# On Multidimensional stable-driven Stochastic Differential Equations with Besov drift

## TL;DR

This paper proves the well-posedness of multidimensional stable-driven SDEs with singular Besov space drifts, using PDE smoothing properties and duality techniques, extending the understanding of such stochastic systems.

## Contribution

It establishes the existence and uniqueness of solutions for multidimensional stable-driven SDEs with singular Besov drifts, a novel result in this context.

## Key findings

- Well-posedness of the martingale problem established.
- Defined a meaningful notion of weak solution for these SDEs.
- Utilized PDE smoothing properties and duality in Besov spaces.

## Abstract

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