle associated with singular covariances are studied . Key words. Stochastic differential equations, Wiener chaos decomposition, coalescence. AMS(MOS) subject classifications. Primary. 60HlO.

Contents 1

2 3

4

5

Introduction . Flows of diffeomorphisms . The Krylov Veretennikov expansion 3.1 Lipschitz case . 3.2 Non-Lipschitz case . 3.3 A flow of infinite matrices . Flows of kernels and flow of maps . 4.1 n-point motions . 4.2 Flow of maps . 4.3 Diffusive flow of kernels 4.4 Diffusive or coalescing? Classification of the solutions of the SDE 5.1 Solutions of th e SDE . Extension of th e noise and weak solutions 5.2

151 152 153 153 154 156 157 157 158 158 158 160 160 160

1. Introduction. The purpose of th is note is to provid e a simple introduction to the papers [6, 7] by considering the example of stochastic flows on the circle which was not treated in these papers. We refer to these papers for the detail of th e proofs, but the ideas can be explained more clearly and more rapidly in this simpler context. We will finally st ate a conject ure on the classification of isotropi c flows of kernels.

Notation. In all the following, we will denote by § the unit circle

1R/21TZ, by m the Lebesgue measure on § and by P(§) the set of Borel probability measures on S. The set P(§) is equipped with a metric compatible with t he weak convergence. Th e er-field of Borel sets of § and of

P(§) are respectively denoted B(§) and B(P(§)). Let (W , F W , Pw) be the canonical probability space of a sequence of independent Wiener processes (Wtk , k ~ 0, t ~ 0). For all s < t, let F~ denote the er-field generated by the random variables W: - W:, s ~ u < v ~ t and k ~ o. Being given (akh~o a sequence of nonnegative *Mathematiques, Bat 425, Universite Par is-Sud , 91405 Ors ay cedex , FRANCE. 151

152

YVES LE JAN , AND OLIVIER RAIMOND

numbers such that Lk>O a~ < 00, we set C(z) = Lk>O a~ cos(kz). Note that all real positive definite functions on § can be written in this form and that C(O) = Ek~O a~. 2. Flows of diffeomorphisms. Assume that Ek>l k2a~ < 00. Then by a stochastic version of Gronwalllemma, it can be shown that for each Xo E § the stochastic differential equation (SDE) (2.1)

Xt = Xo

+ aoW? +

L ak (1 k~l

t sin(kxs)dW;k-l

+

0

it

COS(kxs)dW;k)

0

has a unique strong solution. These solutions can be considered jointly to form a stochastic flow of diffeomorphims ( such that as Z - . 0, C(O) -C(z) = -c2z2log Izl(l + 0(1)) . • When 0: E (0,2), there exists Co: > such that as Z - . 0, C(O) C(z) = co:zO:(l + 0(1)). Proof We prove this lemma only for 0: E (0,2). For Z E (0,21r), G'(z) is well defined and G'(z) = Lk::::1 k-O: sin(kz) . Using the fact that

°

160

YVES LE JAN, AND OLIVIER RAIMOND

k- o =

(00 e-ks s0-1 r(o)' ds

Jo

oo

L r e-kss o- I sin(kz)~ k~IJO r(a) = _2- L r e-kssO-I(eikz _ e-ikz)~

G'(z) = -

oo

2ik~l~

r(a)

1 I" (e-Se iZ e-Se- iZ) ds = - 2i Jo 1 - e-seiz - 1 _ e-se-iz so-I r(a)

r

oo

=- Jo

e- s sin z 0-1 ds 2s s 1-2e-scosz+er(a)

= _zo-I

100 I(t, z)dt

o 1 sin z t Note t h at t here exi h I( were t, ) z = 1-2e-e-tz ere exists a constant G t • cosz+e 2t. r(o)' o3 such that for t > 1, II(t,z)l::; Gxt and for t E (0,1]' II(t,z)l::; Gxt o - I . Thus, using Lebesgue dominated convergence theorem, we prove that there exists a constant Co such that, as z ~ 0, G'(z) = -aco x zo-I + o(zo-I) . This implies the lemma. 0

This implies that m((0 ,2rr)) = 00 for a E [1,21 and m((O,2rr)) < 00 for a E (0,11. When a = 2, 1'1:(0+) = 00 and when a < 2, 1'1:(0+ ) < 00. Th eorem 4.1 implies th e corollary. 0 REMARK 4.1. The case a = 2 has been studied in (1,3, 8J. In fact, it is shown that the maps of the flow are homeomorphisms. 5. Classification of the solutions of the SDE. 5.1. Solutions of the SDE. Let (0, A, P) be an extension of the probability space (W , F W , Pw) . We say that a measurable flow of measurable maps cP = (CPs ,t} is a weak solution of (2.1) if it satisfies (2.1) without being F~-measurable . Similarly, a measurable flow of kernels K = (Ks ,t) will be called weak (generalized) solution of the SDE (2.1) if it satisfies (4.3) without being F~-measurable . We have seen that uniqueness is verified if one assumes in addition is F~ for all s ::; t. Wiener measurability:

«.,

5.2. Extension of the noise and weak solutions. In case (b), a different consistent syst em of Feller semigroups p~n),c can be constructed by considering the coalescing n-point motion Xt( n ),c associated with Xt( n ), the n-point motion of the Wiener solution. A measurable flow of coalescing maps cP~ t, whose n-point motion is n ),c , can be defined on an extension (0 , A, P) of the probability space (W, F W , Pw). This coalescing flow also solves the SDE (2.1). It is a weak solution. For s ::; t, set F~ t = (j(cp~ v' s::; u ::; v::; t) . Then (F~ t)s O. This completes the proof of Proposition 3.3. 0 Remarks. 1. The assumption that the measure V is finite was made for technical simplifications, and seemingly can be removed. 2. The assumption dim(V) > d - 1 is essential for (3.9) to be true. However, one can regularize (3.9) appropriately to include more general V (see [K2]). 3. There is seemingly some overlap of ideas between our construction of the Wiener path integral for the Schrodinger equation and the theory of rough paths of T . Lyons [L]. 4. It is worth noting that the natural topology on the path space CPL is the one induced from the uniform topology of continuous paths (or from the Hilbert space topology of the Cameron-Martin space). This topology enjoys the following properties: (i) it is compatible with the measure, (ii) when reduced to any finite-dimensional simplex it yields its natural Euclidean topology, (iii) any simplex Simr is the boundary for the simplex Simr+l, (iv) the topology is not locally compact, but the whole space is a countable union of locally compact spaces . 4. Two remarks on parabolic equations in momentum representation. As was already mentioned, the first definition of the Feynman path integral representing solutions for the Schrodinger equation as a genuine Lebesgue integral arising from a pure jump Markov process was given in [M], [MCh]. This integral was defined for the Scgrodinger equation in momentum representation with potentials satisfying Ito's complex measure condition. This was an important breakthrough. As for the diffusion equation the familiar Feynman-Kac representation exists, the analogous result for the diffusion equation in momentum representation did not receive much attention. This is also due to the fact that unlike Schrodinger equation the momentum representation for the diffusion equation often does not seem very natural physically, though it does make sense in the study of tunnel effects in quantum mechanics. A recent paper [Ch] is devoted to an interesting detailed analysis of the underlying jump processes for diffusion equations in momentum representation under Ito's complex measure condition for sources (potentials) and drifts. In this section I like to point out two simple observations about this theory. Firstly, in some cases one can get meaningful path integral representation for diffusion equations in momentum representation even when the source does not satisfy Ito's condition (so that the underlying process is not of pure jump type) and when an unbounded source prevents the possibility of using the standard FeynmanKac formula with the Wiener measure. Secondly, a curious asymptotic formula can be obtained by passing to a central limit in a pure jump path integral representation for the diffusion equation. Moreover, instead of just diffusion equations one can directly consider more general parabolic differential and even pseudo-differential equations without any increase in the complexity.

178

VASSILI N. KOLOKOLTSOV

1. Stable laws for parabolic equations. Consider the pseudo-differential parabolic equation

(4.1) in R d , where a > 0, G > 0, (3 E (0,1) are given constants (in fact, instead of 1.6.la one can take even more general operators f(I.6.1) with non-negative continuous function I). Since

with some c depending on d and {3 (see any book discussing stable processes, e.g. [K2J), in momentum representation, Le. for u(p) = I e-ipx'IjJ(x) dx, the equation (4.1) takes the form

(4.2)

~~ (p) = _GlpI2au(p) + c

J

(u(p

+~) - u(p))I~I-d-{3 d~.

As the second operator in this equation generates the Feller semigroup of a (3-stable Levy process, the following result is straightforward. PROPOSITION 4 .1. Solution to (4.2) with an arbitrary bounded initial fun ction Uo is given by the formula u(t,p) =

s,

[exp{ -G

I

t

ly(s)1 2a ds}uo(y(t))] ,

where E p denotes the expectation with respect to the corresponding (3 -stable Levy motion starting at p. 2. A central limit for pure jump path integrals. Consider now the equation (4.3)

with a small parameter h > 0, a, G are again positive constants and V(x) = with some finite positive measure M, Le. V satisfies Ito 's condition. Suppose also that M is symmetric in the sense that I ~M (d~) = 0, and that it has finite moments at least up to the third order. Denote by II(M) = {lIij(M)} the matrix of the second moments I ~i~jM(d~) of M. Performing the h-Fourier transform (which is usual in the theory of semiclassical asymptotics, see e.g. [MFJ), Le. passing to the function

I eix e M (~)

Uh(t,p) =

J

e-ipx /h'IjJ(t,x)dx

{=:::}

'ljJh(t,X) = (21rh)-d

yields for Uh(t,p) the equation (4.4)

ou =

-

ot

(h 0)

- Glpl2a u+ -12 V -;- u h Z op

J

eipx/hu(t,p)dp

CONNECTING PURE JUMP AND WIENER PROCESSES

179

for Uh(t,p). PROPOSITION 4.2. For the solution of (4.4) with the initial condition Uo the following asymptotic formula holds:

lim exp{th- 21IMII}uh(t,p)

h......O

(4.5)

= E w exp { - G

I

t

Ip + W(sWO ds }UO(P + W(t)),

where Ew denotes the expectation with respect to the d-dimensional Wiener process W(s) with the covariance matrix v(M), and where IIMII is, of course, the full measure M(Rd). Proof Using the properties of pure jump processes, one can write the solution to (4.4) with the initial condition Uo as the path integral (4.6)

Uh(t,p) =

exp{th-21IMII}E~ exp { -

G

I

t

jy(sW ds },

where E~ denotes the expectation with respect to the process of jumps which are identically independently distributed according to the probability measure J.1.(dp) = M(d(p/h))/IIMII and which occur at times Sj from [0, t] that are distributed according to the Poisson process of intensity h- 21IMII. Observing that

h- 2V(hx)

= h- 2

J

(eihxe - l)M(de)

+ h- 211MII

d

= -

L

XiXjVij

+ O(h) + h- 21IMII,

i ,j=l

we conclude that the characteristic exponent of our compound Poisson process converges, as h ---+ 0, to the characteristic exponent -(v(M)x,x) of the Wiener process indicated above. This implies the convergence of the corresponding measures on trajectories, and (4.5) follows . 0 Acknowledgements. I am grateful to E. Waymire for inviting me on a very stimulating summer conference in Minnesota on probability and partial differential equations, and to O. Gulinskii and A. Fesenko for useful discussions on the Feynman integral.

REFERENCES [A]

[ABB]

S. ALBEVERIO ET AL. Schrodinger operators with potentials supported by null sets. In: S. Albeverio et a!. (Eds.) Ideas and Methods in Quantum and Statistical Physics, in Memory of R. Hoegh-Krohn, Vo!. 2, Cambridge Univ. Press, 1992, 63-95. S. ALBEVERIO, A. BOUTET DE MONVEL-BERTIER, AND ZD. BRZEZNIAK . Stationary Phase Method in Infinite Dimensions by Finite Approximations: Applications to the Schrodinger Equation. Paten. Anal. 4:5 (1995), 469502.

180 [AKS]

[Ch]

[ChQ]

[K1]

[K2] [K3]

[K4] [K5] [L] [M] [MCh]

[MF] [Me]

[Mey] [PQ] [SS]

VASSILI N. KOLOKOLTSOV S. ALBEVERIO, V.N . KOLOKOLTSOV, AND O.G. SMOLYANOV. Representation des solutions de l'equation de Belavkin pour la mesure quantique par une version rigoureuse de la formule d'integration fonctionnelle de Menski, C.R. Acad. Sci . Paris , Ser . 1, 323 (1996), 661-664 . L. CHEN ET AL. On Ito 's Complex Measure Condition. In: Probability, statistics and their applications: paper in honor of Rabi Bhattacharya, p. 65-80, IMS Lecture Notes Monogr . Ser. 41, Inst. Math. Statist., Beachwood, OH, 2003. A.M . CHEBOTAREV AND R.B . QUEZADA. Stochastic approach to time-dependent quantum tunnelling. Russian J. of Math . Phys . 4:3 (1998), 275-286 . V. KOLOKOLTSOV. Complex measures on path space: an introduction to the Feynman integral applied to the Schrodinger equation. Methodology and Computing in Applied Probability 1:3 (1999), 349-365. V. KOLOKOLTSOV. Semiclassical Analysis for Diffusion and Stochastic Processes. Monograph. Springer Lecture Notes Math. Series , Vol. 1724, Springer 2000. V. KOLOKOLTSOV. A new path integral representation for the solutions of the Schrodinger, heat and stochastic Schrodinger equations. Math. Proc. Cam . Phi!. Soc. 132 (2002), 353-375. V.N. KOLOKOLTSOV. Mathematics of the Feynmann path integral. Proc. of the International Mathematical conference FILOMAT 2001, University of Nis, FILOMAT 15 (2001), 293-312 . V.N . KOLOKOLTSOV. On the singular Schrodinger equations with magnetic fields. Matem. Zbomik 194:6 (2003), 105-126 . T.J . LYONS. Differential equations driven by rough signals. Revista Matematica lberoamericana 14:2, 1998. V .P. MASLOV. Complex Markov Chains and Functional Feynman Integral. Moscow, Nauka, 1976 (in Russian) . V.P . MASLOV AND A.M. CHEBOTAREV. Processus it. sauts et leur application dans la mecanique quantique. In: S. Albeverio et al. (Eds .). Feynm an path integrals. LNP 106, Springer 1979, pp. 58-72. V .P . MASLOV AND M.V . FEDORYUK. Semiclassical approximation in quantum mechanics. Reidel, Dordrecht, 1981. M.B. MENSKI. The difficulties in the mathematical definition of path int egrals are overcome in the theory of continuous quantum measurements. Teor. Mat. Fizika 93:2 (1992), 264-272. P . MEYER. Quantum Probability for Probabilists. Springer Lecture Notes Math. 1538, Springer-Verlag 1991. P . PEREYRA AND R. QUEZADA. Probabilistic representation formulae for the time evolution of quantum systems. J. Math . Phys. 34 (1993), 59-68. O.G . SMOLYANOV AND KT. SHAVGULIDZE. Kontinualniye Integrali. Moscow Univ . Press, Moscow, 1990 (in Russian) .

