Brownian motion with singular drift

Abstract

We consider the stochastic differential equation dXt = dWt + dAt, Where Wt is d-dimensional Brownian motion with d ≥ 2 and the ith component of At is a process of bounded variation that stands in the same relationship to a measure πi as ∫01 f(Xs)ds does to the measure f(x)dx. We prove weak existence and uniqueness for the above stochastic differential equation when the measures πi are members of the Kato class Kd-1. As a typical example, we obtain a Brownian motion that has upward drift when in certain fractal-like sets and show that such a process is unique in law.