SDEs with subcritical Lebesgue--Hölder drift and driven by $\alpha$-stable processes

We obtain the unique weak and strong solvability for time inhomogeneous stochastic differential equations with the drift in subcritical Lebesgue–Hölder spaces $L^p([0,T];{\mathcal{C}}_b^{\beta }({\mathbb{R}}^d;{\mathbb{R}}^d))$ and driven by $\alpha$-stable processes for $\alpha \in (0,2)$. The weak well-posedness is derived for $\beta \in (0,1)$, $\alpha +\beta >1$ and $p>\alpha \mathrm{/}(\alpha +\beta -1)$ through Prohorov’s theorem, Skorohod’s representation and the regularity estimates of solutions for a class of fractional parabolic partial differential equations. The pathwise uniqueness and Davie’s type uniqueness are proved for $\beta >1-\alpha \mathrm{/}2$ by using Itô–Tanaka’s trick. Moreover, we give a counterexample to the pathwise uniqueness for the supercritical Lebesgue–Hölder drifts to explain the present result is sharp.