Small time asymptotics for Brownian motion with singular drift

TL;DR

This paper derives small time large deviation principles and Varadhan asymptotics for Brownian motion with singular drift in higher dimensions, where the drift is given by measures in a Kato class, extending classical results to more singular settings.

Contribution

It establishes small time asymptotics and large deviation principles for Brownian motion with singular drifts represented by measures in a Kato class, in dimensions three and higher.

Findings

Proves a large deviation principle for Brownian motion with singular drift.

Derives Varadhan type asymptotics in the presence of singular drifts.

Extends classical small time asymptotics to measures with singularities in higher dimensions.

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.