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS MUKUL MAJUMDAR" Abstract. Random dynamical systems are useful in modeling the evolution of economic processes subject to exogenous shocks . One obtains strong results on the existence, uniqueness, stability of the invariant distribution of such systems when an appropriate splitting condition is satisfied. Also of importance has been the study of random iterates of maps from the quadratic family. Applications to economi c growth models are reviewed . . Key words. Dynamical systems, Markov processes, iterated random maps, invariant distributions, splitting, quadratic family, estimation, economic growth.

1. Introduction. Consider a random dynamical system (B, I' , Q) where B is the state space (for example, a metric space) , r an appropriate family of maps on B into itself (interpreted as the set of all possible laws of motion) and Q is a probability measure on (some er-field of) r . The evolution of the system can be described as follows: initially, the system is in some state x ; an element a1 of r is chosen randomly according to the probability measure Q and the system moves to the state Xl = a1(x) in period one. Again, independently of aI, an element a2 of r is chosen ac, cording to the probability measure Q and the state of the system in period two is obtained as X 2 = a2(a1(x)). In general, starting from some x in B, one has

(1.1) where the maps (an) are indepenent with the common distribution Q. The initial point x can also be chosen (independently of (an)) as a random variable X o. The sequence X n of states obtained in this manner is a Markov process and has been of particular interest in dynamic economics. It may be noted that every Markov process (with an arbitrary given transition probability) may be constructed in this manner provided B is a Borel subset of a complete separable metric space , although such a construction is not unique [Bhattacharya and Waymire [1] , p. 228]. Hence, random iterates of affine, quadratic or monotone maps provide examples of Markov processes with specific structures that have engaged the attention of probability theorists. Random dynamical systems have been studied in many contexts in economics, particularly in modeling long run evolution of economic systems subject to exogenous random shocks. The framework (1.1) can be interpreted as a descriptive model; but one may also start with a discounted (stochastic) dynamic programming problem , and directly arrive at a stationary optimal policy function, which together with the given law of transition describes the optimal evolution of the states in the form (1.1). "Department of Economics, Cornell University, Ithaca, New York 14853. 181

182

MUKUL MAJUMDAR

Of particular significance are recent results on the "inverse optimal problem under uncertainty" due to Mitra [10] which assert that a very broad class of random systems (1.1) can be so interpreted. To begin with, in order to provide the motivation, I present two examples of deterministic dynamical systems arising in economics. The first is a descriptive growth model that leads to a dynamical system with an increasing law of motion. The second shows how laws of motion belonging to the quadratic family can be generated in dynamic optimization theory. In Section 3 we review some results on random dynamical systems that satisfy a splitting condition, first introduced by Dubins and Freedman [7] in their study of Markov processes. This condition has been recast in more general state spaces (see (3.10)) . The results deal with : (i) The existence, uniqueness and global stability of a steady state (an invariant distribution): a general theorem proved in Bhattacharya and Majumdar [3] is first recalled (Theorem 3.1). The proof relies on a contraction mapp ing argument that yields an estimate of the speed of convergence [see (3.11) and (3.13)]. Corollary 3.1 deals with "split" dynamical systems in which the admissible laws of motion are all monotone.

(ii) Applications of the theoretical results to a few topics : (a) turnpike theorems in the literature on descriptive and optimal growth under uncertainty: when each admissible law of motion is monotone increasing, and satisfies the appropriate Inada-type 'end point' condition, Corollary 3.1 can be applied directly : see Sections 3.2.1-3.2.2 . (b) estimation of the invariant distribution: as noted above, an important implication of the splitting condition is an estimate of the speed of convergence. This estimate is used in Section 3.2.3 to prove a result on JTi-consistency of the sample mean as an estimator of the expect ed long run equilibrium value (Le., the value of the state variable with respect to the invariant distribution) . Next , in Section 4 we briefly turn to qualitative properties of random it erates of quadratic maps : a growing literature has focused on this theme , in view of th e discussion in Section 1.2 and of the privileged status of the quadratic family in understanding complex or chaotic behavior of dynamical systems. 1.1. The Solow model: A dynamical system with an increas-

ing law of motion. Here is a discrete time exposition of Solow's model [11] of economic growth with full employment. There is only one producible commodity which can be either consumed or used as an input along with labor to produce more of itself. When consumed, it simply disappears from the scene. Net output at the "end" of period t, denoted by yt(~ 0) is related to the input of the producible good Kt (called "capit al" ) and

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

183

labor L; employed "at the beginning of" period t according to the following technological rule ("production function") :

(1.2) where Kt ~ 0, L, ~ O. The fraction of output saved (at the end of period t) is a const ant s, so that total saving St in period t is given by

s, = syt, 0 < s < 1.

(1.3)

Equilibrium of saving and investment plans requires

(1.4) where It is the net investment in period t. For simplicity, assume that capital stock does not depreciate over time, so that at the beginning of period t + 1, the capital stock K t+ 1 is given by

(1.5) Suppose that the total supply of labor in period t , denoted by mined completely exogeneously, according to a "nat ural" law:

(1.6) Full employment of the labor force, requires that

(1.7) Hence, from (1.2)-(1.7), we have

K t+ 1 = Kt

+ sF (K t , Lt) .

Assume that F is homogeneous of degree one. We then have

Kt+l . Lt+l Lt+1 Lt A

A

__

Kt Lt A

+ SF (Kt A"

1)

Lt

Writing k« == Kt/ i; we get

(1.8) where

f(k) == F(Kj L, 1). From (1.8)

kt+l = [kt/(l

+ 1])] + [sf(k t)j(l + 1])]

i; is deter-

184

MUKUL MAJUMDAR

or (1.9) where

a(k) == [k/(I

(1.10)

+ 1])] + s[J(k)/(I + 1])].

Equation (1.9) is the fundamental dynamic equation describing the intertemporal behavior of kt when both the full employment condition and the condition of short run savings-investment equilibrium [see (1.4) and (1.7)] are satisfied. We shall refer to (1.9) as the law of motion ofthe Solow model in its reduced form. For any k > 0, the trajectory T(k) from k is given by T(k) == (aj(k))~o where aO(k) == k, a 1(k) == a(k), aj(k) == a(a j- 1(k)) for j ~ 2. Assume that f(O) = 0, f'(k) > 0, f"(k) < 0 for k > 0; and lim f'(k) =

k!O

00,

lim f'(k) = O. Then, using (1.10), we see that a(O) = 0;

kToo

a'(k) = (1 + 1])-1[1 + s!,(k)] > 0 at k > 0; (1.11)

a"(k) = (1 +1])-l sf" (k ) < 0 at k > O.

Also, verify the boundary conditions: lima'(k) (1.12)

k!O

= lim[(I + 1])-1 + (1 + 1])-1 s!,(k)] = 00 . k!O

lima'(k) = (1 + 1])-1 < 1.

kToo

The existence, uniqueness and stability of a steady state k" > 0 of the dynamical system (1.9) can be proved . Here is a summary of the results: PROPOSITION 1.1 . There is a unique k* > 0 such that

k* = a(k*) ; equivalently, (1.13)

k* = [k* /(1

+ 1])] + s[f(k*)/I + 1]].

If k < k", the trajectory T(k) from k is increasing and converges to k*. If k > k", the trajectory T(k) from k is decreasing and converges to k*. 1.2. The quadratic family in dynamic optimization problems. We consider a family of economies indexed by a parameter /-L, where /-LEA = [1,4] . Each economy in this family has the same production function , f : lR+ ---+ lR+ and the same discount factor OE(O, 1). The economies in this family differ in the specification of their return functions, W : lR~ x A ---+ lR+ [depending on the parameter value of /-LEA that is picked].

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

The following assumptions of

f

185

are used:

(F.1) f(O) = O. (F.2) f is non-decreasing, continuous and concave on ~+ . (F.3) There is K > 0, such that f(x) < x for all x > K, and f(x) for all 0 < x < K. A program from an initial input x 2: 0 is a sequence (xt} satisfying

Xo = x , 0:::; Xt :::; f(Xt-d

>x

for t 2: 1.

We interpret Xt as the input in period t, and this leads to the output f(xt} in the subsequent period. The consumption sequence (c.), generated by a program (Xt) is given by

Ct = f(xt-d - Xtk 0) for t 2: 1. It is standard to verify that for any program (Xt) from x 2: 0, we have Xt, == max(K,x) for t 2: O. Given any flEA, the following assumptions on w(·, fL) are used : (W.1) W(C,X,fL) is non-decreasing inc given x, and non-decreasing in x, given c. (W.2) W(C,X,fL) is continuous on ~~ . (W.3) W(C,X,fL) is concave on ~~. In defining "optimality" of a program, we note that the notion has to be economy specific. Since we can keep track of the economies by simply noting its fL value, we find it convenient to rerer to the appropriate notion of optimality as fL-optimality. Given any flEA, a program (Xt) from x 2: 0 is fL-optimal if Ct+l :::; K(x)

00

00

t=O

t=O

I>5 tw(Ct+l , Xt , fL ) 2: Lotw(Ct+l ,Xt,fL) for every program (Xt) from x. Define a set Y

c

~~ by

Y = {(C,X)E~~ : c:::; f(x)}. For much of our discussion of fL-optimal programs, what is crucial is the behavior of w(" fL) on Y (rather than on ~~). We now proceed to assume: (W.4) Given any WA,W(C,X,fL) is strictly increasing and strictly concave

in

C

given x, on the set Y.

Standard arguments ensure that given any flEA, there is a fL-optimal program from every x 2: O. Assumptions (F .2), (W.3), and (W.4) ensure that a fL-optimal program is unique. Since there is a unique fL-optimal program from every x 2: 0, one can define an optimal transition function h : ~+ x A ~ ~+ by

186

MUKUL MAJUMDAR

where (Xt) is the Jl-optimal program from x 2: O. It is easily checked that this definition also implies that for all t 2: 0, we have

Xt+! = hp(xd . Consider now the family of economies, where f,8 and ware numerically specified as follows:

f(x) = (1.14)

(16/3)x - 8x 2 + (16/3)x 4 {

for

XE[O, 0.5]

for x 2: 0.5

1

8 = 0.0025.

The function w is specified in a more involved fashion . To ease writing, denote L == 98, a == 425. Also, denote by I the closed interval [0,1]' and define the function B: I x A --t I by

B(x, Jl) and u : [2 x A (1.15)

--t

~

=

Jlx(1 - x)

for

xel, JlEA

by

u(x, z , Jl) = ax - 0.5Lx 2 + zB(x, Jl) - 0.5z 2 - 8[az - 0.5Lz 2 + 0.5B(z, Jl)2].

Define a set D C [2 by

D= and a function w : D x A

(1.16)

{(C,X)E~+ X --t

~+

L : c::; f(x)}

by

w(c, x , Jl) = u(x,J(x) - c, Jl)

for (c, x)ED and JlEA.

We now extend the definition of w(" Jl) to the domain Y . For (c, x )EY with x> 1 [so that f(x) = 1, and c::; 1], define

(1.17)

w(c,X,Jl) = w(c, I,Jl)'

Finally, we exend the definition of w(' , Jl) to the domain ~~ . For (c,x )E~~ with c> f(x) , define (1.18)

w(c,X,Jl)

=

w(f(x), X,Jl).

It can be checked [see Majumdar and Mitra [9]] that for the above specifications, f satisfies (F.l)-(F.3), and given any JlEA , w(' ,Jl) satisfies (W.l)-

(W.4) . We observe that w(c, x , Jl) 2: w(O, 0, Jl) [by (W.l)] = u(O, f(O) -0, Jl) = 0, for all (c,x )E~~ . Thus w(', Jl) maps from ~~ to ~+ . Also, for all (C,X)E~~, w(c,X,Jl) ::;w(c, I,Jl) = w(l , I,Jl)'

One can verify [see Majumdar and Mitra [9]] the following: PROPOSITION 1.2 . The optimal tmnsition functions for the family of

economics (f,w( ',Jl),8) are given by (1.19)

hp(x) = Jlx(1 - x)

for all xel

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

187

2. Random dynamical systems. Let S be a metric space and S be the Borel a-field of S. Endow r with a a-field E such that the map b, x) - t b(x)) on (I' x S, E 18> S) into (S, S) is measurable. Let Q be a probability measure on (r,E). On some probability space (n,F,p) let ((}n)~=l be a sequence of random functions from r with a common distribution Q. For a given random variable X o (with values in S), independent of the sequence ((}n)~=l' define

Xl == (}1(XO) == (}IXO

(2.1)

Xn+l = (}n+l(X n) == (}n+l(}n ... (}IXO.

(2.2)

We write Xn(x) for the case Xo = x; to simplify notation we write X n = (}IXO for the more general (random) X o. Then X n is a Markov process with a stationary transition probability p(x, dy) given as follows: for x E S, CES,

(}n •••

p(x,C) = Q(b Er: ')'(x) E Cl)·

(2.3)

The stationary transition probability p(x, dy) is said to be weakly continuous or to have the Feller property if for any sequence X n converging to x, the sequence of probability measures p(x n, .) converges weakly to p(x , .). One can show that if r consists of a family of continuous maps, p(x, dy) has the Feller property.

3. Evolution. To study the evolution of the process (2.2), it is convenient to define the map T* [on the space M(S) of all finite signed measures on (S, S)] by (3.1)

T* p,(C) =

is

p(x, C)p,(dx) =

l

p,b-1C)Q(d')'),

P, E

M(S) .

Let P(S) be the set of all probability measures on (S, S). An element 'IT' of P(S) is invariant for p(x, dy) (or for the Markov process X n) if it is a fixed point of T*, i.e.,

(3.2)

'IT'

is invariant

iff

T*'IT' =

'IT' .

