Invariant Measures for Path-Dependent Random Diffusions

TL;DR

This paper establishes the existence, uniqueness, and exponential convergence to invariant measures for path-dependent random diffusions in random environments, including their time discretizations, under certain ergodic conditions.

Contribution

It provides new results on invariant measures and ergodic properties for path-dependent diffusions and their discretizations in random environments, addressing challenges in discretizing Markov chains.

Findings

Unique invariant measures exist under ergodic conditions.

Exponential convergence to equilibrium in Wasserstein distance.

Discrete approximations retain ergodic properties with small stepsizes.

Abstract

In this work, we are concerned with existence and uniqueness of invariant measures for path-dependent random diffusions and their time discretizations. The random diffusion here means a diffusion process living in a random environment characterized by a continuous time Markov chain. Under certain ergodic conditions, we show that the path-dependent random diffusion enjoys a unique invariant probability measure and converges exponentially to its equilibrium under the Wasserstein distance. Also, we demonstrate that the time discretization of the path-dependent random diffusion involved admits a unique invariant probability measure and shares the corresponding ergodic property when the stepsize is sufficiently small. During this procedure, the difficulty arose from the time-discretization of continuous time Markov chain has to be deal with, for which an estimate on its exponential functional is presented.