On multidimensional stable-driven stochastic differential equations with Besov drift

Abstract

We establish well-posedness results for multidimensional non degenerate α-stable driven SDEs with time inhomogeneous singular drifts in ${\mathbb{L}}^r-{\mathbb{B}}_{p,q}^{-1+\mathit{\gamma }}$ with $\mathit{\gamma }<1$ and α in $(1,2]$, where ${\mathbb{L}}^r$ and ${\mathbb{B}}_{p,q}^{-1+\mathit{\gamma }}$ stand for Lebesgue and Besov spaces respectively. Precisely, we first prove the well-posedness of the corresponding martingale problem and then give a precise meaning to the dynamics of the SDE. This allows us in turn to define an ad hoc notion of weak solution, for which well-posedness holds as well. Our results rely on the smoothing properties of the underlying PDE, which is investigated by combining a perturbative approach with duality results between Besov spaces.