Now write p(n)(x , dy) for the n-step transition probability with p(1) = p(x, dy). Then p(n)(x, dy) is the distribution of (}n .....(}IX . Define 'I?" as the n-th iterate of T* :

(3.3)

T*np,

= T*(n-I)(T* f..L)(n ~ 2), T*l = T* , T*0 = Identity.

Then for any C E S, (3.4)

188

MUKUL MAJUMDAR

so that T:" J.L is the distribution of X n when X o has distribution J.L. To express 'I?" in terms of the common distribution Q of the LLd. maps, let I'" denote the usual Cartesian product I' x r x ... x r (n terms), and let Qn be the product probability Q x Q x ... x Q on (I'", s@n) where s @n is the product er-field on I'". Thus Qn is the (joint) distribution of a = (aI, a2,..., an). For 'Y = ("{b 'Y2, ...,'Yn) € I'" let ;Y denote the composition ~

(3.5)

'Y := 'Yn'Yn-I ···'YI·

We suppress the dependence of i' on n for notational simplicity. Then, since T*nJ.L is the distribution of X n = an...aIXO, one has (T*nJ.L)(A) = Prob "", - 1

rv

(X o € a A), where a = anan-I ....al . Therefore, by the independence of and X o,

a

(A€S, J.L€P(S)) .

(3.6)

Finally, we come to the definition of stability. A Markov process X n is stable in distribution if there is a unique invariant probability measure 7f such that Xn(x) converges in distribution to 7f irrespective of the initial state x, Le., if p(n) (x, dy) converges weakly to the same probability measure 7f for all x . 3.1. A general theorem under splitting. Recall that S is the Borel er-field of the state space S . Let A c S , define

(3.7)

d(p" v) := sup IJ.L(A) - v(A)1

(J.L, V€P(S)) .

AeA

1) Consider the following hypothesis (HI) :

(3.8)

(P(S) , d) is a complete metric space;

2) there exists a positive integer N such that for all 'Y e rN , one has

(3.9) 3) there exists 0 > 0 such that 'v'A€A, and with N as in (2), one has

(3.10)

P(a-I(A) = S or o.

THEOREM 3.1. Assume the hypothesis (HI) ' Then there exists a unique invariant probability 7f for the Markov process X n := an...aIXO, where Xo is indepen dent of {an := n ~ I}. Also, one has

(3.11)

d(T*np"

7f)

:S (1 - 0)[nlNI

(p,€P(S))

where T*n is the distribution of X n when X o has distribution J.L, and [n/N] is the integer part of nfN;

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

189

We now state two corollaries of Theorem 3.1 applied to LLd. monotone maps. Corollary 3.1 extends a result of Dubins and Freedman, [1966 Thm. (5.10)], to more general state spaces in ~ and relaxes the requirement of continuity of an ' The set of monotone maps may include both nondecreasing and nonincreasing ones. Let S be a closed subset or an interval of~. Denote by dK(J.l, v) the Kolmogorov distance on P(S) . That is, if Fp. , F; denote the distribution functions (d.f) of J.l and i/, then

dK(J.l, v) = sup 1Fp.(x) - Fv(x)1 (3.12)

X €s

== sup 1Fp.(x) - Fv(x)l , (J.l , VEP(S)). X€!R

It should be noted that convergence in the distance d« on P(S) implies weak convergence in P(S). COROLLARY 3.1. Let S be an interval or a closed subset of~. Suppose an(n ~ 1) is a sequence of i.i.d. monotone maps on S satisfying the splitting condition (H) : (H) There exist XOES, a positive integer Nand a constant fJ > 0 such that

Prob(aNaN-I aIX Prob (aNaN-I aIx

~

~

Xo \fxES) Xo \fxES)

~

fJ

~ fJ

(aJ Then the sequence of distributions T*nJ.l of X n := an...aIXO converges to a probability measure 7l' on S exponentially fast in the Kolmogorov distance dk irrespective of X o. Indeed, (3.13)

where [y] denotes the integer part of y. (bJ tt in (aJ is the unique invariant probability of the Markov process X n . Proofs of Theorem 3.1 and Corollary 3.1 are spelled out in Bhattacharya and Majumdar [3, 4]. 3.2. Applications of splitting. 3.2.1. Stochastic turnpike theorems. We now turn to the problem of economic growth under uncertainty. A complete list of references to the literature - influenced by ther works of Brock and Mirman - is in Majumdar, Mitra and Nyarko [8]. I indicate how the principal results of this literature can be obtained by using Corollary 3.1. Instead of a single law of motion (1.9), we allow for a class of admissible laws with properties suggested by the deterministic Solow model in its reduced form [see (1.10) and (1.11)]. Consider the case where S = ~+; and r = {FI , F2 , .. . , Fe, ..., FN } where the distinct laws of motion F; satisfy:

190

MUKUL MAJUMDAR

(F.1) Pi is strictly increasing, continuous, and there is some r, > 0 such that Fi(x) > x on (0, r i) and Fi(x) < x fOT X > ri. Note that Fi(ri) = Ti for all i = 1, ..., N. Next , assume: (F.2) r, =I rj for i =I j. In other words , the unique positive fixed points r i of distinct laws of motion are all distinct. We choose the indices i = 1,2, ..., N so that

Let Prob (an = Fi) = Pi > O(i ::; i ::; N) . Consider the Markov process {Xn(x)} with the state space (0,00). Ify ~ rI , then Fi(y) ~ Fi(rl) > rl for i = 2, ...N , and F1(TI) = rI, so that Xn(x) ~ TI for all n ~ 0 if x ~ TI. Similarly, if y ::; TN, then F;(y) ::; F;(rN) < TN for i = 1, ..., N - 1 and FN(TN) = r», so that Xn(x) ::; rN for all n ~ 0 if x ::; rn . Hence, if the initial state x is in [rl,rN], then the process {Xn(x) : n ~ O} remains in h, TN] forever. We shall presently see that for a long run analysis we can consider h ,r N] as the effective state space. We shall first indicate that on the state space h , r N] the splitting 2)(x) ::; F1(x) etc. The condition (H) is satisfied. If x ~ rI, F1(x) ::; x, Fi limit of this decreasing sequence F1(n) (x) must be a fixed point of F1 , and therefore must be rl. Similarly, if x ::; rN , then FfJ(x) increases to r N. In particular,

Thus, there must be a positive integer no such that

Fino)(rN) < F~nO)(rl)' This means that if Zo E

[pino) (r N), Fino) (rd]' then

Prob(Xno(x) ::; Zo 'v'xEh,rN]) ~ Probfo., = F 1 for 1 ::; n ::; no) =

P~o

>0

Prob(Xno(x) ~ Zo 'v'xEh, Tn]) ~ Probfo., = FN for 1 ::; n ::; no) = p';Y > O. Hence , considering h, r N] as the state space , and using Theorem 3.1, there is a unique invariant probability 7r with the stability property holding for all initial XEh , rN] ' Now, define m(x) = . min Fi(x) , and fix the initial .=I ,...,N

state XE(O, rr). One can verify that (i) m is continuous; (ii) m is strictly increasing; (iii) m(rI) = rI and m(x) > x for XE(O, TI), and m(x) < x for x > rI. Clearly m(n)(x) increases with n, and m(n)(x) ::; rI. The limit of the sequence m(n)(x) must be a fixed point, and is, therefore "i- Since Fi(rI) > rI for

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

191

i = 2, ..., N , there exists some e > 0 such that Fi(y) > rl (2 :S i :S N) for all YE[rl - c,rI] . Clearly there is some ne such that mn«x) ~ rl - c. If 71 = inf{n ~ 1 : Xn(x) > ri} then it follows that for all k ~ 1

Prcbf r,

> ne + k) :S p~ .

pt

Since goes to zero as k ~ 00 , it follows that 71 is finite almost surely. Also, X T 1 (X) :S rN, since for y:S rI, (i) Fi(y) < Fi(rN) for all i and (ii) Fi(rN) < rn for i = 1,2, ...,N -1 and FN(rN) = rN. (In a single period it is not possible to go from a state less than rl to one larger than rN) . By the strong Markov property, and our earlier result , XT+m( x) converges in distribution to IT as m ~ 00 for all u(O ,rI) . Similarly, one can check that as n ~ 00, Xn( x) converges in distribution to IT for all x > rN. The assumption that I' is finite can be dispensed with if one has additional structures in the model. Here is a simple example.

3.2.2. Uncountable I': an example. Let F: R+ ~ R+ satisfy: (F.1) F is strictly increasing and continuous. We shall keep F fixed. Consider e = [B 1 , B2 ], where 0 < B1 < B2 , and assume the following concavity and "end point" conditions: (F.2) F(x)/x is strictly decreasing in x> 0, 02~~:;") < 1 for some x" > 0

01F~x') > 1 for some x' > O. Since BF(x) is also strictly decreasing x x in x , F .1 and F.2 implies that for each BEe, there is a unique Xo

>0

BF(xo) . BF(x) such that = 1, i.e., BF(xo) = x «. Observe that - - > 1 for Xo x BF(x) 1£ N B' B'" u O < x < Xo , - - < or x > Xo. ow, > impues Xo' > xo" ; x ()'F(xo") ()"F(xo") ()' F(xo') W. --'--:"'-;" > = 1= Xo' > XO". nte Xo" Xo" Xo'

= U : f = BF, BEe}, I, = ()I F I'

12 =

and

()2F.

e

Assume that B is chosen LLd. according to a density function g(B) on which is positive and continuous on e . In our notation, h (xo1) = xo1; 12(xoJ = xo2 • If x ~ XO\l f(x) == BF(x) ~ f(xoJ ~ h(xo 1) = xo1. Hence Xn(x) ~ x01 for all n ~ 0 if x ~ xo1. If x :S X0 2

Hence , if x E [xo 1, xo2] then the process X n (x) remains in [xo 1, xo2 ] forn)(xo = x0 and lim f~n)(X01) = xo . There must ever . Now, lim fi 2 2) 1 n ~oo

n -lo OO

be a positive integer no such that fino) (xo2 ) < f~no) (xo1). Choose some

192

MUKUL MAJUMDAR

no) (X0 ) , fino)(xo ) ). There exist intervals [rh, 0 + m], [0 -m', O 2 1 2] l 2 such that for all 0 € [01,01 + m] and BE[02 - m , O2 ] Zo € ui

Then the splitting condition holds. Now fix x such that 0

< x < XO I , then

Let m be any given positive integer. Since (OIF)(n)(x) ---+ XO I as n ---+ 00, 1 there exists n' == n'(x) such that (OIF)(n)(x) > XO I - - for all n > n' . This m = 1 implies that Xn(x) > XO I - - for all n 2 n'. Therefore, lim infXn(x) 2 m XOI ' We now argue that with probability one, lim infXn(x) > XOI ' For n~~

.

this, note that

. If we choose

n-+oo

~-~

0 = --2- and e

> 0 such that XO

I

-

e>0

then min{OF(y) - fhF(y) : 0 2 (h + 0, y 2 XO I - e} 2 of(xol - e) > O. Write 0' == of(xo l - e) > O. Since with probability one, the LLd. sequence {e(n) : n = 1,2, ...} takes values in [e l + 0, e2 ] infinitely often so that 1 lim infXn(x) > xO I - 1.. + 0'. Choose m so that - < 0'. Then with n ~ oo m m probability one the sequence Xn(x) exceeds XOI' Since x02 = fJn)(xo2 ) 2 X n(xo2 ) 2 Xn(x) for all n, it follows that with probability one, Xn(x) reaches the interval [xo I , xo2 ] and remains in it thereafter. Similarly, one can prove that if x > X0 2 then with probability one, the Markov process Xn( x) will reach [xo l l x o2 ] in finite time and stay in the interval thereafter. REMARK 3.1. The proof of this result holds for any bounded nondegenerate distributioin of e (if E is the support of the distribution of e, define el == inf c < e2 == sup £). 3.2.3. An estimation problem. Consider a Markov chain X n with a unique stationary distribution 71". Some of the celebrated results on the ergodicity and the strong law of large numbers hold for rr-almost every initial condition. However, even with [0,1] as the state space the invariant distribution 71" may be hard to compute explicitly when the laws of motion are allowed to be non-linear, and its support may be difficult to determine or may be a set of zero Lebesgue measure [see Bhattacharya and Rao [2]] . Moreover, in many economic models, the initial condition may be historically given, and there may be little justification in assuming that it belongs to the support of 71". Consider then a random dynamical system with state space [e, dJ (without loss of generality for what follows choose e > 0). Assume r consists of a family of monotone maps from S with S, and the splitting condition (H ) holds. The process starts with a given x. There is, by Corollary 3.1, a unique invariant distribution (a stochastic equilibrium) 71" of the random dynamical system, and (3.13) holds. Suppose

RANDOM DYNAMICAL SYSTEMS IN ECONOMICS

we want to estimate the equilibrium mean ~

n-1

L: X j .

Is Y7f'(dy) by

193

sample means

We say that the estimator ~ L: X, is vn-consistent if

j=O

j=O n-1

~ L x, =

(3.14)

j=O

J

Y7f'(dy)

+ Op(n- 1/ 2 )

where Op(n- 1/ 2 ) is a random sequence en such that len/n- 1/ 21 is bounded in probability. Thus, if the estimator is vn-consistent, the fluctuations of the empirical (or sample-) mean around the equilibrium mean is Op(n- 1/ 2 ) . We shall outline the main steps in the verification (3.14) in our context. For any bounded (Borel) measurable f on rc, dj, define the transition operator T as:

Tf(x) = hf(y)p(x,dY) By using the estimate (3.13), one can show that (see Bhattacharya and Majumdar [4]' pp. 217-219) if f(z) = z - I Y7f'(dy) then 00

00

sup

L

x

n=m+1

ITnf(x)1 ~ (d - c)

L

(1 - 8)[n/N]

--->

0

as m

---> 00

n=m+1 00

Hence, 9 = -

L: T N f

[where TO is the identity operator I] is well-defined,

n=O 00

and g, and Tg are bounded functions. Also, (T - 1)g

= - L: T" f + n=l

00

L: TN f = f . Hence, n=O n-1

n-1

L f(Xj) = L(T - I)g(Xj) j=O

j=O n-1

= L((Tg)(Xj) - g(Xj)) j=O n

= L[(Tg)(Xj-d - g(Xj)] + g(Xn )

-

g(Xo)

j=l

By the Markov property and the definition of Tg it follows that

where Fr is the a-field generated by {Xj : 0 ~ j ::s; r} . Hence, (Tg)(Xj_I)g(Xj)(j ~ 1) is a martingale difference sequence, and are uncorrelated,

194

MUKUL MAJUMDAR

so that n

k

(3.15)

EL [(Tg(Xj-d - g(X j ))]2 = LE((Tg)(Xj-d - g(X j ))2. j=1

j=1

Given the boundedness of 9 and Tg, the right side is bounded by n .a for some constant a . It follows that for all n where r/ is a constant that does not depend on X o. Thus, n-1

E(~ ~ x, J=O

J

Y1r(dy))2 ::; r/ln

which implies, n-1

~ L x, = j=O

J

Y1r(dy)

+ Op(n- 1/ 2).

