  In this paper we study weak solutions for the following type of stochastic
differential equation \[ dX_{t}=dW_{t}+b(t, X_{t})dt, \quad t\ge s, \quad
X_{s}=x, \] where $b: [0,\infty) \times \mathbb{R}^{d} \to \mathbb{R}^{d}$ is a
measurable drift, $W=(W_{t})_{t \ge 0}$ is a $d$-dimensional Brownian motion
and $(s,x)\in [0,\infty) \times \mathbb{R}^{d}$ 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 $|b|$
belongs to the forward-Kato class $\mathcal{F} \mathcal{K}_{d-1}^{\alpha}$ for
some $\alpha \in (0,1/2)$, we prove that the above SDE has a unique weak
solution for every starting point $(s,x)\in [0,\infty) \times \mathbb{R}^{d}$.
