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 ${\mathrm{L}}^p$-Wasserstein distance for a class of irreducible and aperiodic Markov processes. We further discuss these results in the context of Markov L'evy-type processes. In the lack of irreducibility and/or aperiodicity properties, we obtain exponential ergodicity in the ${\mathrm{L}}^p$-Wasserstein distance for a class of It^{o} 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.