Quantitative heat kernel estimates for diffusions with distributional drift

Abstract

We consider the stochastic differential equation on ${\mathbb{R}}^d$ given by $$\mathrm{d}{X}_t=b(t,X_t)\mathrm{d}t+\mathrm{d}{B}_t,$$ where $B$ is a Brownian motion and $b$ is considered to be a distribution of regularity $>-\frac{1}{2}$. We show that the martingale solution of the SDE has a transition kernel $\hat{I}{“}_t$ and prove upper and lower heat kernel bounds for $\hat{I}{“}_t$ with explicit dependence on $t$ and the norm of $b$.