Brownian Motion with Singular Time-Dependent Drift

In this paper we study weak solutions for the following type of stochastic differential equation $$d{X}_t=d{W}_t+b(t,X_t)dt,t\ge s,X_s=x,$$ where $b:[0,\mathrm{\infty })\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ is a measurable drift, $W=(W_t{)}_{t\ge 0}$ is a $d$-dimensional Brownian motion and $(s,x)\in [0,\mathrm{\infty })\times {\mathbb{R}}^d$ is the starting point. A solution $X=(X_t{)}_{t\ge s}$ for the above SDE is called a Brownian motion with time-dependent drift $b$ starting from $(s,x)$. Under the assumption that $\mathrm{\mid }b\mathrm{\mid }$ belongs to the forward-Kato class $\mathcal{F}{\mathcal{K}}_{d-1}^{\alpha }$ for some $\alpha \in (0,1\mathrm{/}2)$, we prove that the above SDE has a unique weak solution for every starting point $(s,x)\in [0,\mathrm{\infty })\times {\mathbb{R}}^d$.