Quantitative Heat-Kernel Estimates for Diffusions with Distributional Drift

Abstract

We consider the stochastic differential equation on Rd given by dXt=b(t,Xt)dt+dBt, where B is a Brownian motion and b is considered to be a distribution of regularity >−12. We show that the martingale solution of the SDE has a transition kernel Γt and prove upper and lower heat-kernel estimates for Γt with explicit dependence on t and the norm of b.