Heat kernel and ergodicity of SDEs with distributional drifts

Xicheng Zhang, Guohuan Zhao

Published 2017-10-28Version 1

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