Stationary Stochastic Perturbation of a Linear Delay Equation

Abstract

Assume we are given a family of Markov processes with a common state space E with generators A_ε, ε>0 such that for each ε the corresponding process possesses a unique invariant probability measure µ_ε. Further assume that the generators converge to some Markov generator A as ε→0 in some appropriate sense and that A corresponds to a deterministic motion on the state space E. Then one may ask whether µ_ε converge to some probability measure µ as ε→0 and if so whether µ is invariant under the dynamics given by A.