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 R d \mathbb {R}^d with d ≥ 3 d\geq 3 whose infinitesimal generator is 1 2 Δ + μ ⋅ ∇ \frac 12 \Delta + \mu \cdot \nabla , where each μ i \mu _i of μ = ( μ 1 , … , μ d ) \mu = (\mu _1, \ldots , \mu _d) is a measure in some suitable Kato class.