Multidimensional SDE with distributional drift and Lévy noise

Abstract

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

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