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Nonlinear Markov Processes and Kinetic Equations


Part of Cambridge Tracts in Mathematics

  • Date Published: July 2010
  • availability: Available
  • format: Hardback
  • isbn: 9780521111843

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About the Authors
  • 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.

    • Presents a multidimensional view of the subject by exploring different methods
    • End-of-chapter exercises enable the reader to test their understanding
    • Applications of these processes include non-equilibrium statistical mechanics, evolutionary biology, population and disease dynamics, and dynamics of economic and social systems
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    Reviews & endorsements

    'This monograph is suitable for graduate students and researchers who have a good background in probability theory and analysis and have mastered such key topics as martingales, stochastic calculus and weak convergence, on the one hand, and the analytic theory of semigroups, on the other hand. … This book is pioneering in developing a new and important type of dynamics for modelling complex stochastic systems. It deserves to be widely read.' Bulletin of the London Mathematical Society

    '… this is an important book. Written with great care by a leading expert, it is accessible to researchers and graduate students in stochastic and functional analysis, with applications in mathematical physics and systems biology.' Mathematical Reviews

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    Product details

    • Date Published: July 2010
    • format: Hardback
    • isbn: 9780521111843
    • length: 394 pages
    • dimensions: 236 x 160 x 21 mm
    • weight: 0.7kg
    • contains: 45 exercises
    • availability: Available
  • Table of Contents

    Basic notations
    1. Introduction
    Part I. Tools From Markov Processes:
    2. Probability and analysis
    3. Probabilistic constructions
    4. Analytic constructions
    5. Unbounded coefficients
    Part II. Nonlinear Markov Processes and Semigroups:
    6. Integral generators
    7. Generators of Lévy–Khintchine type
    8. Smoothness with respect to initial data
    Part III. Applications to Interacting Particles:
    9. The dynamic law of large numbers
    10. The dynamic central limit theorem
    11. Developments and comments
    12. Appendices

  • Author

    Vassili N. Kolokoltsov, University of Warwick
    Vassili N. Kolokoltsov is a Reader in Probability in the Department of Statistics at the University of Warwick.

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