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 $(s,x)\in \mathbb{ R}_+ \times \mathbb{ R}^{ d}$.
Categories: math.PR