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Multidimensional SDE with distributional drift and Lévy noise

Helena Kremp, Nicolas Perkowski

Abstract

We solve multidimensional SDEs with distributional drift driven by symmetric, $α$-stable Lévy processes for $α\in (1,2]$ 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. Keywords: Singular diffusions, stable Lévy noise, distributional drift, paracontrolled distributions, Brox diffusion

Multidimensional SDE with distributional drift and Lévy noise

Abstract

We solve multidimensional SDEs with distributional drift driven by symmetric, -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. Keywords: Singular diffusions, stable Lévy noise, distributional drift, paracontrolled distributions, Brox diffusion

Paper Structure

This paper contains 9 sections, 17 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. The Kolmogorov backward equation
  5. Existence and uniqueness for the martingale problem
  6. Brox diffusion with Lévy noise
  7. Appendix
  8. Commutator estimates
  9. An application of Campbell's formula

Key Result

Lemma \oldthetheorem

For $\varphi \in C^\infty_b$ we have $-\mathscr{L}^{\alpha}_{\nu} \varphi = A\varphi$, where $A$ is the generator of the symmetric, $\alpha$-stable Lévy process $L$ with characteristic exponent $\mathds{E}[\exp(2\pi i\langle z,L_{t}\rangle )]=\exp(-t\psi^{\alpha}_{\nu}(z))$. The process $L$ has the

Theorems & Definitions (43)

  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • ...and 33 more