Multidimensional SDE with distributional drift and Lévy noise

Abstract

We solve multidimensional SDEs with distributional drift driven by symmetric, $\hat{I}\pm$-stable Lévy processes for $\hat{I}\pm \in (1,2]$ by studying the associated (singular) martingale problem and by solving the Kolmogorov backward equation. We allow for drifts of regularity $(2-2\hat{I}\pm )/3$, and in particular we go beyond the by now well understood "Young regime", where the drift must have better regularity than $(1-\hat{I}\pm )/2$. 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 Lévy noise. Keywords: Singular diffusions, stable Lévy noise, distributional drift, paracontrolled distributions, Brox diffusion

25 pages