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.
Author(s): Vassili N. Kolokoltsov
Series: Cambridge Tracts in Mathematics
Publisher: CUP
Year: 2010
Language: English
Pages: 395
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Readers and prerequisites......Page 13
Plan of the book......Page 14
Sets and numbers......Page 16
Functions......Page 17
Measures......Page 18
Standard abbreviations......Page 19
1.1 Nonlinear Markov chains......Page 21
1.2 Examples: replicator dynamics, the Lotka–Volterra equations, epidemics, coagulation......Page 26
1.3 Interacting-particle approximation for discrete mass exchange processes......Page 28
1.4 Nonlinear Lévy processes and semigroups......Page 31
1.5 Multiple coagulation, fragmentation and collisions; extended Smoluchovski and Boltzmann models......Page 33
1.6 Replicator dynamics of evolutionary game theory......Page 44
1.7 Interacting Markov processes; mean field and kth-order interactions......Page 48
1.8 Classical kinetic equations of statistical mechanics: Vlasov, Boltzmann, Landau......Page 52
1.9 Moment measures, correlation functions and the propagation of chaos......Page 54
1.10 Nonlinear Markov processes and semigroups; nonlinear martingale problems......Page 59
Part I Tools from Markov process theory......Page 61
2.1 Semigroups, propagators and generators......Page 63
2.2 Feller processes and conditionally positive operators......Page 74
2.3 Jump-type Markov processes......Page 84
2.4 Connection with evolution equations......Page 87
3.1 Stochastic integrals and SDEs driven by nonlinear Lévy noise......Page 93
3.2 Nonlinear version of Ito's approach to SDEs......Page 102
3.3 Homogeneous driving noise......Page 109
3.4 An alternative approximation scheme......Page 110
3.5 Regularity of solutions......Page 112
3.6 Coupling of Lévy processes......Page 116
4.1 Comparing analytical and probabilistic tools......Page 122
4.2 Integral generators: one-barrier case......Page 124
4.3 Integral generators: two-barrier case......Page 131
4.4 Generators of order at most one: well-posedness......Page 134
4.5 Generators of order at most one: regularity......Page 137
4.6 The spaces…......Page 140
4.7 Further techniques: martingale problem, Sobolev spaces, heat kernels etc.......Page 141
5.1 A growth estimate for Feller processes......Page 151
5.2 Extending Feller processes......Page 155
5.3 Invariant domains......Page 158
Part II Nonlinear Markov processes and semigroups......Page 165
6.1 Overview......Page 167
6.2 Bounded generators......Page 169
6.3 Additive bounds for rates: existence......Page 174
6.4 Additive bounds for rates: well-posedness......Page 180
6.5 A tool for proving uniqueness......Page 185
6.6 Multiplicative bounds for rates......Page 189
6.7 Another existence result......Page 190
6.8 Conditional positivity......Page 193
7.1 Nonlinear Lévy processes and semigroups......Page 195
7.2 Variable coefficients via fixed-point arguments......Page 200
7.3 Nonlinear SDE construction......Page 204
7.4 Unbounded coefficients......Page 206
8.1 Motivation and plan; a warm-up result......Page 208
8.2 Lévy–Khintchine-type generators......Page 212
8.3 Jump-type models......Page 221
8.4 Estimates for Smoluchovski's equation......Page 228
8.5 Propagation and production of moments for the Boltzmann equation......Page 236
8.6 Estimates for the Boltzmann equation......Page 239
Part III Applications to interacting particles......Page 243
9.1 Manipulations with generators......Page 245
9.2 Interacting diffusions, stable-like and Vlasov processes......Page 252
9.3 Pure jump models: probabilistic approach......Page 256
9.4 Rates of convergence for Smoluchovski coagulation......Page 265
9.5 Rates of convergence for Boltzmann collisions......Page 270
10.1 Generators for fluctuation processes......Page 272
10.2 Weak CLT with error rates: the Smoluchovski and Boltzmann models, mean field limits and evolutionary games......Page 283
10.3 Summarizing the strategy followed......Page 287
10.4 Infinite-dimensional Ornstein–Uhlenbeck processes......Page 288
10.5 Full CLT for coagulation processes (a sketch)......Page 290
11.1 Measure-valued processes as stochastic dynamic LLNs for interacting particles; duality of one-dimensional processes......Page 295
11.2 Discrete nonlinear Markov games and controlled processes; the modeling of deception......Page 299
11.3 Nonlinear quantum dynamic semigroups and the nonlinear Schrödinger equation......Page 302
11.4 Curvilinear Ornstein–Uhlenbeck processes (linear and nonlinear) and stochastic geodesic flows on manifolds......Page 313
11.5 The structure of generators......Page 320
11.6 Bibliographical comments......Page 330
Appendix A Distances on measures......Page 339
Appendix B Topology on càdlàg paths......Page 344
Appendix C Convergence of processes in Skorohod spaces......Page 349
Appendix D Vector-valued ODEs......Page 354
Appendix E Pseudo-differential operator notation......Page 357
Appendix F Variational derivatives......Page 358
Appendix G Geometry of collisions......Page 363
Appendix H A combinatorial lemma......Page 367
Appendix I Approximation of infinite-dimensional functions......Page 369
Appendix J Bogolyubov chains, generating functionals and Fock-space calculus......Page 372
Appendix K Infinite-dimensional Riccati equations......Page 375
References......Page 380
Index......Page 393
