SDE driven by cylindrical $\hat{I}\pm$-stable process with distributional drift

Abstract

For $\hat{I}\pm \in (1,2)$, we study the following stochastic differential equation driven by a non-degenerate symmetric $\hat{I}\pm$-stable process in ${\mathbb{R}}^d$: \begin{align*} {\rm d} X_t=b(t,X_t){\mathord{\rm d}} t+σ(t,X_{t-}){\mathord{\rm d}} L_t^{(α)},\ \ X_0 =x \in \mathbb{R}^d, \end{align*} where $b$ belongs to $L^{\infty }({\mathbb{R}}_{+};{\mathbf{C}}^{-{\hat{I}}^2}({\mathbb{R}}^d))$ with some ${\hat{I}}^2\in (0,\hat{I}\pm -1)$, and ${\mathbf{C}}^2$ denotes a Besov space (see Definition (2.2) below). The coefficient $\ddot{I}ƒ:{\mathbb{R}}_{+}\times {\mathbb{R}}^d\to {\mathbb{R}}^d\otimes {\mathbb{R}}^d$ is a measurable matrix-valued function. The noise $L_t^{(\hat{I}\pm )}=(L_t^{(\hat{I}\pm ),1},...,L_t^{(\hat{I}\pm ),d})$ consists of independent $1$-dimensional symmetric $\hat{I}\pm$-stable processes, and is referred to as a cylindrical $\hat{I}\pm$-stable process. We establish the well-posedness of weak solutions to the SDE, and provide quantitative stability estimates with respect to the drift coefficients.