Multidimensional SDE with distributional drift and Lévy noise

Abstract: We solve multidimensional SDEs with distributional drift driven by symmetric, $\alpha$-stable L\'evy processes for $\alpha\in (1,2]$ by studying the associated (singular) martingale problem and by solving the Kolmogorov backward equation. We allow for drifts of regularity $(2-2\alpha)/3$, and in particular we go beyond the by now well understood "Young regime", where the drift must have better regularity than $(1-\alpha)/2$. The analysis of the Kolmogorov backward equation in the low regularity regime is based on paracontrolled distributions. As an application of our results we construct a Brox diffusion with L\'evy noise. Keywords: Singular diffusions, stable L\'evy noise, distributional drift, paracontrolled distributions, Brox diffusion