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Abstract

We obtain the unique weak and strong solvability for time inhomogeneous stochastic differential equations with the drift in subcritical Lebesgue--H\"{o}lder 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 $p>\alpha/(\alpha+\beta-1)$ 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 $\beta>1-\alpha/2$ by using It\^{o}--Tanaka's trick. Moreover, we give a counterexample to the pathwise uniqueness for the supercritical Lebesgue--H\"{o}lder drifts to explain the present result is sharp.