Subexponential Upper and Lower Bounds in Wasserstein Distance for Markov Processes

In this article, relying on Foster-Lyapunov drift conditions, we establish subexponential upper and lower bounds on the rate of convergence in the L-p-Wasserstein distance for a class of irreducible and aperiodic Markov processes. We further discuss these results in the context of Markov Levy-type processes. In the lack of irreducibility and/or aperiodicity properties, we obtain exponential ergodicity in the L-p-Wasserstein distance for a class of Ito processes under an asymptotic flatness (uniform dissipativity) assumption. Lastly, applications of these results to specific processes are presented, including Langevin tempered diffusion processes, piecewise Ornstein-Uhlenbeck processes with jumps under constant and stationary Markov controls, and backward recurrence time chains, for which we provide a sharp characterization of the rate of convergence via matching upper and lower bounds.