Nonlinear Markov Processes and Kinetic Equations

395 Pages·2010·1.9 MB·English

by  Vassili N. Kolokoltsov

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CAMBRIDGE TRACTS IN MATHEMATICS
GeneralEditors
B. BOLLOBÁS, W. FULTON, A. KATOK, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
182 NonlinearMarkovProcessesandKineticEquations

Nonlinear Markov Processes
and Kinetic Equations
VASSILI N. KOLOKOLTSOV
UniversityofWarwick
CAMBRIDGE UNIVERSITY PRESS
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Published in the United States of America by Cambridge University Press, New York
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Information on this title: www.cambridge.org/9780521111843
© V. N. Kolokoltsov 2010
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provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2010
ISBN-13 978-0-511-78802-4 eBook (EBL)
ISBN-13 978-0-521-11184-3 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
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and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
Contents
Preface pageix
Basicdefinitions,notationandabbreviations xiv
1 Introduction 1
1.1 NonlinearMarkovchains 1
1.2 Examples: replicator dynamics, the Lotka–Volterra
equations,epidemics,coagulation 6
1.3 Interacting-particle approximation for discrete mass-
exchangeprocesses 8
1.4 NonlinearLévyprocessesandsemigroups 11
1.5 Multiple coagulation, fragmentation and collisions;
extendedSmoluchovskiandBoltzmannmodels 13
1.6 Replicatordynamicsofevolutionarygametheory 24
1.7 InteractingMarkovprocesses;meanfieldandkth-order
interactions 28
1.8 Classical kinetic equations of statistical mechanics:
Vlasov,Boltzmann,Landau 32
1.9 Moment measures, correlation functions and the
propagationofchaos 34
1.10 NonlinearMarkovprocessesandsemigroups;nonlinear
martingaleproblems 39
PartI ToolsfromMarkovprocesstheory 41
2 Probabilityandanalysis 43
2.1 Semigroups,propagatorsandgenerators 43
2.2 Fellerprocessesandconditionallypositive
operators 54
vi Contents
2.3 Jump-typeMarkovprocesses 64
2.4 Connectionwithevolutionequations 67
3 Probabilisticconstructions 73
3.1 Stochastic integrals and SDEs driven by nonlinear
Lévynoise 73
3.2 NonlinearversionofIto’sapproachtoSDEs 82
3.3 Homogeneousdrivingnoise 89
3.4 Analternativeapproximationscheme 90
3.5 Regularityofsolutions 92
3.6 CouplingofLévyprocesses 96
4 Analyticalconstructions 102
4.1 Comparinganalyticalandprobabilistictools 102
4.2 Integralgenerators:one-barriercase 104
4.3 Integralgenerators:two-barriercase 111
4.4 Generatorsoforderatmostone:well-posedness 114
4.5 Generatorsoforderatmostone:regularity 117
4.6 Thespaces(C∞l (Rd))(cid:2) 120
4.7 Further techniques: martingale problem, Sobolev
spaces,heatkernelsetc. 121
5 Unboundedcoefficients 131
5.1 AgrowthestimateforFellerprocesses 131
5.2 ExtendingFellerprocesses 135
5.3 Invariantdomains 138
PartII NonlinearMarkovprocessesandsemigroups 145
6 Integralgenerators 147
6.1 Overview 147
6.2 Boundedgenerators 149
6.3 Additiveboundsforrates:existence 154
6.4 Additiveboundsforrates:well-posedness 160
6.5 Atoolforprovinguniqueness 165
6.6 Multiplicativeboundsforrates 169
6.7 Anotherexistenceresult 170
6.8 Conditionalpositivity 173
7 GeneratorsofLévy–Khintchinetype 175
7.1 NonlinearLévyprocessesandsemigroups 175
7.2 Variablecoefficientsviafixed-pointarguments 180
Contents vii
7.3 NonlinearSDEconstruction 184
7.4 Unboundedcoefficients 186
8 Smoothnesswithrespecttoinitialdata 188
8.1 Motivationandplan;awarm-upresult 188
8.2 Lévy–Khintchine-typegenerators 192
8.3 Jump-typemodels 201
8.4 EstimatesforSmoluchovski’sequation 208
8.5 Propagation and production of moments for the
Boltzmannequation 216
8.6 EstimatesfortheBoltzmannequation 219
PartIII Applicationstointeractingparticles 223
9 Thedynamiclawoflargenumbers 225
9.1 Manipulationswithgenerators 225
9.2 Interactingdiffusions,stable-likeandVlasovprocesses 232
9.3 Purejumpmodels:probabilisticapproach 236
9.4 RatesofconvergenceforSmoluchovskicoagulation 245
9.5 RatesofconvergenceforBoltzmanncollisions 250
10 Thedynamiccentrallimittheorem 252
10.1 Generatorsforfluctuationprocesses 252
10.2 Weak CLT with error rates: the Smoluchovski and
Boltzmannmodels,meanfieldlimitsandevolutionary
games 263
10.3 Summarizingthestrategyfollowed 267
10.4 Infinite-dimensionalOrnstein–Uhlenbeckprocesses 268
10.5 FullCLTforcoagulationprocesses(asketch) 270
11 Developmentsandcomments 275
11.1 Measure-valuedprocessesasstochasticdynamicLLNs
for interacting particles; duality of one-dimensional
processes 275
11.2 Discrete nonlinear Markov games and controlled
processes;themodelingofdeception 279
11.3 Nonlinear quantum dynamic semigroups and the
nonlinearSchrödingerequation 282
11.4 CurvilinearOrnstein–Uhlenbeckprocesses(linearand
nonlinear)andstochasticgeodesicflowsonmanifolds 293
11.5 Thestructureofgenerators 300
11.6 Bibliographicalcomments 310
viii Contents
Appendices 319
A Distancesonmeasures 319
B Topologyoncàdlàgpaths 324
C ConvergenceofprocessesinSkorohodspaces 329
D Vector-valuedODEs 334
E Pseudo-differentialoperatornotation 337
F Variationalderivatives 338
G Geometryofcollisions 343
H Acombinatoriallemma 347
I Approximationofinfinite-dimensionalfunctions 349
J Bogolyubov chains, generating functionals and
Fock-spacecalculus 352
K Infinite-dimensionalRiccatiequations 355
References 360
Index 373