Brownian motion with singular drift

Abstract

We consider the stochastic differential equation \[ dX_t=dW_t+dA_t, \] where $W_t$ is $d$-dimensional Brownian motion with $d\geq 2$ and the $i$th component of $A_t$ is a process of bounded variation that stands in the same relationship to a measure $\pi^i$ as $\int_0^t f(X_s)\, ds$ does to the measure $f(x)\, dx$. We prove weak existence and uniqueness for the above stochastic differential equation when the measures $\pi^i$ are members of the Kato class $\K_{d-1}$. 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.