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

Abstract

We obtain the unique weak and strong solvability for time inhomogeneous stochastic differential equations with the drift in subcritical Lebesgue--Hölder spaces $L^p([0,T];{\mathcal C}_b^β({\mathbb R}^d;{\mathbb R}^d))$ and driven by $α$-stable processes for $α\in (0,2)$. The weak well-posedness is derived for $β\in (0,1)$, $α+β>1$ and $p>α/(α+β-1)$ 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 $β>1-α/2$ 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.