On Weak Solutions of SDEs with Singular Time-Dependent Drift and Driven by Stable Processes

Abstract

Let $d \ge 2$. In this paper, we study weak solutions for the following type of stochastic differential equation \[ dX_{t}=dS_{t}+b(s+t, X_{t})dt, \quad X_{0}=x, \] where $(s,x)\in \mathbb{R}_+ \times \mathbb{R}^{d}$ is the initial starting point, $b: \mathbb{R}_+ \times \mathbb{R}^{d} \to \mathbb{R}^{d}$ is measurable, and $S=(S_{t})_{t \ge 0}$ is a $d$-dimensional $\alpha$-stable process with index $\alpha \in (1,2)$. We show that if the $\alpha$-stable process $S$ is non-degenerate and $b \in L_{loc}^{\infty}(\mathbb{R}_{+};L^{\infty}(\mathbb{R}^{d}))+ L_{loc}^{q}(\mathbb{R}_{+};L^{p}(\mathbb{R}^{d}))$ for some $p,q>0$ with $d/ p+\alpha/q <\alpha-1$, then the above SDE has a unique weak solution for every starting point $(s,x)\in \mathbb{R}_+ \times \mathbb{R}^{d}$.