Nonlinear Markov processes and kinetic equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2010
|
| Edition: | 1. publ. |
| Series: | Cambridge tracts in mathematics
182 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 375 S. 23 cm |
| ISBN: | 9780521111843 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Titel: Nonlinear Markov processes and kinetic equations
Autor: Kolokolʹcov, Vassilij N.
Jahr: 2010
Contents
Preface page ix
Basic definitions, notation and abbreviations xiv
1 Introduction 1
1.1 Nonlinear Markov chains 1
1.2 Examples: replicator dynamics, the Lotka-Volterra
equations, epidemics, coagulation 6
1.3 Interacting-particle approximation for discrete mass-
exchange processes 8
1.4 Nonlinear Levy processes and semigroups 11
1.5 Multiple coagulation, fragmentation and collisions;
extended Smoluchovski and Boltzmann models 13
1.6 Replicator dynamics of evolutionary game theory 24
1.7 Interacting Markov processes; mean field and th-order
interactions 28
1.8 Classical kinetic equations of statistical mechanics:
Vlasov, Boltzmann, Landau 32
1.9 Moment measures, correlation functions and the
propagation of chaos 34
1.10 Nonlinear Markov processes and semigroups; nonlinear
martingale problems 39
Parti Tools from Markov process theory 41
2 Probability and analysis 43
2.1 Semigroups, propagators and generators 43
2.2 Feller processes and conditionally positive
operators 54
vi Contents
2.3 Jump-type Markov processes 64
2.4 Connection with evolution equations 67
3 Probabilistic constructions 73
3.1 Stochastic integrals and SDEs driven by nonlinear
Levy noise 73
3.2 Nonlinear version of Ito s approach to SDEs 82
3.3 Homogeneous driving noise 89
3.4 An alternative approximation scheme 90
3.5 Regularity of solutions 92
3.6 Coupling of Levy processes 96
4 Analytical constructions 102
4.1 Comparing analytical and probabilistic tools 102
4.2 Integral generators: one-barrier case 104
4.3 Integral generators: two-barrier case 111
4.4 Generators of order at most one: well-posedness 114
4.5 Generators of order at most one: regularity 117
4.6 The spaces (C^CR^))* 120
4.7 Further techniques: martingale problem, Sobolev
spaces, heat kernels etc. 121
5 Unbounded coefficients 131
5.1 A growth estimate for Feller processes 131
5.2 Extending Feller processes 135
5.3 Invariant domains 138
Part ? Nonlinear Markov processes and semigroups 145
6 Integral generators 147
6.1 Overview 147
6.2 Bounded generators 149
6.3 Additive bounds for rates: existence 154
6.4 Additive bounds for rates: well-posedness 160
6.5 A tool for proving uniqueness 165
6.6 Multiplicative bounds for rates 169
6.7 Another existence result 170
6.8 Conditional positivity 173
7 Generators of Lévy-Khintchine type 175
7.1 Nonlinear Levy processes and semigroups 175
7.2 Variable coefficients via fixed-point arguments 180
Contents vii
7.3 Nonlinear SDE construction 184
7.4 Unbounded coefficients 186
8 Smoothness with respect to initial data 188
8.1 Motivation and plan; a warm-up result 188
8.2 Lévy-Khintchine-type generators 192
8.3 Jump-type models 201
8.4 Estimates for Smoluchovski s equation 208
8.5 Propagation and production of moments for the
Boltzmann equation 216
8.6 Estimates for the Boltzmann equation 219
Part III Applications to interacting particles 223
9 The dynamic law of large numbers 225
9.1 Manipulations with generators 225
9.2 Interacting diffusions, stable-like and Vlasov processes 232
9.3 Pure jump models: probabilistic approach 236
9.4 Rates of convergence for Smoluchovski coagulation 245
9.5 Rates of convergence for Boltzmann collisions 250
10 The dynamic central limit theorem 252
10.1 Generators for fluctuation processes 252
10.2 Weak CLT with error rates: the Smoluchovski and
Boltzmann models, mean field limits and evolutionary
games 263
10.3 Summarizing the strategy followed 267
10.4 Infinite-dimensional Ornstein-Uhlenbeck processes 268
10.5 Full CLT for coagulation processes (a sketch) 270
