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

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+σ(t,X_{t-}){\mathord{\rm d}} L_t^{(α)},\ \ X_0 =x \in $\mathbb{R}$^d, \end{align*} where $b$ belongs to $L^{\mathrm{\infty }}({\mathbb{R}}_{+};{\mathbf{C}}^{-\beta }({\mathbb{R}}^d))$ 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 $L_t^{(\alpha )}=(L_t^{(\alpha ),1},\mathrm{.}\mathrm{.}\mathrm{.},L_t^{(\alpha ),d})$ 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.