## Stochastic Systems

### Asymptotics of the invariant measure in mean field models with jumps

#### 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.

#### Article information

Source
Stoch. Syst., Volume 2, Number 2 (2012), 322-380.

Dates
First available in Project Euclid: 24 February 2014

https://projecteuclid.org/euclid.ssy/1393252026

Digital Object Identifier
doi:10.1214/12-SSY064

Mathematical Reviews number (MathSciNet)
MR3354770

Zentralblatt MATH identifier
1296.60258

#### Citation

Borkar, Vivek S.; Sundaresan, Rajesh. Asymptotics of the invariant measure in mean field models with jumps. Stoch. Syst. 2 (2012), no. 2, 322--380. doi:10.1214/12-SSY064. https://projecteuclid.org/euclid.ssy/1393252026

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