Stability and Approximation of Random Invariant Measures of Markov Chains in Random Environments

We consider finite-state Markov chains driven by a P-stationary ergodic invertible process σ : Ω → Ω, representing a random environment. For a given initial condition ω ∈ Ω, the driven Markov chain evolves according to A(ω)A(σω) · · ·A(σn−1), where A : Ω → Md is a measurable d × d stochastic matrix-valued function. The driven Markov chain possesses P-a.e. a measurable family of probability vectors v(ω) that satisfy the equivariance property v(σω) = v(ω)A(ω). Writing vω = δω × v(ω), the probability measure ν(·) = ∫ vω(·) dP(ω) on Ω × {1, . . . , d} is the corresponding random invariant measure for the Markov chain in a random environment. Our main result is that ν is stable under a wide variety of perturbations of σ and A. Stability is in the sense of convergence in probability of the random invariant measure of the perturbed system to the unperturbed random invariant measure ν. Our proof approach is elementary and has no assumptions on the transition matrix functionA except measurability. We also develop a new numerical scheme to construct rigorous approximations of ν that converge in probability to ν as the resolution of the scheme increases. This new numerical approach is illustrated with examples of driven random walks and an example where the random environment is governed by a multidimensional nonlinear dynamical system.