For other examples of vn-consistent estimation, see Bhattacharya and Majumdar [4] . 4. Iterates of quadratic maps. On the state space S = (0,1) consider the Markov process defined recursively by

X n +1 = a g n + 1 X n (n = 0, 1,2, ...)

(4.1)

where {en : n ~ 1} is a sequence of LLd. random variables with values in (0,4) and , for each value (}£(0,4), a() is the quadratic function (on S):

(4.2)

aox

== ao(x) = Ox(l

- x)

o and d. A Lyapunov junction for the problem under examination is the kinetic energy 2:k IUkI 2 . 4. Regularity of the transition probabilities. The first main point of the proof of Theorem 3.3 is to prove that the transition semigroup has the strong Feller property. It means that for each bounded measurable function ep defined on the state space U, the function Ptep is bounded continuous. First assume that the noise excites all modes. In our notations, K contains all indices Ikl oo :s; N or, equivalently, the covariance f5j of the noise is a non-singular matrix. The Bismut-Elworthy-Li formula (see Bismut [2] and Elworthy and Li [10]) gives

(D(Pt'P)(UO),V)

=

1 -E['P(u(t,uo» t

it 0

(..P-1Du(s,uo)v,dB s ) ] ,

where u(t, uo) is the process , solution to (2.6), starting at Uo (one can see also Cerrai [3]). It turns out that P; has a regularising effect, in particular it implies the strong Feller property. In the general case, where K contains just a few modes.f one needs to have better information on the non-linear dynamics, which is ultimately responsible of the spreading of the noise forcing to all modes. Here we need the probabilistic version of Hormander celebrated theorem. THEOREM 4.1 (Malliavin [26], Stroock [35]). Consider the Stratonovich SDE in IRffi n

«x, =

F(Xt ) dt +

L Fi(X

t)

0

dW;,

i=l

Xo=x, 2 As it has been proved in E and Mattingly [11], for the spectral approximation of 2D Navier-Stokes equations, two modes, namely (1,0) and (1,1) are enough. In the 3D case, as it can be found in Romito [33], the three modes (1,0,0), (0,1,0), (0,0,1) are sufficient .

A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 203

where the vector fields F , F i , . . . , Fn satisfy suitable boundedness conditions, and assume that Hormander's condition holds: the Lie algebra generated by the vector fields F , F i , ... , F n , once evaluat ed at x , spans IR m . Then, for all t > 0, the random variable X, has an absolutely continuous distribution with a sm ooth COO density. Such a result has been established by Malliavin [26], and in its full strength by Stroock [34], [35]. A simplified argument is given in Norris [32]. In order to analyse how the non-linearity spreads the effect of the noise, we notice that the state space U, defined in (3.1), can be written as a direct sum of linear subspaces, namely,

where each point of Uk has all coordinates equal to zero , but the coordinates Uk corresponding to k, and k . Uk = O. In the same way we can define the Lie algebra U of vector fields 9 = L Gk a~k such that k . Gk = O. Moreover, we will consider the subspaces Uk of U of constant vector fields whose coefficients are in Uk . Now the main technical lemma follows. LEMMA 4.1. Let m and n be indices, such that Iml oo , [n] 00 ~ N , and consider two vector fields V E Urn , WE Un' Then (i) if m 1\ n = 0, then [[F,V], W] = 0; (ii) [[F,V] , W] = ~[[F, V + W], V + W] . Moreover, if m 1\ n i= 0, [m], i= Inl2 and [m + nl oo ~ N ,3 then the Li e algebra Urn +n is contained in the Lie algebra generated by the vecto r fields [[F, V], W], with V E Urn and W E Un . The last part of the lemma is the main point. We can interpret it in the following way: if the modes m and n are forced by the noise, the mod es corresponding to m + n are in the Lie algebra generated by the vector fields F and Xk. It means that the (m + n)-component of the system of SDEs is forced indirectly by the noise. Those components, again, can transmit the noise force to other components, and so on. If the set N of directly forced mod es generates the set of all modes, the equ ations behave as if some kind of forcing happens at each component. This is truly a geometric cascade for the Fourier modes of the Navier-Stokes equations. In order to substantiate the idea, we introduce the set A(N) with the following rules:

Ne A(N) , if k E A(N) , -k E A(N) , [> if m , nE A(N), m 1\ n i= 0, Iml2 i= Inl2 and [m + nl oo ~ N , then m+n E A(N). DEFINITION 4.1 (Determining sets of indices) . We say that a set [> [>

le c {k Ilkl oo ~ N} is a determining set of indices for the threshold N if the set A(N) contains all indices Ikloo ~ N . 3Here

Irnl2

is the Euclidean norm of the vector

rn,

namely Irnl ~ =

L rn;.

204

M. ROMITO

Consequently, we have the following result. THEOREM 4.2 . If the set N is a determining set of indices, then the Markov semigroup associated to the SDE (2.6) has the strong Feller property. 5. Irreducibility and the control problem. A Markov process is said to be irreducible if for each time T and each open set A of the state space, the probability that the process is in A at time T is positive. The irreducibility property is implied (see for example Stroock and Varadhan [36]) by a controllability property of the control problem which is obtained from the original system (2.6) by replacing the noise with some controls . In our case, we obtain the system

where Pk is a quadratic polynomial in the Uk and Vk are the controls. We say that the system is controllable if for each time T and each pair of points 1 2 U , u , there exists a control v such that the corresponding solution of the control problem starts at time 0 in u 1 and stops at time T in u 2 • Intuitively, the irreducibility is related to the control problem because, if the system is controllable, then this means that there are realizations of the noise such that the Markov process solution of the SDE can move from u 1 to u 2 . A necessary condition for such a property is indeed the hypo-ellipticity (see Jurdjevic [18] for generalities on the geometric control theory) . Heuristically, if at each point of the state space , our system is allowed to follow any direction in any way (only the first assumption is true in the hypoellipticity proof) , then the system is controllable. Roughly speaking, the Lie brackets that have been evaluated in the previous section represents all the possible directions that the process can follows . Indeed, the drift of the equations would lead the process to follow the field :F of the deterministic dynamics . The kicks of the noise allow the process to follow other directions, which are a combination of the original dynamics :F and the directions Xk of the kicks. If the polynomial P has odd degree, the hypo-ellipticity condition is also sufficient (see Jurdjevic and Kupka [19]), since by changing u ----+ -u , the system can always follows both ways of each direction. The problem is, the Navier-Stokes system gives rise to a polynomial of even degree. Such quadratic terms in some sense give a preferential direction to be followed by the system. In order to give a clearer idea of such phenomena, one can see the following section , where an extremely simple SDE is presented, which gives a hypo-elliptic diffusion, and such that the upwards direction is preferential. The key property that the spectral approximation of the Navier-Stokes equations enjoys is the absence of such dangerous quadratic terms, and it is a consequence of properties (i) and (ii) of Lemma 4.1.

A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 205

/

" " , , ..,. ""

""

."

... . . ".

tI ; ~""., of If'

,

.,., ,,

of".,"

, , • , ,. ,,, t

..

, ,

..

r ,.

"

..

", , ,. ,,, , ,.

..

" "",,,.,,,,, .,,,,,,, ,,,,. ...... i.," , "".,,, , , , ,, ,, , t, ,,.,. "\. .. ... , "''', ,''' " , f' .. ::~ :::::::;:;: :: ~ ~ :: : ~ ~ ~ ~ ~ ~ : ~ : ...

... ;

...

·"

jj

· ,·"."

I I

11 '

·

'

..

,.

'

,.

\\

,

... ", " " ,,,,,., " " 1"""""'4",,,,, ,·,1" """, •• , ,"' ".. 1"""".,,· ."." ••" .,, ". ' """ . . "'1."" ", ", ,."

..

~ • I I ' 'r ••••• ••• I' ~ . t j ".,. ,' r

........

• • • • 14"

.,."""""""

J "'''''i''"ij .,..,.",,,,,,,, " i I. ' iI j

j

j

j

j j

j

t

.,..,."'''''·,, · ' , ; ,

Jjji~~~~~~~~~~~~~~~~~~~~~~Il~~ FIG. 2. Th e directions of the deterministic dynamics F. THEOREM 5.1. If the set N is a determining set of indices, then the Markov semigroup associated to the SOE (2.6) is irreducible. REMARK 5.1. In [1], Agrachev and Sarychev prove something more for the 2D case. Namely, they prove controllability in observed projection, that is, given a spectral threshold N , an initial point uO E L2([0, 27r]2) , with divergence equal to zero, and a time T > 0, there is a control that steers the solution of the infinite dimensional Navier-Stokes equations from uO to a point with assigned finite dimensional projection . As a consequence, approximate controllability holds for the infinite dimensional stochastic POE.

6. A toy model. As an example of how such things go on, we shall examine the following toy model, where we will see how the ideas of th e previous section can work. Consider th e system of stochastic differential equations dx; = - Xt + y; - XtYt + dB t , { dYt = -Yt + x; - XtYt,

(6.1)

where B, is a one dimensional Brownian motion . We will see that the diffusion given by the SDE is hypo-elliptic (so that strong Feller prop erty holds), but the syst em is not controllable. Hence the controllability results on the Navier-Stokes dynamics depend heavily on the geometrical structure of the non-linearity. We can have a rough idea of the deterministic dynamics from Figure 2. It is easy to see that the system has a global in time solution (just apply + Y;). Define the following vector fields Ito formula to

x;

2

a

a

2

F= (-X+Y - xY)ax +(-y+x -xY)ay

and

x=

a ax '

where F is the vector fields given by the deterministic dynamics (the drift) and X is the one given by the direction of the noise. Since

UT,Xl, x]

a

= 2 ay'

206

M. ROMITO

it follows that that the Lie algebra generated by F and X has full rank, once it is evaluated at each point of the state space IR 2 . In other words, Hormander's condition holds. On the other hand, the system is not globally controllable. Indeed, if one consider the solution at the starting point (xo, Yo) = (0,0), it is easy to solve the equation for Yt (it is a linear equation with random coefficients), thus obtaining

which is almost surely non negative. In view of the control theory issues explained in the previous section, one can explain the above phenomenon in the following way: since the direction Ox is the one forced by the noise, both Ox and -Ox are directions that the system is allowed to follow, and Oy = [[F, X], X] as well. On the contrary, the vector field -Oy (which is in the Lie algebra, thus ensuring the regularity of the transition probability densities) is a forbidden direction for the system. REMARK 6.1. The Markov process solution of the system (6.1) is anyway ergodic. Hence, controllability arguments are just sufficient for the proof of uniqueness of the invariant measure. One can use the same argument that is used in E and Mattingly [l l], namely it is sufficient to show that the neighbours of the origin, which is the stable point aitracior of the dynamics, are recurrent for the process. 7. Some numerical results. In this last section we aim to investigate what happens in the case where the hypo-ellipticity condition does not hold. The Navier-Stokes equations, without any external forcing, converge to 0 as the time goes to infinity. If we have an external forcing, namely a white noise forcing, which excites only a few number of modes, it may happen that the other modes are indirectly forced by means of the nonlinearity. This is what essentially happens under the assumptions of the previous sections. Assume now that the forced modes do not constitute a determining set of indices (as defined in Definition 4.1). Intuitively, we believe that in this case the system experiences a sort of decoupling, where the (directly and indirectly) forced modes behave as in the former case, while the others behave as if there were no forcing, hence collapsing to zero. If this picture turns out to be true, it follows that the invariant measure concentrates on a hyper-plane of the state space . 7.1. The numerical simulation. In order to lower the number of equations of the system to be numerically solved, we have chosen to examine the approximation of the 2D Navier-Stokes equations in the simplified

A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION ... 207

vorticity" formulation that has been introduced in E and Mattingly [11] . Namely, we consider the following system of SDEs 2

dCk = - vJk l ck +

h.L · I ""' h.L · I k W(ShSl - ChCi) + L W (ChCl + shsd + O"k d{3~' , h+l=k h-l=k ""'

L

2 ""' h.L · I h.L · I k dsk=-vlkl Sk- L W(rhS1+S hCi)+ ""' L W(ShCi-ChSd+'Ykd.B:' , h+l=k h-cleek

where the vorticity is given by

~(t, x) =

L Ck(t) cos(k· x) + Sk(t) sin(k · x) k

and the sums are extended to all pairs k = (k l , k 2 ) of positive integers, each one being less than the spectral threshold N . In order to solve numerically the system of stochastic equation, we have used the backward Euler method (a standard reference for the approximation of stochastic equations is Kloeden and Platen [20]) . In few words , say that the equation to be approximated is y' = f (y) + a dWt , where y is a vector. Then at each time-step,

where b. Wn are independent centred Gaussian random variables, having variance equal to the size of the time step. The implicit equation in the unknown Yn+! is solved using the Newton's method. The problem is stiff, due mainly to the linear part (there are the large negative coefficients -vlkI 2 ) . In order to obtain relevant results we have taken a small viscosity and a comparable intensity of the noise. Different realizations of the simulation have been run , with different initial conditions. Each initial condition has been chosen with a centred st andard normal distribution. Notice th at, even though we know from the theoretical analysis that the Markov process converges for each init ial condition, the same claim can be incorrect if the size of the initial condition is too large (see for example Higham, Mattingly and Stuart [17]) . All the numerical experiments in the following have been run with the following value of the parameters: t> t> t>

Grashof number'' Gr = 50, spectral threshold N = 8, tim e increment b.t = 0.05.

4The vorticity, in the 2D case , is defined as the scalar field given by € = curl u , wher e u is the velocity field. In the Fourier coord inates, if (Uk)kE71 d are the Fourier coefficients of the velocity field, the coefficients of the vorticity are given by €k = ik X Uk . 5The Grashof number is defined (in the 2D case) as the ratio between the intensity of the force and the square of the viscosity (see for example Foias , Manley, Rosa and Tem am [13]).

