Multidimensional SDE with distributional drift and L\'evy noise

Helena Kremp; Nicolas Perkowski

doi:10.3150/21-bej1394

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arXiv · 2008.05222

Multidimensional SDE with distributional drift and L\'evy noise

Helena Kremp·Nicolas Perkowski

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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

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Helena Kremp, Nicolas Perkowski. 2020-08-12. Multidimensional SDE with distributional drift and L\'evy noise. https://doi.org/10.3150/21-bej1394

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