Invariant probability measures for path-dependent random diffusions

In this work, we are concerned with path-dependent random diffusions. Under certain ergodic condition, we show that the path-dependent random diffusion under consideration has 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 (numerical) invariant probability measure and preserves the corresponding ergodic property when the step size is sufficiently small. Moreover, we provide an estimate on the exponential functional of the discrete observation for a Markov chain, which may be interesting by itself.