Multidimensional SDE with distributional drift and Lévy noise

Abstract

We solve multidimensional SDEs with distributional drift driven by symmetric, α-stable Lévy processes for $\mathit{\alpha }\in (1,2]$ by studying the associated (singular) martingale problem and by solving the Kolmogorov backward equation. We allow for drifts of regularity $(2-2\mathit{\alpha })∕3$, and in particular we go beyond the by now well understood “Young regime”, where the drift must have better regularity than $(1-\mathit{\alpha })∕2$. 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 Lévy noise.