Small time asymptotics for Brownian motion with singular drift

Abstract

We establish a small time large deviation principle and a Varadhan type asymptotics for Brownian motion with singular drift on $\bR^d$ with $d\geq 3$ whose infinitesimal generator is $\frac12 \Delta + \mu \cdot \nabla$, where each $\mu_i$ of $\mu = (\mu_1, \cdots, \mu_d)$ is a measure in some suitable Kato class.