208

M. ROMITO

'"

17~

'os

'"

'2 5

~ tl t

t

tP

t 11 11

J I

JlJ5

175

t

10!

FIG. 3. The errors graphic of the (1,1) are forced.

Ck

coefficients. The three modes (1,0) , (0,1) and

The first data obtained have been discarded, they should correspond to the transient regime and they don't give any useful information concerning the invariant measure. The time increment is rather small , because of the stiffness of the problem. REMARK 7.1. For the implementation of the numerical method (especially for the random number generator), we have heavily used some libraries from the GNU Scientific Library [37]. 7.2. Conclusions. According to the results of the numerical simulations, the picture we have figured out , seems to be reasonable. The first case we examine is the one where the three modes (1,0), (0,1) and (1,1) are excited. According to the theoretical result (See Theorem 3.3), the invariant measure is fully supported. The empirical averages of the process hence converge to the expectations with respect to the invariant measure. We see that all the modes are excited (see Figures 3 and 4), even though with different intensity (Notice that the size of the standard deviations has been rescaled by a factor .3 for the sake of readability). In the second case, only the mode (1,0) is excited. Most of the energy then stays in the forced mode and in a few others (see Figures 5 and 6). Notice that in this case the dissipation scale is larger than in the previous case, so that the system dissipates the energy faster than in the previous case. Figure 7 summarises and compare the two cases. It represents the intensity of each J c~ + component in the two cases examined, depending on the position in the Fourier space . The largest values correspond to the modes being excited by the noise.

sa

A GEOMETRIC CASCADE FOR THE SP ECT RAL APPROXIMATIO N... 209

t It. tt 1

F IG . 4. Th e errors gmphic of the Sk coefficients. Th e three modes (1, 0), (0, 1) an d (1, 1) are f orced.

FIG. 5 . Th e error s gm phic of the

Ck

coeffi cien ts. Only the mode (1, 0) is f orced.

M. ROMITO

210

FIG. 6. The errors graphic of the

Sk

coeffi cients. Only th e mode (1,0) is fo rced.

FIG. 7. The values of the means for Jc~ + s~. On th e left , only the mode (1, 0 ) is forced. On the right, the three mod es (1,0), (0,1) and (1, 1) are forced.

A GEOMETRIC CASCADE FOR THE SPECTRAL APPROXIMATION... 211

7.2. It has to be remarked that the numerical simulations we have run , are of little interest from the point of view of turbulence , and their only aim is to give a support to our conjecture. Indeed, the spectral threshold is too small and the main parameter, the Grashof number, corresponds to a not so turbulent regime. REMARK

Acknowledgements. The author wishes to thank D. Blomker, P. Constantin, F . Flandoli, M. Hairer, J. Mattingly, A. Sarychev for the helpful conversations on the subject of this paper . The author wishes also to thank the Institute for Matbematics and its Applications, at the University of Minnesota, for its warm hospitality during the Summer Program on Probability and Partial Differential Equations in Modern Applied Matbematics. REFERENCES [1] A.A. AGRACHEV AND A.V . SARYCHEV, Navier-Stokes equation controlled by degenerate forcing: controllability in finite-dimensional projections, to appear on J . Math. Fluid Mech. [2] J.M . BISMUT, Martingales , the Malliavin calculus and hypo-ellipticity under general Hormander 's conditions, Z. Wahrsch. Verw. Gebiete 56 (1981) , 469-505 [3] S . CERRAI, Second Order PDE's in Finite and Infinite Dimension, Lecture Notes in Math. 1762, Springer Verlag, 2001. [4] P . CONSTANTIN , C . FOlAS, AND R . TEMAM , On the large time Galerkin approximation of the Navier-Stoke s equations, SIAM J. Numer. Anal. 21 (1984) , no. 4, 615--634. [5J P . CONSTANTlN, Geometric statistics in turbulence, SIAM Re view 36(1) (1994) , 73-98. [6] G . DA PRATO AND A. DEBUSSCHE, Ergodicity for the 3D stochastic Navier-Stokes equations, preprint (2003) . [7] G . DA PRATO AND J . ZABCZYK, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Not e Series , 229. Cambridge Unive rsity Press, Cambridge, 1996. [8] J .L. DOOB, Asymptotic properties of Markov transition probabilities, Trans. Amer. Math. Soc. 63 (1948) , 393-421. [9] J .P . ECKMANN AND M. HAIRER, Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise, Comm. Math. Phys. 219 (2001) , no . 3, 523-565. [10] D.K. ELWORTHY AND X.M . LI, Formulae for the derivatives of heat semigroups, J . Func. Anal. 125 (1994) , 252-286. [11] W . E AND J .C . MATTlNGLY, Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation , Comm. Pure Appl. Math. 54 (2001), no . 11, 1386-1402. [12] F . FLANDOLI, Irreducibility of the 3-D stochastic Navier-Stokes equation, J. Funct. Anal. 149 (1997), no. 1, 160-177. [13] C . FOlAS, O . MANLEY, R . Rosx , AND R. TEMAM, Navier-Stokes equation s and turbulence, Cambridge University Press, Cambridge, 2001. [14] G. GALLAVOTTI, Foundations of fluid dynamics, translated from the Italian. Texts and Monographs in Physics. Springer Verlag, Berlin, 2002. [15] M . HAIRER, Exponential mixing for a stochastic partial differential equation driven by degenerate noise, Nonlinearity 15 (2002), no. 2, 271-279. [161 R .Z. HAS'MINSKif, Ergodic properties of recurrent diffusion processes and stabilization of the solutions of the Cauchy problem for parabolic equations, Theory Probab. Appl. 5 (1960) ,179-196.

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[17] D.J . HIGHAM, J .C. MATTINGLY, AND A. M. STUART, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl. 101 (2002), no. 2, 185-232. [18] V. JURDJEVIC, Geometric control theory, Cambridge Studies in Advanced Mathematics 51, Cambridge University Press, Cambridge, 1997. [19] V. JURDJEVIC AND I. KUPKA, Polynomial control systems Math. Ann . 272 (1985), no. 3, 361-368. [20] P .E. KLOEDEN AND E . PLATEN, Numerical solution of stochastic differential equations , Applications of Mathematics 23, Springer Verlag, Berlin, 1992. [21] A.N . KOLMOGOROV, Local structure of turbulence in an incompressible fluid at a very high Reynolds number, Dokl. Akad. Nauk SSSR 30 (1941), 299-302 ; English transls., C.R. (Dokl.) Acad . ScL URSS 30 (1941), 301-305, and Proc. Ray. Soc. London Ser . A 434 (1991), 9-13 . [22] L.D . LANDAU AND E .M . LIFSHITZ, Fluid mechanics, Course of Theoretical Physics, VoI. 6 . Pergamon Press, London , Paris, Frankfurt, 1959. [23] J. LERAY, Etude de diverses equations integrales non lineaires et de quelques probiemes que pose l'hydrodynamique, J. Math. Pures Appl. 12 (1933), 1-82 . [24] J . LERAY, Essei sur les mouvements plans d'un liquide visqueux que limitent des parois, J . Math. Pures Appl. 13 (1934), 331-418. [25] J . LERAY, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248 . [26] P . MALLlAVIN Stochastic calculus of variation and hypoelliptic operators, Proceedings of the International Symposium on Stochastic Differential Equations (Res . Inst . Math. ScL, Kyoto Univ., Kyoto (1976), 195-263 , WHey, New York Chichester Brisbane, 1978. [27J J .C. MATTINGLY, E. PARDOUX, Malliavin calculus and the randomly forced NavierStokes equat ions, preprint (2003). [28] J .C . MATTINGLY, On recent progress for the stochastic Navier-Stokes equations, to appear on Journees Equations aux derivees partielles. [29] S.P . MEYN AND R .L. TWEEDlE, Markov chains and stoch astic stability, Communications and Control Engineering Series. Springer-Verlag London, Ltd ., London, 1993. [30] S.P . MEYN AND R .L. TWEEDlE, Stability of Markovian processes. Il . Continuoustime processes and sampled chains, Adv . in AppI. Probab. 25 (1993), no. 3, 487-517. [31] S.P . MEYN AND R .L. TWEEDlE, Stability of Markovian processes. IlI. FosterLyapunov criteria for continuous-time processes, Adv . in AppI. Probab. 25 (1993), no. 3, 518-548. [32J J . NORRlS, Simplified Malliavin calculus, Seminaire de Probabilites, XX, 1984/85, 101-130, Lecture Notes in Math., 1204, Springer, Berlin, 1986. [33] M. ROMITO, Ergodicity of the finite dimensional approximation of the 3D NavierStokes equations forced by a degenerate noise, J . Stat. Phys. 114 (2004), Nos. 1/2,155-177. [34] D.W . STROOCK, The Malliavin calculus, a function al analytic approach, J. Funct. Anal. 44 (1981), no. 2, 212-257 . [35] D.W. STROOCK, Some applications of stochastic calculus to partial differential equations, Eleventh Saint Flour probability summer school-1981 (Saint Flour, 1981), 267-382, Lecture Notes in Math. 976, Springer, Berlin, 1983. [36] D.W . STROOCK AND S.R.S . VARADHAN, On the support of diffusion processes with applications to the strong maximum principle , Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ . California, Berkeley, Calif., 1970/1971), Vol. Ill: Probability theory, pp. 333-359. Univ. California Press, Berkeley, Calif. , 1972. [37] VV . AA ., The GNU Scientific Library, http;//VWIo7.gnu.org/software/gsl/ .

INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS BJORN SCHMALFUSS· Abstract . The intention of the article is t o show the existence of inert ial manifolds for random dyn amical systems generated by infinite dimensional random evolution equations. To find these manifolds we formulate a random graph transform . This transform allows us to introduce a random dynamical system on graphs. A random fixed point of this system defines the graph of the inertial manifold. In contrast to other publications dealing with these objects we also suppose th at th e linear part of such an evolut ion equation contains random operators. To deal with these objects we apply th e multiplicative ergodic theorem. The key assumption for the existence of an inertial manifold is an w-wise gap condition. Key words. Inertial manifolds, random dyn amical systems, stochastic partial de 's, mult iplicative ergodic theorem. AMS(MOS) subject classifications.

Primary: 37HlO ; secondary: 37H15 ,

60H15, 35B42 .

1. Introduction. Inertial manifolds are objects that allow to interpret the dimension of the long-time dynamics of ordinary or partial differential equations. In the case of a partial differential equation this dyn amics will be finite dimensional. Inertial manifolds are Lipschitz continuous manifolds in the phas e space. On the graph of these manifolds the dynamics is given by an ordinary differential equation of smaller dimension than th e dimension of the original equation. States outside these manifolds will be attracted by th e inertial manifolds exponentially fast . A st andard method to show th e existence of th ese manifolds is the Lyapunov-Perron method. Using this technique the graph of the inertial manifold is given as a fixed point of an operator equation related to our original differential equation, see Chow and Lu [7] , Chueshov [8], Constantin et al.[13], Foias et al. [17] and Temam [23] . Similar methods has been used to find inertial manifolds for sto chasti c (partial) differenti al equations, see Bensoussan and Flandoli [3]' Da Prato and Debussche [14]' Chueshov and Girya [10]' [11]' Duan et al. [16] and for delay equations see Chueshov and Scheutzow [12] and references therein. Another technique for non-autonomous dyn amical syste ms has been introduced by Koksch and Siegmund [18] . In contrast to these applications we will use another method. This method is called graph transform method. This transform allows us to introduce a random dyn amical syst em defined on graphs. A random fixed point for this system defines t he random graph of the random inertial manifold. • Mathematical Institute, University of Paderborn, Warburger StraBe 100, 33098 Pad erborn, Germany (schmalfuss0upb . de) . 213

214

BJORN SCHMALFUSS

This method has been introduced by SchmalfuB [211 and by Duan et al. [15] but only for invariant manifolds. In contrast to all these applications we consider more general random evolution equations. Especially, we are able to treat equations with a random linear part and with non-linearities having a random Lipschitz constant. The curial assumption to find invariant manifolds is the gap condition. We will formulate an w-wisegap condition containing the random Lipschitz constant and random coefficients of the exponential dichotomy condition of the linear part. As a main tool to achieve our results we have to apply the multiplicative ergodic theorem. Since this article is base on techniques from the theory or random dynamical systems we introduce in the next section basic terms from this theory. In Section 3 we introduce the random evolution equation for which we will study random inertial manifolds. Section 4 contains the definition of the random dynamical system on graphs . The fixed point argument giving us the random inertial manifold is described in Section 5. The last section contains examples. 2. Random dynamical systems. In the following we are going to describe the dynamics of systems under the influence of random perturbations. Such a perturbation is given by a noise. The mathematical model for a noise is a metric dynamical system. DEFINITION 2.1. Let (0, .1', JP» be a probability space. Suppose that the measurable mapping B : (IR x 0, B(IR) 0 F)

-+

(0, F)

forms a flow :

Bo = ido , for t, r E IR. The measure JP> is supposed to be invariant (ergodic) with respect to (J. Then the quadro-tuple (0, .1', JP>, (J) is called a metric dynamical system. Metric dynamical systems are generated by the Brownian motion (see Arnold [2] Appendix A) or by the fractional Brownian motion (see Maslowski et al. [19]) or by random impulses (see Schmalfuf [22]). For instance we can choose for the set Co(lR, U) of continuous functions on IR which are zero at zero with values in a separable Hilbert space U. For .1' we choose the associated Borel-a-algebra with respect to the compact open topology, for JP> we choose the Wiener measure Le. the distribution of a Wiener process for some covariance Q and B is given by the Wiener shift :

°

Btw( ·) = w(· + t) - w(t)

for w E 0.

Then we obtain the metric dynamical system (0, .1', JP>, B) which is called the Brownian motion. Indeed JP> is ergodic with respect to B.

INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS 215

In the following we will always suppose that we have a metric dynamical system such that JPl is ergodic with respect to B. Let H be some topological space. A random dynamical system with phase space H and with respect to a metric dynamical system is a measurable mapping

cjJ : (]R+ x D x H,B(]R+) ®F ®B(H)) ---. (H,B(H)) satisfying the cocycle property

cjJ(t + T, W, x) = cjJ(t, (}rw, cjJ( T, W, x)),

cjJ(O ,W, x) = x.

