Nonlinear Markov processes and kinetic equations / 182

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Bibliographic Details
Author / Creator:Kolokolʹt͡sov, V. N. (Vasiliĭ Nikitich)
Imprint:Cambridge ; New York : Cambridge University Press, 2010.
Description:xvii, 375 p. : ill. ; 24 cm.
Language:English
Series:Cambridge tracts in mathematics ; 182
Cambridge tracts in mathematics ; 182.
Subject:Markov processes.
Nonlinear theories.
Kinetic theory of matter.
Markov processes.
Nonlinear theories.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/8157851
Hidden Bibliographic Details
ISBN:9780521111843 (hbk.)
0521111846 (hbk.)
Notes:Includes bibliographical references (p. [360]-372) and index.
Table of Contents:
  • Preface
  • Basic definitions, notation and abbreviations
  • 1. Introduction
  • 1.1. Nonlinear Markov chains
  • 1.2. Examples: replicator dynamics, the Lotka-Volterra equations, epidemics, coagulation
  • 1.3. Interacting-particle approximation for discrete mass-exchange processes
  • 1.4. Nonlinear Lévy processes and semigroups
  • 1.5. Multiple coagulation, fragmentation and collisions; extended Smoluchovski and Boltzmann models
  • 1.6. Replicator dynamics of evolutionary game theory
  • 1.7. Interacting Markov processes; mean field and kth-order interactions
  • 1.8. Classical kinetic equations of statistical mechanics: Vlasov, Boltzmann, Landau
  • 1.9. Moment measures, correlation functions and the propagation of chaos
  • 1.10. Nonlinear Markov processes and semigroups; nonlinear martingale problems
  • Part I. Tools from Markov process theory
  • 2. Probability and analysis
  • 2.1. Semigroups, propagators and generators
  • 2.2. Feller processes and conditionally positive operators
  • 2.3. Jump-type Markov processes
  • 2.4. Connection with evolution equations
  • 3. Probabilistic constructions
  • 3.1. Stochastic integrals and SDEs driven by nonlinear Lévy noise
  • 3.2. Nonlinear version of Ito's approach to SDEs
  • 3.3. Homogeneous driving noise
  • 3.4. An alternative approximation scheme
  • 3.5. Regularity of solutions
  • 3.6. Coupling of Lévy processes
  • 4. Analytical constructions
  • 4.1. Comparing analytical and probabilistic tools
  • 4.2. Integral generators: one-barrier case
  • 4.3. Integral generators: two-barrier case
  • 4.4. Generators of order at most one: well-posedness
  • 4.5. Generators of order at most one: regularity
  • 4.6. The spaces (C 1 ∞(R d ))*
  • 4.7. Further techniques: martingale problem, Sobolev spaces, heat kernels etc.
  • 5. Unbounded coefficients
  • 5.1. A growth estimate for Feller processes
  • 5.2. Extending Feller processes
  • 5.3. Invariant domains
  • Part II. Nonlinear Markov processes and semigroups
  • 6. Integral generators
  • 6.1. Overview
  • 6.2. Bounded generators
  • 6.3. Additive bounds for rates: existence
  • 6.4. Additive bounds for rates: well-posedness
  • 6.5. A tool for proving uniqueness
  • 6.6. Multiplicative bounds for rates
  • 6.7. Another existence result
  • 6.8. Conditional positivity
  • 7. Generators of Lévy-Khintchine type
  • 7.1. Nonlinear Lévy processes and semigroups
  • 7.2. Variable coefficients via fixed-point arguments
  • 7.3. Nonlinear SDE construction
  • 7.4. Unbounded coefficients
  • 8. Smoothness with respect to initial data
  • 8.1. Motivation and plan; a warm-up result
  • 8.2. Lévy-Khintchine-type generators
  • 8.3. Jump-type models
  • 8.4. Estimates for Smoluchovski's equation
  • 8.5. Propagation and production of moments for the Boltzmann equation
  • 8.6. Estimates for the Boltzmann equation
  • Part III. Applications to interacting particles
  • 9. The dynamic law of large numbers
  • 9.1. Manipulations with generators
  • 9.2. Interacting diffusions, stable-like and Vlasov processes
  • 9.3. Pure jump models: probabilistic approach
  • 9.4. Rates of convergence for Smoluchovski coagulation
  • 9.5. Rates of convergence for Boltzmann collisions
  • 10. The dynamic central limit theorem
  • 10.1. Generators for fluctuation processes
  • 10.2. Weak CLT with error rates: the Smoluchovski and Boltzmann models, mean field limits and evolutionary games
  • 10.3. Summarizing the strategy followed
  • 10.4. Infinite-dimensional Ornstein-Uhlenbeck processes
  • 10.5. Full CLT for coagulation processes (a sketch)
  • 11. Developments and comments
  • 11.1. Measure-valued processes as stochastic dynamic LLNs for interacting particles; duality of one-dimensional processes
  • 11.2. Discrete nonlinear Markov games and controlled processes; the modeling of deception
  • 11.3. Nonlinear quantum dynamic semigroups and the nonlinear Schrödinger equation
  • 11.4. Curvilinear Ornstein-Uhlenbeck processes (linear and nonlinear) and stochastic geodesic flows on manifolds
  • 11.5. The structure of generators
  • 11.6. Bibliographical comments
  • Appendices
  • A. Distances on measures
  • B. Topology on càdlàg paths
  • C. Convergence of processes in Skorohod spaces
  • D. Vector-valued ODEs
  • E. Pseudo-differential operator notation
  • F. Variational derivatives
  • G. Geometry of collisions
  • H. A combinatorial lemma
  • I. Approximation of infinite-dimensional functions
  • J. Bogolyubov chains, generating functionals and Fock-space calculus
  • K. Infinite-dimensional Riccati equations
  • References
  • Index