SDE driven by cylindrical $α$-stable process with distributional drift and application

Abstract: For $\alpha \in (1,2)$, we study the following stochastic differential equation driven by a non-degenerate symmetric $\alpha$-stable process in ${\mathbb R}^d$: \begin{align*} {\mathord{{\rm d}}} X_t=b(t,X_t){\mathord{{\rm d}}} t+\sigma(t,X_{t-}){\mathord{{\rm d}}} L_t^{(\alpha)},\ \ X_0 =x \in {\mathbb R}^d, \end{align*} where $b$ belongs to $ L^\infty({\mathbb R}+;{\mathbf B}{\infty,\infty}^{-\beta}({\mathbb R}^d))$ with some $\beta\in(0,\frac{\alpha-1}{2})$, and $\sigma:{\mathbb R}_+\times {\mathbb R}^d \to {\mathbb R}^d \otimes {\mathbb R}^d$ is a $d \times d $ matrix-valued measurable function. We point out that the noise could be a cylindrical $\alpha$-stable process. We first show the generalized martingale problems and then establish the stability estimates of solutions. As an application, we give the weak convergence rate of the Euler scheme for additive noises with drift coefficient $b=b(x)$.