On multidimensional stable-driven stochastic differential equations with Besov drift

We establish well-posedness results for multidimensional non degenerate alpha-stable driven SDEs with time inhomogeneous singular drifts in L-r - B-p,q(-1+gamma) with gamma < 1 and alpha in (1, 2], where L-r and B-p,q(-1+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.