Ergodicity of stochastic differential equations with jumps and singular coefficients

TL;DR

This paper establishes the strong ergodic properties and well-posedness of stochastic differential equations driven by Lévy noise with singular, dissipative coefficients, including cases with large jumps.

Contribution

It introduces a general approach for analyzing SDEs with jumps and singular coefficients, focusing on Krylov's a priori estimates to prove ergodicity and well-posedness.

Findings

Proves strong well-posedness for SDEs with Lévy noise and singular coefficients.

Shows exponential ergodicity and strong Feller property under certain conditions.

Allows for large jumps in the stochastic differential equations.

Abstract

We show the strong well-posedness of SDEs driven by general multiplicative L\'evy noises with Sobolev diffusion and jump coefficients and integrable drift. Moreover, we also study the strong Feller property, irreducibility as well as the exponential ergodicity of the corresponding semigroup when the coefficients are time-independent and singular dissipative. In particular, the large jump is allowed in the equation. To achieve our main results, we present a general approach for treating the SDEs with jumps and singular coefficients so that one just needs to focus on Krylov's {\it apriori} estimates for SDEs.