Nonlinear Markov processes and kinetic equations

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Nonlinear Markov processes and kinetic equations / Vassili N. Kolokoltsov

資料種別:: 図書
出版情報:: Cambridge, UK ; Tokyo : Cambridge University Press, 2010
形態:: xvii, 375 p. ; 23 cm
シリーズ名:: Cambridge tracts in mathematics ; 182 <BA00134372>
著者名:: Kolokoltsov, Vassili N. <DA10792277>
ISBN:: 9780521111843 [0521111846] (: Hardback)
書誌ID:: BB02734720

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注記:

Includes bibliographical references (p. [360]-372) and index

主題:

Markov processes; Nonlinear theories

言語:

英語

目次情報:

Preface

Basic definitions, notation and abbreviations

Introduction / 1：

Nonlinear Markov chains / 1.1：

Examples: replicator dynamics, the Lotka-Volterra equations, epidemics, coagulation / 1.2：

Interacting-particle approximation for discrete mass-exchange processes / 1.3：

Nonlinear LÃ©vy processes and semigroups / 1.4：

Multiple coagulation, fragmentation and collisions; extended Smoluchovski and Boltzmann models / 1.5：

Replicator dynamics of evolutionary game theory / 1.6：

Interacting Markov processes; mean field and kth-order interactions / 1.7：

Classical kinetic equations of statistical mechanics: Vlasov, Boltzmann, Landau / 1.8：

Moment measures, correlation functions and the propagation of chaos / 1.9：

Nonlinear Markov processes and semigroups; nonlinear martingale problems / 1.10：

Tools from Markov process theory / Part I：

Probability and analysis / 2：

Semigroups, propagators and generators / 2.1：

Feller processes and conditionally positive operators / 2.2：

Jump-type Markov processes / 2.3：

Connection with evolution equations / 2.4：

Probabilistic constructions / 3：

Stochastic integrals and SDEs driven by nonlinear LÃ©vy noise / 3.1：

Nonlinear version of Ito's approach to SDEs / 3.2：

Homogeneous driving noise / 3.3：

An alternative approximation scheme / 3.4：

Regularity of solutions / 3.5：

Coupling of LÃ©vy processes / 3.6：

Analytical constructions / 4：

Comparing analytical and probabilistic tools / 4.1：

Integral generators: one-barrier case / 4.2：

Integral generators: two-barrier case / 4.3：

Generators of order at most one: well-posedness / 4.4：

Generators of order at most one: regularity / 4.5：

Further techniques: martingale problem, Sobolev spaces, heat kernels etc. / 4.6：

Unbounded coefficients / 5：

A growth estimate for Feller processes / 5.1：

Extending Feller processes / 5.2：

Invariant domains / 5.3：

Nonlinear Markov processes and semigroups / Part II：

Integral generators / 6：

Overview / 6.1：

Bounded generators / 6.2：

Additive bounds for rates: existence / 6.3：

Additive bounds for rates: well-posedness / 6.4：

A tool for proving uniqueness / 6.5：

Multiplicative bounds for rates / 6.6：

Another existence result / 6.7：

Conditional positivity / 6.8：

Generators of LÃ©vy-Khintchine type / 7：

Variable coefficients via fixed-point arguments / 7.1：

Nonlinear SDE construction / 7.3：

Smoothness with respect to initial data / 7.4：

Motivation and plan; a warm-up result / 8.1：

LÃ©vy-Khintchine-type generators / 8.2：

Jump-type models / 8.3：

Estimates for Smoluchovski's equation / 8.4：

Propagation and production of moments for the Boltzmann equation / 8.5：

Estimates for the Boltzmann equation / 8.6：

Applications to interacting particles / Part III：

The dynamic law of large numbers / 9：

Manipulations with generators / 9.1：

Interacting diffusions, stable-like and Vlasov processes / 9.2：

Pure jump models: probabilistic approach / 9.3：

Rates of convergence for Smoluchovski coagulation / 9.4：

Rates of convergence for Boltzmann collisions / 9.5：

The dynamic central limit theorem / 10：

Generators for fluctuation processes / 10.1：

Weak CLT with error rates: the Smoluchovski and Boltzmann models, mean field limits and evolutionary games / 10.2：

Summarizing the strategy followed / 10.3：

Infinite-dimensional Ornstein-Uhlenbeck processes / 10.4：

Full CLT for coagulation processes (a sketch) / 10.5：

Developments and comments / 11：

Measure-valued processes as stochastic dynamic LLNs for interacting particles; duality of one-dimensional processes / 11.1：

Discrete nonlinear Markov games and controlled processes; the modeling of deception / 11.2：

Nonlinear quantum dynamic semigroups and the nonlinear SchrÃ¶dinger equation / 11.3：

Curvilinear Ornstein-Uhlenbeck processes (linear and nonlinear) and stochastic geodesic flows on manifolds / 11.4：

The structure of generators / 11.5：

Bibliographical comments / 11.6：

Appendices

Distances on measures / A：

Topology on cÃ dlÃ g paths / B：

Convergence of processes in Skorohod spaces / C：

Vector-valued ODEs / D：

Pseudo-differential operator notation / E：

Variational derivatives / F：

Geometry of collisions / G：

A combinatorial lemma / H：

Approximation of infinite-dimensional functions / I：

Bogolyubov chains, generating functionals and Fock-space calculus / J：

Infinite-dimensional Riccati equations / K：

References

Index

Preface

Basic definitions, notation and abbreviations

Introduction / 1：

Nonlinear Markov chains / 1.1：

Examples: replicator dynamics, the Lotka-Volterra equations, epidemics, coagulation / 1.2：

Interacting-particle approximation for discrete mass-exchange processes / 1.3：

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