On Sub-Geometric Ergodicity of Diffusion Processes

TL;DR

This paper investigates conditions under which diffusion processes exhibit sub-geometric ergodicity, providing theoretical results on their convergence rates in total variation and Wasserstein distances, including processes with jumps.

Contribution

It establishes new criteria for sub-geometric ergodicity of diffusion processes and extends the analysis to certain jump processes, enhancing understanding of their long-term behavior.

Findings

Conditions for sub-geometric ergodicity in total variation distance.

Sub-geometric contractivity in Wasserstein distances.

Ergodicity results for diffusion processes with jumps.

Abstract

In this article, we discuss ergodicity properties of a diffusion process given through an It\^{o} stochastic differential equation. We identify conditions on the drift and diffusion coefficients which result in sub-geometric ergodicity of the corresponding semigroup with respect to the total variation distance. We also prove sub-geometric contractivity and ergodicity of the semigroup under a class of Wasserstein distances. Finally, we discuss sub-geometric ergodicity of two classes of Markov processes with jumps.