We solve multidimensional SDEs with distributional drift driven by symmetric, α-stable Lévy processes for by studying the associated (singular) martingale problem and by solving the Kolmogorov backward equation. We allow for drifts of regularity , and in particular we go beyond the by now well understood “Young regime”, where the drift must have better regularity than . 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.
H.K. was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH (EXC-2046/1, project ID: 390685689). Part of the work was done while N.P. was employed at Max-Planck-Institute for Mathematics in the Sciences, Leipzig, and Humboldt-Universität zu Berlin. N.P. gratefully acknowledges funding by DFG through the Heisenberg program.
Helena Kremp. Nicolas Perkowski. "Multidimensional SDE with distributional drift and Lévy noise." Bernoulli 28 (3) 1757 - 1783, August 2022. https://doi.org/10.3150/21-BEJ1394