Nonlinear Markov Processes and Kinetic Equations

by Vassili N. Kolokoltsov

• ISBN13: 9780521111843
• ISBN10: 0521111846
• Format: Hardcover
• Copyright: 2010-09-06
• Publisher: Cambridge University Press

Summary

A nonlinear Markov evolution is a dynamical system generated by a measure-valued ordinary differential equation with the specific feature of preserving positivity. This feature distinguishes it from general vector-valued differential equations and yields a natural link with probability, both in interpreting results and in the tools of analysis. This brilliant book, the first devoted to the area, develops this interplay between probability and analysis. After systematically presenting both analytic and probabilistic techniques, the author uses probability to obtain deeper insight into nonlinear dynamics, and analysis to tackle difficult problems in the description of random and chaotic behavior. The book addresses the most fundamental questions in the theory of nonlinear Markov processes: existence, uniqueness, constructions, approximation schemes, regularity, law of large numbers and probabilistic interpretations. Its careful exposition makes the book accessible to researchers and graduate students in stochastic and functional analysis with applications to mathematical physics and systems biology.

Table of Contents

Preface	p. ix
Basic definitions, notation and abbreviations	p. xiv
Introduction	p. 1
Nonlinear Markov chains	p. 1
Examples: replicator dynamics, the Lotka-Volterra equations, epidemics, coagulation	p. 6
Interacting-particle approximation for discrete mass-exchange processes	p. 8
Nonlinear Lévy processes and semigroups	p. 11
Multiple coagulation, fragmentation and collisions; extended Smoluchovski and Boltzmann models	p. 13
Replicator dynamics of evolutionary game theory	p. 24
Interacting Markov processes; mean field and kth-order interactions	p. 28
Classical kinetic equations of statistical mechanics: Vlasov, Boltzmann, Landau	p. 32
Moment measures, correlation functions and the propagation of chaos	p. 34
Nonlinear Markov processes and semigroups; nonlinear martingale problems	p. 39
Tools from Markov process theory	p. 41
Probability and analysis	p. 43
Semigroups, propagators and generators	p. 43
Feller processes and conditionally positive operators	p. 54
Jump-type Markov processes	p. 64
Connection with evolution equations	p. 67
Probabilistic constructions	p. 73
Stochastic integrals and SDEs driven by nonlinear Lévy noise	p. 73
Nonlinear version of Ito's approach to SDEs	p. 82
Homogeneous driving noise	p. 89
An alternative approximation scheme	p. 90
Regularity of solutions	p. 92
Coupling of Lévy processes	p. 96
Analytical constructions	p. 102
Comparing analytical and probabilistic tools	p. 102
Integral generators: one-barrier case	p. 104
Integral generators: two-barrier case	p. 111
Generators of order at most one: well-posedness	p. 114
Generators of order at most one: regularity	p. 117
The spaces (C1∞(Rd))*	p. 120
Further techniques: martingale problem, Sobolev spaces, heat kernels etc.	p. 121
Unbounded coefficients	p. 131
A growth estimate for Feller processes	p. 131
Extending Feller processes	p. 135
Invariant domains	p. 138
Nonlinear Markov processes and semigroups	p. 145
Integral generators	p. 147
Overview	p. 147
Bounded generators	p. 149
Additive bounds for rates: existence	p. 154
Additive bounds for rates: well-posedness	p. 160
A tool for proving uniqueness	p. 165
Multiplicative bounds for rates	p. 169
Another existence result	p. 170
Conditional positivity	p. 173
Generators of Lévy-Khintchine type	p. 175
Nonlinear Lévy processes and semigroups	p. 175
Variable coefficients via fixed-point arguments	p. 180
Nonlinear SDE construction	p. 184
Unbounded coefficients	p. 186
Smoothness with respect to initial data	p. 188
Motivation and plan; a warm-up result	p. 188
Lévy-Khintchine-type generators	p. 192
Jump-type models	p. 201
Estimates for Smoluchovski's equation	p. 208
Propagation and production of moments for the Boltzmann equation	p. 216
Estimates for the Boltzmann equation	p. 219
Applications to interacting particles	p. 223
The dynamic law of large numbers	p. 225
Manipulations with generators	p. 225
Interacting diffusions, stable-like and Vlasov processes	p. 232
Pure jump models: probabilistic approach	p. 236
Rates of convergence for Smoluchovski coagulation	p. 245
Rates of convergence for Boltzmann collisions	p. 250
The dynamic central limit theorem	p. 252
Generators for fluctuation processes	p. 252
Weak CLT with error rates: the Smoluchovski and Boltzmann models, mean field limits and evolutionary games	p. 263
Summarizing the strategy followed	p. 267
Infinite-dimensional Ornstein-Uhlenbeck processes	p. 268
Full CLT for coagulation processes (a sketch)	p. 270
Developments and comments	p. 275
Measure-valued processes as stochastic dynamic LLNs for interacting particles; duality of one-dimensional processes	p. 275
Discrete nonlinear Markov games and controlled processes; the modeling of deception	p. 279
Nonlinear quantum dynamic semigroups and the nonlinear Schrödinger equation	p. 282
Curvilinear Ornstein-Uhlenbeck processes (linear and nonlinear) and stochastic geodesic flows on manifolds	p. 293
The structure of generators	p. 300
Bibliographical comments	p. 310
Appendices	p. 319
Distances on measures	p. 319
Topology on càdlàg paths	p. 324
Convergence of processes in Skorohod spaces	p. 329
Vector-valued ODEs	p. 334
Pseudo-differential operator notation	p. 337
Variational derivatives	p. 338
Geometry of collisions	p. 343
A combinatorial lemma	p. 347
Approximation of infinite-dimensional functions	p. 349
Bogolyubov chains, generating functionals and Fock-space calculus	p. 352
Infinite-dimensional Riccati equations	p. 355
References	p. 360
Index	p. 373
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