The eocycle property is a generalization of the semi-group property. It reflexes the fact that the deterministic dynamics is perturbed by a noise. For our applications we will always suppose that H is a separable Hilbert space with norm 11 . 11 . Later we will allow a slight modification of the measurability of such a system. Generators for random dynamical systems are solution operators for random or stochastic differential equations. For instance , consider the random differential equation with phase space: H = ]Rd

dcjJ dt = f((}tw, cjJ), Suppose that this equation has a globally unique solution for any noise path wand for any initial condition x E ]Rd. This solution at time t is denoted by cjJ(t, w, x). These operators form a cocyclejrandom dynamical system . The measurability of this mapping then follows by some regularity assumptions about the right hand side f . Another tool we need is the multiplicative ergodic theorem. We consider the linear differential equation (2.1) where B is a random matrix. Then this equation generates a linear random dynamical system with time set ]R, see Arnold [2] 3.4.15. We can describe the dynamics of this random dynamical system by the following theorem which is the multiplicative ergodic theorem: THEOREM 2.2. Let B E ]Rd ®]Rd be a random matrix contained in L 1 (D, F, JPl, (}) where (} is supposed to be ergodic. Then there exists a {(}t hEIR -invariant set of full measure, non random numbers pEN, P ::; d, -00 ::; Ap < Ap-l < ... < Al and d 1 , ' " , dp E N and random linear spaces E 1(w),·· · ,Ep(w) of deterministic dimension d1 , ' " ,dp such that

n

(2.2)

]Rd =

E 1(w) EB,, · EB Ep(w)

for wEn. The spaces E, are invariant with respect to the random dynamical system 'IjJ generated by (2.1): 'IjJ(t,w,Ei(w)) = Ei((}tW)

fori = 1,'" ,p,

wEn,

t E]R

216

BJORN SCHMALFUSS

and

)\{} 11'm logll7/J(t,w)xll __ /I,.. if an d only if x E E( i W 0

t_±oo

t

for wEn . The spaces E; depend mea surably on w. In particular, there exist measurable projections on these spaces.

The numbers Ap , ' " ,AI are called Lyapunov exponents to (2.1). The spaces E, are the Oseledets spaces to (2.1). The assertions of the multiplicative ergodic theorem are only true with respect to a {B t hEIR-invariant set of full measure. We can restrict our original metric dynamical system to this set such that B is B(IR) ® F n,F n measurable where the F n trace (J algebra with respect to We will denote this new metric dynamical system by the old symbols (0, F , IP', B). If we can modify our original metric dynamical system in this sense such that some property holds for the new metric dynamical system we say that a property holds B-almost surely, B-a.s. The intention of this article is to find sufficient conditions for the existence of random inertial manifolds for random dynamical systems . We start with basic notations for this purpose . We call a family of random parametrized sets M {M(W)}wEn, M(w) cH a random set if the mapping

n

n.

0:3 w -+ dist(y, M(w)):=

inf

xEM(w)

Ilx -

yll

is a random variable for any y EH. Such a set is called a positively invariant random set with respect to a random dynamical system cP if

cP(t,w ,M(w)) c M(Btw) for t

~

O,W E O.

Let HI, H 2 be a splitting of the Banach space H :

In addition we suppose that there exist continuous projections HI, H 2 . Let

1fl , 1f2

onto

such that w -+ ')'(w ,x) is measurable for fixed x E HI. For fixed w the mapping x -+ ')'( w, x) is Lipschitz continuous . Then we call the set

INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS

217

a random Lipschitz manifold. Indeed M is a random set what follows from Castaing and Valadier [6] Lemma 111.14. In addition, we call a random positively invariant set M exponentially attracting with respect to the random dynamical system 4> if lim

inf

t-oo zEM(8 t w)

114>(t,w,x) - z] = 0

with exponential speed for any x E H. DEFINITION 2.3. An exponentially attracting random Lipschitz manifold which is positively invariant for a random dynamical system 4> is called a random inertial manifold for 4>. The dimension of this manifold is defined by the dimension of HI . Let X be a mapping on 0 with values in jR+. Such a mapping is called tempered if the mapping t --t X (Btw) is subexponentially growing for t --t ±oo B-a.s.: lim log+ X(Btw) = O. t

t-±oo

Note that if X is a random variable on the ergodic metric dynamical system (0, F , P, B) then there exists only one alternative for the above property. Suppose that the random variable X is not tempered. Then we have

(2.3)

. log+ X(Btw) 1im sup =

B- a.s.

00

t

t-±oo

The following lemma is not hard to prove . LEMMA 2.1. (i) Sufficient for temperedness of the random variable X is that lE sup log" X(Btw)
if 4>(t,w,X(w)) for t 2: 0 B-a.s.

= X(Btw)

218

BJORN SCHMALFUSS

3. random evolution equations. The objective of our interest will be the dynamics of the following random evolution equation du dt + A(Btw)u = F(lhw, u).

(3.1)

Our task is to formulate conditions which ensure the existence of an inertial manifold . We start with the random linear differential equation

du dt

(3.2)

+ A(Btw)u =

O.

To treat the non-linear evolution equation by the variation of constant formula we have to assume that t ---? -A(Btw) generates a fundamental solution. For the definition see Amann [11 Chapter 11.2. More precisely, we assume that A generates a random dynamical system of linear continuous mappings on H:

U(t + r,w) = U(t,Brw)U(r,w). Details about the existence of such a random dynamical system can be found in Caraballo et al. [41 . For the following we need the exponential dichotomy condition: There exist continuous projections 7rl, 7r2 related to the splitting of the phase space H = HI Ef7 H 2 commuting with U:

We suppose that U (t, W)7rl defined on the finite dimensional space HI is invertible where these inverse mappings are denoted by a negative first argument:

In addition, we assume that there exist random variables aI, a2 E i, (n, F , JID) such that (3.3)

IIU(t,w)7rl ll

::; eJ~ al(O.w)ds

for t ::; 0

IIU(t, W) 7r2 11

::;

eJ~a2(O.w)ds

for t ~ O.

(It follows from assumptions given below that al(w) > a2(w).) We now consider the complete non-linear equation (3.1). Let us assume that W ---?

and that

F(w, x)

is measurable for any x

INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS 219

is Lipschitz continuous

(3.4)

117r IF (w, x) - 1r IF(w , y)11

:::; L(w)llx - YII,

111r2F(W, x) - 1r2F(W, y)11

:::; L(w) llx - yll

for any wEn where the Lipschitz constant L(w) depends on w. In particular we assume that this Lipschitz constant is locally integrable:

l

b

L(Bsw)ds < 00 for -

< a < b < 00 .

00

The Lipschitz continuity and the existence of a fundamental solution of the linear problem ensure that the equation

T(w , ~(w)) we conclude that T = T(w , ~(w)) + T, T ~ T(OT(w,l« w)) , ~(OT(w ,l«w)))) . On account of Lemma 4.4 (iii) we know that (T(w , ~(w)),w,,) E G(OT(w,l«w))w) , , E G(w) (T,w,,) E G(OTW) exists by Lemma 4.2. More precisely, for the parameters T, w" the problem (4.1) has a unique solution. But rewriting this system as we have seen in the proof of Lemma 4.2 we obtain that

We have to use the bijection property of Corollary 4.3. Similarly, if we have the eocycle property on [0, Ti-1(W)] then we can extend the eocycle property to [0,Ti(w)]. At first we show for the measurability property that for T ~ 0, , E Q and y+ E H+ the expression (Tj(w)I\T,w,,(w))(y+) is an w-wise limit of random variables hence measurable. Setting wO == y+ , VO == ,(w, y+) and generating

(w i ,Vi) := rr .l-y(w),T1(w)AT,w,y+ ( W i-l ,V i-l) where'Ly ,T,w,y+ has been introduced in the proof of Lemma 4.1. From the convergence of the sequence (wi , vi) we obtain for any y+ E H 1 v i - 1 )(T1(w) 1\ T)), Hm l-ti(W) = (T1(w) 1\ T, W, ,(w))(y+) = : I-t(w, y+) E G(OTAT1(w)W) .

l-ti(W, y+) :=

• _00

7l'2 ('Ly(W),Tl (w) AT,w,y+

(wi -

1

,

228

BJORN SCHMALFUSS

We now repeat this iteration for the setting p, T 2: T 1 (w):

="

W

=

(}Tdw)W for

W :

is again a limit of random variables, hence measurable. Extending this procedure we have that W - t iI>(Tj(w) 1\ T, w, I(W))(Y+) is measurable. Letting j - t 00 we get that W - t iI>(T,w,,(w))(y+) is measurable with respect w. We mention that the limit is just arrived for j 2: jo(w) because T 1\ Tj(w) = T for j 2: jo(w). From Lemma 4.1 we obtain that T - t iI>(T,w,,(w))(y+) is continuous. The assertion of the theorem then follows by Castaing and Valadier [6] Lemma III.14. We also note that y+ - t iI>(T, w,I(w))(y+) as a composition of (finitely many) continuous mappings is continuous. 0

5. random fixed points and random inertial manifolds. To see that the random dynamical system generated by (3.1) has an inertial manifold we will show that the random graph transform has a random fixed point. To obtain the existence of a random fixed point we are going to apply a version of the fixed point theorem from Schmalfuf [21] . As a preparation we need the following lemma: LEMMA 5.1. Suppose that the assumptions (4.8), (4.9) are satisfied. Then there exists a process

(t,w)

-t

K(t,w)

such that B-a.s.

11iI>(T,w,'1(W)) - iI>(T,w, 12(W)) lie ::; K(T,w)II'1(W) - 12(w)lle for /1, 12 E 9 and for any e

>0

lim K(T,w)e-(A2+(n,{}_nw ,"'/({}_nw))) is a Cauchy sequence {}-a.s. This follows by the exponential convergence of

lliI>(n, (}-nw,"'/({}- nw)) - iI>(n + 1, {}-n-1W, "'/({}-n-lw))lle (5.2)

:::; 11iI>(n, (}-nw,"'/({}-nw)) - iI>(n, {}-nw, iI>(1 , {}-n-1W , "'/({}-n-lw))lle

:::; K(n ,(}-nw)II"Y({}-nw) - iI>(1 , {}-n-1W, "'/ ({}- n- l w))lle -. O. We have used the temperedness of W -. 11iI>(1, (}-IW,"'/({}-IW)) lie, see Lemma 5.2. The limit of this sequence is denoted by "'/* such that ",/*(w) E G(w). We note that by the properties of F we have

232

BJORN SCHMALFUSS

but the right hand side defines a tempered random variable, see Lemma 5.3. The convergence in (5.2) also holds for T E lR:

IlrP(T, O_TW, ')'(O_TW)) - rP([T], O_[TjW, ')'(O_[TJW)) lie

:::; K([T], O_[TjW) IiI'(O-[TjW) - (T - [T], O-T+[TjW, ')'(O_T+[TjW))

lie

where [.] is the integer part of a real number . To see that ')'* is a random fixed point we can apply the eocycle property and the continuity of ')' ---+

(S,w, ')') E C(H l , H2): (S,w,')'*(w)) = (S,w, lim (n,O_nw,')'(O_nw))) n-oo

=

lim (S + n, O-S-nOSw, ,),(O-n-sOSw)) =

n-oo

')'* (Osw)

for S 2: 0 where we have to use the convergence in (5.2). As a limit in Cb(Hl, H2 ) we have that ,),*(w) E Cb(Hl , H2 ) . In addition for any x+ E H, the mapping W ---+ ')'*(w,x+) is the pointwise limit of random variables, hence measurable. We obtain directly from Lemma 5.1:

11(T,w,')'(w)) -')'*(OTW)I/e = 11(T,w,')'(w)) - (T,w,')'*(w)))lle :::; K(T,w)lII'(w) -')'*(w)I/e

o Applying Lemma 5.2 we have the following main result: 5 .2. Suppose that the inequalities (4.8), (4.9) are satisfied. Then the random dynamical system generated by (3.1) has a random THEOREM

inertial manifold. Proof. We just know that the random fixed point for the graph transform defines a graph of an invariant manifold for the random dynamical system rP. We have to show that this manifold is exponentially attracting. Let w ---+ X(w) be a (tempered) random variable in H . We introduce the graph ')'x(w,x+) == 7T2X(W) which is contained in g. We set for every

x+ E H, y+=y+(T,w, x+) =7TlrP(t , w, x++7T2X(W)),

Y(T,w)=y+(T,w, 7Tl X (W))

such that

7T2rP(T,W, 7Tl X (W) + 7T2X(W)) = (T, w, ')'X )(Y(T, w)). Notice that by Corollary 4.3 the mapping and the other hand we have

:=: exist for any')' E g. On

I/7T2rP(T, W, 7Tl X (W) + 7T2X(W)) -')'*(OTW ,Y(T, w))1/ :::; 11(T,w,')'x) -')'*(OTw)lle where the right hand side tends to zero exponentially fast by Lemma 5.3. 0 This formula is also true for X (w) == x EH.

INERTIAL MANIFOLDS FOR RANDOM DIFFERENTIAL EQUATIONS 233

6. Examples. In this section we will study two examples . For the first example a similar equation is just formulated in Chueshov and Girya [9] (see also Chueshov and Scheutzow [12] for delay equations and Duan et al. [16]). These considerations are based on the Lyapunov-Perron method. However, our first example demonstrates how the graph method works for those equations. Our second example is completely different to these equations because the linear part of the evolution equation is given by a stationary process. We consider the sto chasti c reaction diffusion equation (6.1)

au at

= t::.u + g(Btw, 1/, u) + ~(t, 1/),

u(O ,1/)

= x(1/) ,

1/ E (0, n) .

with homogeneous Dirichlet boundary conditions on the interval [0, n]. To formulate this equation as an evolution equation we introduce the phase space H = L 2 (0, n) . It is well known that t::. can be interpreted as the generator -A of a strongly cont inuous semigroup U(·) . The eigenvalues, eigenvectors of A are (i 2 ,sin(i1/)) for i E N. Suppose that ~ is a noise term given as the generalized temporal derivative of a two-sided Wiener process W with values in H defined on a filtered probability space (0, F , {Ft}tEIR, JPl) . The covariance of this process is supposed to be of trace class. For the exact definition of such an associated filtered metric dynamical system we refer to Arnold [2] Appendix A. For g we choose a mapping such

(0 x JR, Fo 0 B(JR)) 3 (w, x)

-t

g(w,1/, u)

is measurable for every u E JR, W

sup

-t

Ig(w ,1/, u)1 < 00

1]E[O,7rJ, uEIR

is tempered. We have the Lipschitz condition Ig(w, 1/, u) - g(w,1/, v)1:::; Llu -

vi

for u, v E JR uniformly for w, 1/. Then

H

3

v(·) - t g(w, ., v(·)) E H

defines a Lipschitz continuous mapping on H denoted by G(w, .). In oder to apply our general results we formulate (6.1) as a stochastic evolution equation on H:

(6.2)

du + Audt = G(Btw)dt + dW(t ,w).

