Brownian Motion with Singular Time-Dependent Drift

Abstract

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 $(s,x)\in [0,\infty) \times \mathbb{R}^{d}$.