Invariant probability measures for path-dependent random diffusions

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