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2012 Asymptotics of the invariant measure in mean field models with jumps
Vivek S. Borkar, Rajesh Sundaresan
Stoch. Syst. 2(2): 322-380 (2012). DOI: 10.1214/12-SSY064

Abstract

We consider the asymptotics of the invariant measure for the process of the empirical spatial distribution of $N$ coupled Markov chains in the limit of a large number of chains. Each chain reflects the stochastic evolution of one particle. The chains are coupled through the dependence of the transition rates on this spatial distribution of particles in the various states. Our model is a caricature for medium access interactions in wireless local area networks. It is also applicable to the study of spread of epidemics in a network. The limiting process satisfies a deterministic ordinary differential equation called the McKean-Vlasov equation. When this differential equation has a unique globally asymptotically stable equilibrium, the spatial distribution asymptotically concentrates on this equilibrium. More generally, its limit points are supported on a subset of the $\omega$-limit sets of the McKean-Vlasov equation. Using a control-theoretic approach, we examine the question of large deviations of the invariant measure from this limit.

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Vivek S. Borkar. Rajesh Sundaresan. "Asymptotics of the invariant measure in mean field models with jumps." Stoch. Syst. 2 (2) 322 - 380, 2012. https://doi.org/10.1214/12-SSY064

Information

Published: 2012
First available in Project Euclid: 24 February 2014

zbMATH: 1296.60258
MathSciNet: MR3354770
Digital Object Identifier: 10.1214/12-SSY064

Subjects:
Primary: 34H05, 49J15, 60F10, 60K35, 68M20, 90B18

Rights: Copyright © 2012 INFORMS Applied Probability Society

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