Exponential Mixing for SDEs under the total variation

TL;DR

This paper develops a new ergodic theory for stochastic differential equations on  with Levy noise, using coupling methods and weaker irreducibility conditions, leading to exponential mixing results.

Contribution

It introduces a novel ergodic framework for SDEs with Levy noise under weak irreducibility, expanding the applicability of ergodic theory in stochastic processes.

Findings

Established an abstract ergodic result on  with weak irreducibility.

Applied the result to Levy-driven SDEs to prove exponential mixing.

Used coupling methods to achieve the main results.

Abstract

First, we establish an abstract ergodic result on $\mR^d$. Classical ergodic results on $\mR^d$ require that the process is irreducible, we weaken it to some weak form of irreducibility in this article. The main method used in this article is coupling. Then, we apply our abstract ergodic result to stochastic differential equations driven by a L\'evy noise and obtain a new result.