SDE driven by cylindrical $\alpha$-stable process with distributional drift

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*} {\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{C}^{-\beta}(\mathbb{R}^d))$ with some $\beta\in(0,\alpha-1)$, and $\mathbf{C}^\beta$ denotes a Besov space (see Definition (2.2) below). The coefficient $\sigma:\mathbb{R}_+\times \mathbb{R}^d \to \mathbb{R}^d \otimes \mathbb{R}^d$ is a measurable matrix-valued function. The noise $L_t^{(\alpha)}=(L_t^{(\alpha),1},...,L_t^{(\alpha),d})$ consists of independent $1$-dimensional symmetric $\alpha$-stable processes, and is referred to as a cylindrical $\alpha$-stable process. We establish the well-posedness of weak solutions to the SDE, and provide quantitative stability estimates with respect to the drift coefficients.