Coupling and exponential ergodicity for stochastic differential equations driven by L\'{e}vy processes

TL;DR

This paper introduces a new coupling method for solutions of SDEs driven by Lévy noise, leading to exponential convergence results in various distances, advancing understanding of their long-term behavior.

Contribution

It proposes a novel coupling approach inspired by optimal transportation theory to establish exponential ergodicity for Lévy-driven SDEs.

Findings

Exponential contractivity of semigroups in Kantorovich distance

Exponential convergence in total variation distance

Exponential convergence in $L^1$-Wasserstein distance

Abstract

We present a novel idea for a coupling of solutions of stochastic differential equations driven by L\'{e}vy noise, inspired by some results from the optimal transportation theory. Then we use this coupling to obtain exponential contractivity of the semigroups associated with these solutions with respect to an appropriately chosen Kantorovich distance. As a corollary, we obtain exponential convergence rates in the total variation and standard $L^1$-Wasserstein distances.