Well-posedness and long time behavior of singular Langevin stochastic differential equations

TL;DR

This paper proves the well-posedness and exponential ergodicity of singular Langevin stochastic differential equations, demonstrating their long-term stability and behavior under singular velocity fields.

Contribution

It establishes the strong well-posedness and exponential ergodicity for Langevin SDEs with singular velocity fields, combining Lyapunov functions and Krylov's estimate.

Findings

Strong well-posedness of singular Langevin SDEs

Exponential ergodicity of solutions

Methodology combining Lyapunov functions and Krylov's estimate

Abstract

In this paper, we study damped Langevin stochastic differential equations with singular velocity fields. We prove the strong well-posedness of such equations. Moreover, by combining the technique of Lyapunov functions with Krylov's estimate, we also establish the exponential ergodicity for the unique strong solution.