Quantitative heat kernel estimates for diffusions with distributional drift

TL;DR

This paper establishes explicit upper and lower bounds for the heat kernel of solutions to stochastic differential equations with distributional drifts of regularity greater than -1/2, extending understanding of such diffusions.

Contribution

It provides the first quantitative heat kernel estimates for SDEs with distributional drifts of regularity greater than -1/2, including explicit dependence on drift norms.

Findings

Derived upper and lower bounds for the transition kernel.

Established explicit dependence of heat kernel bounds on drift regularity.

Extended heat kernel estimates to SDEs with distributional drifts.

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 $ > -\frac12$. We show that the martingale solution of the SDE has a transition kernel $\Gamma_t$ and prove upper and lower heat kernel bounds for $\Gamma_t$ with explicit dependence on $t$ and the norm of $b$.