Brownian motion with singular drift

We consider the stochastic differential equation \[ dX_t=dW_t+dA_t, \] 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 ∫₀ᵗ 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 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.