Numerical computations of geometric ergodicity for stochastic dynamics

Abstract

A probabilistic approach to compute the geometric convergence rate of a stochastic process is introduced in this paper. The goal is to quantitatively compute both the upper and lower bounds for rate of the exponential convergence to the stationary distribution of a stochastic dynamical system. By applying the coupling method, we derive an algorithm which does not rely on the discretization of the infinitesimal generator. In this way, our approach works well for many high-dimensional examples. We apply this algorithm to the random perturbations of both iterative maps and differential equations. We show that the rate of geometric ergodicity of a random perturbed system can, to some extent, reveal the degree of chaoticity of the underlying deterministic dynamics. Various SDE models including the ones with degenerate noise or living on the high-dimensional state space are also explored.