E-Book Overview
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.
E-Book Content
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CAMBRIDGE TRACTS IN MATHEMATICS General Editors B . B O L L O B Á S , W. F U LTO N , A . K ATO K , F. K I RWA N , P. S A R NA K , B . S I M O N , B . TOTA RO
182 Nonlinear Markov Processes and Kinetic Equations
Nonlinear Markov Processes and Kinetic Equations VA S S I L I N . KO L O KO LT S OV University of Warwick
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521111843 © V. N. Kolokoltsov 2010 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN-13
978-0-511-78802-4
eBook (EBL)
ISBN-13
978-0-521-11184-3
Hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
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 massexchange 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
page ix xiv 1 1 6 8 11 13 24 28 32 34 39
Part I Tools from Markov process theory
41
2
43 43
Probability and analysis 2.1 Semigroups, propagators and generators 2.2 Feller processes and conditionally positive operators
54
vi
Contents 2.3 2.4
3
4
5
Jump-type Markov processes Connection with evolution equations
64 67
Probabilistic constructions 3.1 Stochastic integrals and SDEs driven by nonlinea