## Stochastic Systems

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

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

Vivek S. Borkar and Rajesh Sundaresan

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/12-SSY064

**Mathematical Reviews number (MathSciNet)**

MR3354770

**Zentralblatt MATH identifier**

1296.60258

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F10: Large deviations 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B18: Communication networks [See also 68M10, 94A05] 49J15: Optimal control problems involving ordinary differential equations 34H05: Control problems [See also 49J15, 49K15, 93C15]

**Keywords**

Decoupling approximation fluid limit invariant measure McKean-Vlasov equation mean field limit small noise limit stationary measure stochastic Liouville equation

#### 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