Author(s): Vivek Shripad Borkar | Rajesh Sundaresan
Journal: Stochastic Systems
ISSN 1946-5238
Volume: 2;
Issue: 2;
Start page: 322;
Date: 2012;
Original page
Keywords: Decoupling approximation | fluid limit | invariant measure | McKean-Vlasov equation | mean field limit | small noise limit | stationary measure | stochastic Liouville equation
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 ω-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.
Journal: Stochastic Systems
ISSN 1946-5238
Volume: 2;
Issue: 2;
Start page: 322;
Date: 2012;
Original page
Keywords: Decoupling approximation | fluid limit | invariant measure | McKean-Vlasov equation | mean field limit | small noise limit | stationary measure | stochastic Liouville equation
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 ω-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.