11 Developments and comments 275
11.1 Measure-valued processes as stochastic dynamic LLNs
for interacting particles; duality of one-dimensional
processes 275
11.2 Discrete nonlinear Markov games and controlled
processes; the modeling of deception 279
11.3 Nonlinear quantum dynamic semigroups and the
nonlinear Schrödinger equation 282
11.4 Curvilinear Ornstein-Uhlenbeck processes (linear and
nonlinear) and stochastic geodesic flows on manifolds 293
11.5 The structure of generators 300
11.6 Bibliographical comments 310
viii Contents
Appendices 319
A Distances on measures 319
? Topology on càdlàg paths 324
C Convergence of processes in Skorohod spaces 329
D Vector-valued ODEs 334
E Pseudo-differential operator notation 337
F Variational derivatives 338
G Geometry of collisions 343
H A combinatorial lemma 347
I Approximation of infinite-dimensional functions 349
J Bogolyubov chains, generating functionals and
Fock-space calculus 352
? Infinite-dimensional Riccati equations 355
References 360
Index 373
|
| any_adam_object | 1 |
| author | Kolokolʹcov, Vassilij N. 1959- |
| author_GND | (DE-588)121728633 |
| author_facet | Kolokolʹcov, Vassilij N. 1959- |
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| author_sort | Kolokolʹcov, Vassilij N. 1959- |
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| bvnumber | BV036614231 |
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| ctrlnum | (OCoLC)699591512 (DE-599)OBVAC08193727 |
| discipline | Mathematik |
| edition | 1. publ. |
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| id | DE-604.BV036614231 |
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| indexdate | 2024-07-09T22:44:13Z |
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| isbn | 9780521111843 |
| language | English |
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| physical | XVII, 375 S. 23 cm |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge tracts in mathematics |
| series2 | Cambridge tracts in mathematics |
| spelling | Kolokolʹcov, Vassilij N. 1959- Verfasser (DE-588)121728633 aut Nonlinear Markov processes and kinetic equations Vassili N. Kolokoltsov 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2010 XVII, 375 S. 23 cm txt rdacontent n rdamedia nc rdacarrier Cambridge tracts in mathematics 182 Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Kinetische Gleichung (DE-588)4030667-7 gnd rswk-swf Nichtlinearer Prozess (DE-588)4304582-0 gnd rswk-swf Markov-Prozess (DE-588)4134948-9 s Nichtlinearer Prozess (DE-588)4304582-0 s Kinetische Gleichung (DE-588)4030667-7 s DE-604 Cambridge tracts in mathematics 182 (DE-604)BV000000001 182 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020534414&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kolokolʹcov, Vassilij N. 1959- Nonlinear Markov processes and kinetic equations Cambridge tracts in mathematics Markov-Prozess (DE-588)4134948-9 gnd Kinetische Gleichung (DE-588)4030667-7 gnd Nichtlinearer Prozess (DE-588)4304582-0 gnd |
| subject_GND | (DE-588)4134948-9 (DE-588)4030667-7 (DE-588)4304582-0 |
| title | Nonlinear Markov processes and kinetic equations |
| title_auth | Nonlinear Markov processes and kinetic equations |
| title_exact_search | Nonlinear Markov processes and kinetic equations |
| title_full | Nonlinear Markov processes and kinetic equations Vassili N. Kolokoltsov |
| title_fullStr | Nonlinear Markov processes and kinetic equations Vassili N. Kolokoltsov |
| title_full_unstemmed | Nonlinear Markov processes and kinetic equations Vassili N. Kolokoltsov |
| title_short | Nonlinear Markov processes and kinetic equations |
| title_sort | nonlinear markov processes and kinetic equations |
| topic | Markov-Prozess (DE-588)4134948-9 gnd Kinetische Gleichung (DE-588)4030667-7 gnd Nichtlinearer Prozess (DE-588)4304582-0 gnd |
| topic_facet | Markov-Prozess Kinetische Gleichung Nichtlinearer Prozess |
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