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Nonlinear Markov semigroups and interacting Lévy type processes

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Kolokoltsov, V. N. (Vasiliĭ Nikitich). (2007) Nonlinear Markov semigroups and interacting Lévy type processes. Journal of Statistical Physics, Vol.126 (No.3). pp. 585-642. ISSN 0022-4715

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Official URL: http://dx.doi.org/10.1007/s10955-006-9211-y

Abstract

Semigroups of positivity preserving linear operators on measures of a measurable space X describe the evolutions of probability distributions of Markov processes on X. Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions B(X) on X describe the evolutions of averages over the trajectories of these Markov processes. In this paper we introduce and study the general class of semigroups of non-linear positivity preserving transformations on measures that is non-linear Markov or Feller semigroups. An explicit structure of generators of such groups is given in case when X is the Euclidean space R-d (or more generally, a manifold) showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for Markov models of interacting particles. Well posedness results for these equations are given together with applications to interacting particles: dynamic law of large numbers and central limit theorem, the latter being new already for the standard coagulation-fragmentation models.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Markov processes, Semigroups
Journal or Publication Title:	Journal of Statistical Physics
Publisher:	Springer New York LLC
ISSN:	0022-4715
Date:	February 2007
Volume:	Vol.126
Number:	No.3
Number of Pages:	58
Page Range:	pp. 585-642
Identification Number:	10.1007/s10955-006-9211-y
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
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URI:	http://wrap.warwick.ac.uk/id/eprint/32389