Heat kernel and ergodicity of SDEs with distributional drifts

TL;DR

This paper investigates the existence, uniqueness, and properties of solutions to stochastic differential equations with distributional drifts, establishing heat kernel estimates and ergodic behavior under certain regularity and dissipativity conditions.

Contribution

It introduces new conditions for well-posedness and ergodicity of SDEs with distributional drifts, providing sharp heat kernel bounds and invariant measure regularity results.

Findings

Existence and uniqueness of martingale solutions.

Sharp two-sided heat kernel estimates.

Ergodicity and regularity of invariant measures.

Abstract

In this paper we consider the following SDE with distributional drift $b$: $$ {\rm d} X_t=\sigma(X_t){\rm d} B_t+b(X_t){\rm d} t,\ X_0=x\in{\mathbb R}^d, $$ where $\sigma$ is a bounded continuous and uniformly non-degenerate $d\times d$-matrix-valued function, $B$ is a $d$-dimensional standard Brownian motion. Let $\alpha\in(0,\frac{1}{2}]$, $p\in(\frac{d}{1-\alpha},\infty)$ and $\beta\in[\alpha,1]$, $q\in(\frac{d}{\beta},\infty)$. Assume $\|({\mathbb I}-\Delta)^{-\alpha/2}b\|_p+\|(-\Delta)^{\beta/2}\sigma\|_q<\infty$. We show the existence and uniqueness of martingale solutions to the above SDE, and obtain sharp two-sided and gradient estimates of the heat kernel associated to the above SDE. Moreover, we study the ergodicity and global regularity of the invariant measures of the associated semigroup under some dissipative assumptions.