We also note that the evolution equation dz + Az dt = dW(t)

234

BJORN SCHMALFUSS

has the unique stationary solution

Z(w) =

[~ U(-T)dW(T,w)

B- a.s.

The random variable Z is tempered. In order to show the existence of an inertial manifold we study the random evolution equation

dv (6.3)

dt

+ Av =

F(Btw ,v)

F(w,v):= G(Btw , v + Z(Btw)). Note that F has the same Lipschitz constant as G above. This equation is defined B-a.s. THEOREM 6.1. Choose an n such that

(6.4)

(n

+ 1)2 -

n2

= 2n + 1> 4L .

Then the random dynamical system

0 such that c2(x) :::: E, where C;(lR) is the set of functions on lR having bounded, continuous derivatives up to order k inclusive. (2) h(x) E Cr'+l(lR)nlHlm+l(lR). (3) O"(x) E Cr'+l(lR) and there exist two positive numbers 0 < o; < O"b such that 0"a :S 0"( x) :S O"b holds for all x E lR. For a given initial function II;"

sE[r,Tj

holds. Then, by the Picard iterative scheme, the equation (3.11) has a unique , strong solution 'l/Jr,t, which satisfies (3.14)

lE sup II 'I/Jr,sll;" ~

sE[r,Tj

K(m)IEIIif>II;n.

Since under the basic condition and if> E Cb(lR)+ nlHIm(lR) with m ~ 1, Theorem 1.1 can guarantee the existence of a nonnegative solution and the solution of Equation (1.1) has uniqueness, above 'l/Jr,t, thus, is just the nonnegative solution of Equation (1.1). The remaining parts of the conclusion follow from an argument similar to the proofs of Proposition 3 and Theorem 3 on page 139 of Rozovskii [5] . 0

REFERENCES [1] DA PRATO G . AND ZABCZYK J . (1992) . Stochastic Equations in Infinite Dimensions. Cambridge University Press, 1992. [2] DAWSON D.A ., LI Z., AND WANG H. (2001) . Superprocesses with dependent spatial motion and general branching densities. Electron. J. Probab., V6, 25: 1-33, 200l. [3] KURTZ T .G . AND XIONG J . (1999) . Particle representations for a class of nonlinear SPDEs. Stoch. Proc. Appl., 83: 103-126, 1999. [4] LI Z.H., WANG H ., AND XIONG J . (2004) . Conditional Log-Laplace functionals of immigration superprocesses with dependent spatial motion. Submitted, 2004. (Available at http ://darkwing . uoregon . edu/ ~haowang/research/pub.html.) [5] ROZOVSKII B.L. (1990). Stochastic Evolution Systems - Linear Theory and Applications to Non -linear Filtering. Kluwer Academic Publishers, 1990. [6] WALSH J .B. (1986). An introduction to stochast ic partial differential equations. Lecture Notes in Math., 1180: 265-439, 1986. [7J WANG H. (1998). A class of measure-valued branching diffusions in a random medium. Sto chastic Anal. Appl., 16(4) : 753-786, 1998.

NONLINEAR PDE'S DRIVEN BY LEVY DIFFUSIONS AND RELATED STATISTICAL ISSUES WOJBOR A. WOY CZYNSKI* Abstract . Recent work on nonlinear evolution equat ions of the form

Ut

= £u -

N u, wher e N is a nonlinear and , perhaps , nonloc al operator, and £ is an infinitesimal generator of a Levy pro cess is reviewed , and futu re challen ges are discussed. The role of selfsimilar solutions and th eir connect ion to t he study of scaling limits of random solutions is explored . The latter results are fundament al for development of statistical estimation procedures for parameters appearing in the original evoluti on equations.

Introduction. This paper is a written version of the talk delivered at the 2003 IMA Workshop on Probability and Partial Differential Equations. Most of the work described is a result of joint work with several colleagues, including T . Funaki , S.A. Molchanov, D. Surgailis. N. Leonenko, P. Biler and G. Kar ch. Th e relevant cit ations can be found in the Bibliography section. Let v : JR -. JR be a function in th e domain of operator 1 (1)

[.cv](x) =

J

(v(x+ y) - v(x) -l{IYI::>1} v'( x)y) L(dy) ,

an infinitesimal generator of a Levy process defined by t he Levy-Khin chin measure L satisfying condition J(1 I\x 2 ) L(dx) < 00. In the particular case of the symmetric a -stable process with L(dx) = LO:(dx) :=

Ixfl:O: '

0 < a < 2,

the infinitesimal operator .c = .co: can be simply expressed in the Fourier domain as a symmetric derivative (fractional Laplacian) of order a :

(2) Here, F stands for the Fourier transform and the multiplicative const ant required to transform (1) into (2) is omitted. For a = 2, operator .c 2 is local and represents the standard Laplacian,

the infinitesimal generator of th e stand ard Brownian diffusion. • Department of St atistics and Cente r for Sto chastic and Cha otic Processes in Science and Techn ology, Case Western Reserve University, Cleveland , OH 44106 (wawCDcase . edu) . 1 Int egral s without specific limits ar e assumed to be t aken over th e whole spa ce. 247

248

WOJBOR A. WOYCZYNSKI

We are interested in the initial-value problem for the one-dimensional (and multi-dimensional, with obvious adjustment of above definitions and notation) evolution equations

(3) (4)

Otu(t,x) = v.Cxu(t;x) - oxNu(t,x)), u(O, x) = uo(x),

where (t, x) E JR+ x JR, v > 0, and N is a nonlinear operator to be specified later. As usual, subscrips t and x indicate the variable an operator acts upon. Note that the nonlinear part in equation (4) is of the inertial (gradient) type. Thus the existence, uniqueness and properties of these equations depend on the interplay and competition between the smoothing influence of the diffusive operator C and the gradient-steepening properties of the inertial operator a.N. Nonlinear equations of this type appear in the study of growing interfaces in the presence of selfsimilar hopping anomalous surface diffusion. Developed in [5] model for the evolution of the elevation of such growing surface included equation like (3) and was based on a continuum formulation of mass conservation at the interface, including reactions. The experimental data for the diffusion of palladium atoms on a tungsten substrate indicated that the surface transport could be by a hopping mechanism of a Levy flight. For the same data, the best-fit algorithm suggested that the diffusive term in (3) is of the form C = ca, with a = 1.25. The existence and uniqueness results for such nonlinear and nonlocal equations were established, via analytic tools , in [1] and [2] (see also literature cited therein). Of course, linear equations involving fractional Laplacian have been extensively studied both in the physical and mathematical communities for a long time, see, e.g., [6, 8] and references listed there. The general goal of our program is to develop information about evolution of the solution random fields u( t, x) in the situation when the initial data uo(x) are random. Then, assuming knowledge of the field u(t, x) at "large" times, that information can be used to identify structural parameters (such as v, a, and, say, parameters describing uo(x)) involved in (3-4) so that the equation can be used for modeling purposes in real physical situations. A good illustration of this issue is the problem of statistical estimation of parameters in the homogeneous and isotropic "primordial" random field representing original distribution of mass in the Universe on the basis of quasi-Voronoi tasselation-like, cellular , large-scale mass distribution in the currently observed Universe, see [7]. In the so-called "adhesion" approximation of Zeldovich, evolution of the mass is described by the Burgers equation, a special case of (3), see Section 1. Our conclusions are: Given that the nonlinear evolution dictated by equation (3) is in general very complicated, the explicit solution of the above problem cannot be achieved directly by attacking the "inverse" problem; the basic approach is to study the scaling limits of solution random

NONLINEAR PDE'S DRIVEN BY LEVY DIFFUSIONS

249

fields using selfsimilarity properties of equations themselves. In a sense, this approach is analogous to the method of using the central limit theorem in classical statistical parameter estimation problems. Propositions 1.2, 4.3, and 4.5 give new statistical interpretations of previously obtained analytic results. The paper is composed as follows : In the first section we discuss selfsimilar solutions as limits of general solutions . Then , in Section 2, for the special case of the Burgers equation, we report on parablic scaling limits for initial data with short-range and long-range dependence . The resulting parameter estimation techniques are discussed in Section 3. Fractional Burgers-type equations, called here fractional conservation laws, will be introduced in Section 4, where the role of their selfsimilar solutions and scaling limits is also described. 1. Selfsimilar solutions of evolutions equations as limits of general solutions. A desirable scaling limit result for solutions of equation (3) with random initial data (4) would have a generic form

lim ,\au(,\bt , ,\C) = U(t, x ), >'->00

where the limit, for some choice of scaling constants a, b, and c, is understood in the sense of the weak convergence of finite dimensional probability distributions of random fields on the left-hand side to those of the random field on the right-hand side (the convergence of infinite dimiensional distrbutions has not been studied but is a desirable goal). The above result would be useful only if the limiting random field U (t, x) were a sort-of a standard object (like the Gaussian law in the central limit theorem) which had some kind of usable description either through finite-dimensional distributions, characteristics functions, n-point correlations, stochastic integrals, or other similar convenient descriptors. However, before we produce results of this kind let us begin with a statement of a th eorem due to Zuazua [11] which shows, in the case of nonrandom initial data, how a scaling limit of an arbitrary solution of the Burgers equation can be seen as such a standard selfsimilar object. Recall that the Burgers equation (see, e.g., [10]) corresponds to the choice of the Laplacian as a diffusive operator, E = £ 2 = b., and a simple quadr atic nonlinearity Nu = u 2 and is thus , in one dimension, of the form

(5) (6)

u (O, x )

= uo(x ),

In what follows , unless explicitely stated otherwise , we will take IJ = 1. THEOREM 1.1. Let p 2: 1 and let u (t , x ) be a solution of the initial value problem (5-6) with an integrable initial condition with

J

uo(x ) dx = M .

250

WOJBOR A. WOYCZYNSKI

Then lim t!(l-i)lIu(t, .) - U(t , ·)lIp = 0

(7)

t-+oo

where U(t , x) is the unique selfsimilar solution

U(t, x) = t-! e-:: (K(M) -

(8)

['-'I' e- 4 dZ) ,

such that JU(t,x)dx = M and U(t,.) ~ M80 as t It can be immediately verified that

(9)

~ O.

U(t , x ) = C! U(1,t-!x) .

In other words, if the similarity transformation is defined by the formula

(10) then the solution of the Burgers equation described in Theorem 1.1 is selfsimilar, that is,

(11)

U>.=U,

and the theorem implies that, for each t > 0,

J lu>.(t , x) - U(t, x )IP dx

= x-lJ l>.u(>.2 t, >.x) - >.U(>.2t, >.x)IP d(>.x) (12) =

>.p-l J lu(>.2 t, >.x) - U(>.2t , >.x)IP d(>.x)

= C!(P-l)(>.2 t)!(p-l) J lu(>.2 t, >.x) - U(>.2 t, >.x)IP d(>.x)

~0

as >. ~ 00. Thus, if solutions depend on random initial condit ion one gets, under obvious boundedness conditions, that for the random fields u(t, x ) and U(t , x) the following scaling limit result holds true: PROPOSITION 1.2. Let p 21 , and u(t,x) be a solution of the Burgers equation corresponding to integrable random initial data uo. Then, for the rescaled solution random field u>. ,

(13)

lim u>. = U,

>'-+00

where the limit is understood in the following sense: for each t > 0, the expected value (14)

as >.

E

Jlu>.(t,x)-U(t,X)IPdX~O,

~ 00 .

This is just an heuristic start. More subtle scaling limit results will be described in the next section.

NONLINEAR PDE'S DRIVEN BY LEVY DIFFUSIONS

251

2. Parabolic scaling limits for Burgers turbulence. The Burgers turbulence problem, see, e.g., [10], corresponds to the situation where the initial data Uo in the Burgers equation are gradients uo(x) = \7~(x) , where ~ is a stationary (homogeneous) potential field on JRd which , as a rule , is not integrable on the whole space. Thus a different approach from the one described in Section 1 is necessary. The methods and results described below have been developed in a series of papers with D. Surgailis [9] and with N. Leonenko [4], see also [10]. The fundamental observation is that if the initial potential field satisfies the limit condition (15)

V/3(Y) := B(,6)(exp[~(,6y)jv]- A(,6)) => V(y),

,6

---+ 00,

in the sense of weak convergence (=» of finite-dimensional distributions of the random fields (possibly generalized), then

(16)

,6d+1B(,6) u(,62t ,,6x) => const- (V( .), \7g(t,x - .)),

,6

---+ 00 ,

where (.,.) stands for the usual Hilbertian inner product and g(t,x) (27rt)-1/2 exp[-x 2 j2t] is the standard Gaussian kernel. This result is a direct consequence of the Hopf-Cole formula for the Burgers equation. It follows from the classical Dobrushin's theory that if potential field ~ (x) is strictly stationary and ergodic then, necessarily,

(17)

B(,6) = ,6" L(,6) ,

for some real exponent K, and slowly varying function L. If, additionally, A(,6) == A is independent of ,6 then the limit random field V in (2.15) is selfsimilar and, for each ,6 > 0, and all "smooth" test functions

-(j.

(W( .),\7g(t,x - .)),

where W is the standard white noise.

as

,6 ---+

00,

252

WOJBOR A. WOYCZYNSKI

(ii) Let the initial potential random field ~(x) in the Burgers turbulence problem in JRd be a shot-no ise field of the form

~(x) =

L 1]ih(x - ( i), i

with 1Ji being i.i.d., ((i) being a Poisson point process, and hE £l n Loo. Then the limit behavior (19) also obtains with

and

B(x) :=

J

E(e1) l h (u ) - l)(e1) l h (u + x )

-

1) duo

For initial data with long-range dependence, that is data where the covariance function is not integrable and the spectrum is singular, the situation is more complicated and not completely understood. If the only singularity is at the origin then the following result holds true: THEOREM 2.2. Let the initial potential field ~(x) on md be of zeromean and variance one, and possess a singular spectral density of the form (20)

0
1. This evolution equation has been extensively studied in a joint work with Biler and Karch, see [2], and the main theorem of this section is taken from that paper. Recall that solutions of such equation have to be understood as mild or weak solutions . It is clear that if the similarity transformation is defined now by the formula

a-I "( = --1' r-

oX

> 0,

then, if u is a solution of conservation law (23), then u>. is also a solution of the same conservation law (23). The question of existence of selfsimilar solutions for such conservation laws, that is solutions such that u == u>. , is answered in the following theorem. Observe that the inital data for such selfsimilar solutions have to be homogeneous functions of degree -"(. THEOREM 4 .1. Ifm+"«d,rrJ.N,m~ lrJ. and Uo E

{v E Cm(JRd) : IDfjv(x)1 ~ C!xl--y-Ifjl, LBI ~ m}

for sufficiently small C, then there exists a junction

such that

NONLINEAR PDE'S DRIVEN BY LEVY DIFFUSIONS

255

is a solution of the fractional conservation law (23). However, the asymptotic large-tim e behavior of solutions of such fractional conservation laws is seldom dictated by th e asymptotics of special selfsimilar soluti ons as was the case, in view of Zuazua's result, for th e Burgers equation. It turns out th at if th e nonlinearity exponent r exceeds the critical value

0: - 1 r e := 1 + -dthen th e asymptotics of solutions is essentially that of th e linearized equation Ut = £Ot u . Indeed, we have THEOREM 4.2. If r > r«, and u(t, x) is a solution of the fractional conservation law (23), then

where ex p[- t £ Ot ] is the exponential semigroup generated by the infinitesimal generator £ Ot . The fundament al solution POt (t , x) of th e linear equation Ut = £ Ot u , th at is th e marginal density of the o:-stable Levy process, satisfies th e selfsimilarity condition

that is, it is invariant under the similarity transformation (24) so that th e solution of the linearized equation U(t, x ) := ex p[- t £Ot l uo(x) =

J

POt(t , x - y )uo(u ) dy

enjoys th e same selfsimilarity prop erty as long as th e initi al condit ion is a homogeneous function of degree - d, that is UO(A X) = A-duo(X). In this case th e assertion of Th eorem 4.2 suggests the following scaling limit result for initi al random dat a: PROPOSITIO N 4 .3 . Let r > r e and u(t,x ) be a solution of the fractional conservation law (23) corresponding to the suitably integrable random initial data Uo with sample paths being homogeneous functions of degree -d. Then, for the rescaled solution random field u>. given by (4.24), (25)

lirn u>. = U, >'-+00

256

WOJBOR A. WOYCZYNSKI

where the limit is understood in the following sense: for each t > 0, the expected value (26)

E

J

lu,\(t,x) - U(t,x)1 2dx ----; 0,

as..\ ----; 00 . Of course, not many interesting initial random fields satisfy stringent conditions of Proposition 4.3 so it is an open question how Theorem 4.2 can be further exploited to get, in case r > re, more subtle scaling limit results for random inital data and , eventually, statistical estimation procedures based on them. In the case when the nonlinearity exponent is equal to the critical value, r = re, the situation becomes totally analogous to that of the Burgers equation. For that reason it is only in this case when we can truly talk about the "fractional Burgers turbulence". The basic result, also found in a joint work with Biler and Karch [2], is as follows: THEOREM 4.4 . Let r = re = 1 + (Q - l)/d and let u(t , x) be a solution of the fractional conservation law (23) with initial data uo(x) such that J Uo (x) dx = M < 00. Then, there exists a unique selfsimilar source solution

U(O, .) = M80 ,

(27)

such that t d/ 2ollu(t, .) - U(t, .)112 ----; 0,

(28)

as

t ----;

00 .

Thus solutions of the critical fractional conservation laws display a true nonlinear asymptotic behavior. In this case Proposition 4.3 can be replaced by a more useful statement since the demand that inital random data are homogeneous of degree -d can be removed. Indeed, condition (4.27) implies that, for any ..\ > 0, under the similarity transformation (4.24) ,

(29) Hence, in view of (4.28), for each t

J

> 0,

lu,\(t, x) - U(t, xW dx

J J

= ,\-d (30) = ,\d =

I..\du('\°t ,'\x) - ,\dU(..\°t, '\x)1 2 d('\x)

lu('\°t, '\x) - U('x°t, 'xxW d('xx)

t-~('x°t)~

J

lu('x°t,'xx) - U('x°t ,'xxWd('xx) ----;

°

257

NONLINEAR PDE'S DRIVEN BY LEVY DIFFUSIONS

as

>. - t

This calculation suggests the following PROPOSITION 4.5 . Let r = r c and u(t,x) be a solution of the fractional conservation law (23) corresponding to the suitably integrable random initial data Uo. Then, for the rescaled solution random field u,\ given by 00.

(4·24), (31)

lim

u,\

= U,

'\-->00

where the limit is understood in the following sense : for each t expected value

(32) as

E

J

lu,\(t , x) - U(t, x)12 dx

-t

> 0,

the

0,

>. - t 00 .

Of course, the result suggested by Proposition 4.5 does not fully address the issue of finding scaling limit results that would provide convergence in the finite-dimensional distributions, along the lines of Section 2. But it is a step in the right direction. However, the problem of finding statistical estimation procedures analogous to those explained in Section 3, remains a challenge.

REFERENCES [IJ P . BILER AND W .A. WOYCZYNSKI. Global and exploding solutions for non local quadratic evolution problems. SIAM J. Appl. Math., 59 (1999), 845-869.

[2] P. BILER, G. KARCH , AND W .A . WOYCZYNSKI. Critical non linearity exponent and [3]

[4] [5]

[6] [7]

[8]

[9]

self-similar asymptotics for Levy conservation laws. Annales d'Institute H. Poincare-Analyse Nonlineaire (Paris), 18 (2001), 613-637. P . BILER, T . FUNAKI, AND W .A. WOYCZYNSKI. Fractal Burgers equations. Journal of Differential Equations, 148 (1998), 9-46. N. LEONENKO AND W .A. WOYCZYNSKI. Parameter identification for stochastic Burgers' flows via parabolic rescaling. Probability and Mathematical Statistics, 21(1) (2001) , 1-55 . J .A. MANN AND W .A. WOYCZYNSKI. Growing fractal interfacces in the presence of self-similar hopping surface diffusion. Physica A : Statistical Mechanics, 291 (2001) , 159-183. R. METZLER AND J . KLAFTER. The random walk 's guide to anomalous diffusion: a fractional dynamics approach. Physics Reports, 339 (2000), 1-77. S.A. MOLCHANOV, D . SURGAILIS , AND W .A. WOYCZYNSKI. The large-scale structure of the Universe and quasi-Voronoi tessellation of shock fronts in forced inviscid Burgers' turbulence in Rd , Annals of Applied Probability, 7 (1997), 200-228. A. PIRYATINSKA , A.I. SAICHEV, AND W.A . WOYCZYNSKI. Models of anomalous diffusion: the subdiffusive case, CWRU Statistics Department Preprint (2003) , pp. 58. D. SURGAILIS AND W .A . WOYCZYNSKI. Limit theorems for the Burgers equation initialized by data with long-range dependence, in Theory and Applications of Long-Range Dependence, P. Doukhan, G. Oppenheim, and M. Taqqu, Eds., Birkhauser-Boston 2003, pp. 507-524.

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[1OJ W.A . WOYCZYNSKI. Burgers-KPZ Turbulence. Lecture Notes in Mathematics 1700, Springer-Verlag 1998. [11] E . ZUAZUA. Weakly nonlinear large time behavior in scalar convection-diffusion equations. Differential and Integml Equations 6 (1993), 1481-1491.

LIST OF WORKSHOP PARTICIPANTS

• Hassan Allouba, Department of Mathematical Sciences, Kent State University • Anna Amirdjanova, Department of Statistics, University of Michigan • Douglas N. Arnold , IMA, University of Minnesota • Kr ishna Athreya, Operations Research and Industrial Engineering, Cornell University • Siva Athreya, Statistical Mathematical Unit, Indian Statistical Institute • Paul Atzberger, Courant Institute of Mathematical Sciences, New York University • Gerard Awanou , Department of Mathematics, University of Georgia • Michele Baldini, Department of Physics, New York University • Rabi Bhattacharya, Department of Mathematics, University of Arizona • Dirk Blomker, Mathematics Research Center, University of Warwick • Maury Bramson, School of Mathematics, University of Minnesota • Susanne C. Brenner, Department of Mathematics, University of South Carolina • Maria-Carme T . Calderer, School of Mathematics, University of Minnesota • Marco Cannone, Laboratoire d' Analys e et de Mathematiques Applique , Universite de Marne-la-Vallee • Rene Carmona, Operations Research & Financial Engineering, Princeton University • Fernando Carreon, Department of Mathematics, University of Texas - Austin • Panagiotis Chatzipantelidis, Department of Mathematics, Texas A&M University • M. Aslam Chaudhry, Department of Mathematical Sciences, King Fahd University of Petroleum & Minerals • Larry Chen , Department of Mathematics, Oregon State University • Long Chen , Department of Mathematics, Pennsylvania State University • Zhenqing Chen , Department of Mathematics, University of Washington • Lan Cheng , Department of Mathematics, University of Pittsburgh

259

260

LIST OF WORKSHOP PARTICIPANTS

• Erhan Cinlar, Department of Operations Research & Financial Engineering, Princeton University • Michael Cranston, Department of Mathematics, University of Rochester • Ian M. Davies, Department of Mathematics, University of Wales Swansea • Hongjie Dong, School of Mathematics, University of Minnesota • Jinqiao Duan, Department of Applied Mathematics, Illinois Institute of Technology • Valdo Durrleman, Bendheim Center for Finance • Maria Emelianenko , Department of Mathematics, Pennsylvania State University • William Faris , Department of Mathematics, University of Arizona • Mark Freidlin, Department of Mathematics, University of Maryland • Peter K. Friz, Courant Institute of Mathematical Sciences, New York University • Victor Goodman, Department of Mathematics, Indiana University • Priscilla E. Greenwood, Department of Mathematics, Arizona State University • Martin Greiner , Information & Communications, Siemens AG • Ernesto Gutierrez-Miravete, Department of Engineering and Science, Rensselaer Polytechnic Institute • Naresh Jain, School of Mathematics, University of Minnesota • Siwei Jia, Department of Statistics, Oregon State University • Yu-Juan Jien, Department of Mathematics, Purdue University • Yoon Mo Jung, School of Mathematics, University of Minnesota • G. Kallianpur, Department of Statistics, University of North Carolina • Rolf Moritz Kassmann, Department of Mathematics, University of Connecticut • Markus Keel, School of Mathematics, University of Minnesota • Djivede Kelome, Department of Mathematics and Statistics, University of Massachusetts • Eun Heui Kim, Department of Mathematics, California State University, Long Beach • Kyounghee Kim, Department of Mathematics, Indiana University • Panki Kim, Department of Mathematics, University of Washington • Vassili N. Kolokoltsov, School of Computing and Technology, Nottingham Trent University • Yuriy Kolomiyets, Department of Mathematical Sciences, Kent State University • Robert Krasny, Department of Mathematics, University of Michigan • Yves LeJan, Departement de Mathematiques, University Paris Sud

LIST OF WORKSHOP PARTICIPANTS

261

• Seung Lee, Department of Mathematics, Ohio State University • Guang-Tsai Lei, Physiology and Bio-Physics, Mayo Clinic • Runchang Lin, Department of Mathematics, Wayne State University • Yuping Liu, Department of Mathematics, Purdue University • Kening Lu, Department of Mathematics, Michigan State University • Mukul Majumdar, Department of Economics, Cornell University • Rogemar Mamon, Department of Statistics and Actuarial Science, University of Waterloo • Sylvie Meleard, UFR Segmi, Universite Paris X • Oana Mocioalca, Department of Mathematics, Purdue University • Salah Mohammed, Department of Mathematics, Southern Illinois University • Charles M. Newman, Department of Mathematics, New York University • Mahdi Nezafat, Department of Electrical and Computer Engineering, University of Minnesota • Keith Nordstrom, C4-CIRES, University of Colorado , Boulder • Chris Orum, Department of Mathematics, Oregon State University • Mina Ossiander, Department of Mathematics, Oregon State University • Chetan Pahlajani, University of Illinois Urbana-Champaign • Veena Paliwal, Department of Mathematics, Southern Illinois University • Jun Hyun Park, Talbot Laboratory, University of Illinois - UrbanaChampaign • Cecile Penland, NOAA-CIRES, University of Colorado • Lea Popovic, Department of Statistics, University of California Berkeley • Jorge M. Ramirez, Department of Mathematics, Oregon State University • Vivek Ranjan, Department of Mathematics, Indiana University • Marco Romito, Dipartimento di Matematica, Universita di Firenze • Boris Rozovskii, Department of Mathematics, University of California - Los Angeles • David Saunders, Department of Mathematics, University of Pittsburgh • Michael Scheutzow, Fakultiit Il, Institut fur Mathematik Technische, Universitiit Berlin • Bjorn Schmalfuss, Mathematical Institute, University of Paderborn • Rongfeng Sun, Courant Institute of Mathematical Sciences, New York University

262

LIST OF WORKSHOP PARTICIPANTS

• Li-Yeng Sung, Department of Mathematics, University of South Carolina • Michael Tehranchi, Department of Mathematics, University of Texas, Austin • Enrique Thomann, Department of Mathematics, Oregon State University • Ilya Timofeyev, Department of Mathematics, University of Houston • Daniell Toth, Department of Mathematics, J uniata College • Hao Wang, Department of Mathematics, University of Oregon • Jing Wang, The Spectacle Lens Group of Johnson and Johnson • Li Wang, Department of Probability and Statistics, Michigan State University • Lixin Wang, Operations Research and Financial Engineering, Princeton University • Edward C. Waymire, Department of Mathematics, Oregon State University • Hans Weinberger, School of Mathematics, University of Minnesota • Andrew Westmeyer, Department of Mathematics, University of Wyoming • Wojbor A. Woyczynski, Department of Statistics and Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University • Jian Yang, Department of Mathematics, University of Illinois Urbana-Champaign • Zhihui Yang, Department of Mathematics , University of Maryland • Aaron Nung Kwan Yip, Department of Mathematics, Purdue University • Toshio Yoshikawa, Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong • Jianfeng Zhang, School of Mathematics, University of Minnesota • Tao Zhang, Department of Mathematics, Purdue University • Yongcheng Zhou, Department of Mathematics, Michigan State University