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Stochastic transport equation with Lévy noise

Zdzisław Brzeźniak, Enrico Priola, Jianliang Zhai, Jiahui Zhu

TL;DR

This work addresses the well-posedness of a stochastic transport equation with a Hölder continuous and bounded drift under non-degenerate pure-jump Lévy noise, extending the Flandoli–Gubinelli–Priola Brownian framework to α-stable jump noise. The authors develop a sharp C^{1+δ} regularity theory for the stochastic flow and establish new regularity results for the Jacobian determinant, enabling existence and pathwise uniqueness of weak* L∞-solutions via a stochastic flow representation. Their analysis demonstrates regularization by noise in a Lévy-driven PDE, including the supercritical regime α<1, and introduces a robust Itô–Wentzell formula in the jump setting. They also prove stability of the stochastic flow under drift perturbations and provide a rigorous framework for weak solutions in the Marcus sense, with precise flow-based constructions and renormalization arguments. Overall, the paper generalizes key Brownian results to non-degenerate jump Noise, offering novel techniques for flow regularity, Jacobian analysis, and PDE well-posedness under Lévy dynamics with potential numerical implications via Wong–Zakai-type limits.

Abstract

We study the stochastic transport equation with globally $β$-Hölder continuous and bounded vector field driven by a non-degenerate pure-jump Lévy noise of $α$-stable type. Whereas the deterministic transport equation may lack uniqueness, we prove the existence and pathwise uniqueness of a weak solution in the presence of a multiplicative pure jump noise, assuming $\fracα{2}+β>1$. Notably, our results cover the entire range $α\in (0,2)$, including the supercritical regime $α\in(0,1)$ where the driving noise exhibits notoriously weak regularization. A key step of our strategy is the development of a \emph{sharp} $C^{1+δ}$-diffeomorphism and new regularity results for the Jacobian determinant of the stochastic flow associated to its stochastic characteristic equation. These novel probabilistic results are of independent interest and constitute a substantial component of our work. Our results are the first full generalization of the celebrated paper by Flandoli, Gubinelli, and Priola [Invent. Math. 2010] from the Brownian motion to the pure jump Lévy noise. To the best of our knowledge, this appears to be the first example of a partial differential equation of fluid dynamics where well-posedness is restored by the influence of a non-degenerate pure-jump noise.

Stochastic transport equation with Lévy noise

TL;DR

This work addresses the well-posedness of a stochastic transport equation with a Hölder continuous and bounded drift under non-degenerate pure-jump Lévy noise, extending the Flandoli–Gubinelli–Priola Brownian framework to α-stable jump noise. The authors develop a sharp C^{1+δ} regularity theory for the stochastic flow and establish new regularity results for the Jacobian determinant, enabling existence and pathwise uniqueness of weak* L∞-solutions via a stochastic flow representation. Their analysis demonstrates regularization by noise in a Lévy-driven PDE, including the supercritical regime α<1, and introduces a robust Itô–Wentzell formula in the jump setting. They also prove stability of the stochastic flow under drift perturbations and provide a rigorous framework for weak solutions in the Marcus sense, with precise flow-based constructions and renormalization arguments. Overall, the paper generalizes key Brownian results to non-degenerate jump Noise, offering novel techniques for flow regularity, Jacobian analysis, and PDE well-posedness under Lévy dynamics with potential numerical implications via Wong–Zakai-type limits.

Abstract

We study the stochastic transport equation with globally -Hölder continuous and bounded vector field driven by a non-degenerate pure-jump Lévy noise of -stable type. Whereas the deterministic transport equation may lack uniqueness, we prove the existence and pathwise uniqueness of a weak solution in the presence of a multiplicative pure jump noise, assuming . Notably, our results cover the entire range , including the supercritical regime where the driving noise exhibits notoriously weak regularization. A key step of our strategy is the development of a \emph{sharp} -diffeomorphism and new regularity results for the Jacobian determinant of the stochastic flow associated to its stochastic characteristic equation. These novel probabilistic results are of independent interest and constitute a substantial component of our work. Our results are the first full generalization of the celebrated paper by Flandoli, Gubinelli, and Priola [Invent. Math. 2010] from the Brownian motion to the pure jump Lévy noise. To the best of our knowledge, this appears to be the first example of a partial differential equation of fluid dynamics where well-posedness is restored by the influence of a non-degenerate pure-jump noise.

Paper Structure

This paper contains 27 sections, 58 theorems, 739 equations.

Table of Contents

  1. Introduction
  2. Motivation for the study of stochastic transport equations.
  3. Our first main results: The well-posedness results for \ref{['eqn-transport-Markus']}.
  4. Our second main result: Existence and stability of stochastic flow of diffeomorphisms.
  5. New contributions of the paper.
  6. Plan of the paper and additional comments.
  7. Preliminaries
  8. Notation
  9. Lévy Process
  10. Hypotheses on the noise in the transport equation
  11. A regular stochastic flow
  12. The homeomorphism property
  13. The differentiability property
  14. The diffeomorphism property
  15. Stability
  16. ...and 12 more sections

Key Result

Theorem 1.1

Assume that $\alpha \in (0,2)$ and Assume that $L$ is a rotationally invariant $\alpha$-stable Lévy process whose symbol $\psi$ is of the following form, for some constant $C_{\alpha}>0$, and a vector field $b: \mathbb{R}^d \to \mathbb{R}^d$ satisfies conditions eqn-g-beta and Then for every Borel measurable and essentially bounded function $u_0:\mathbb{R}^d \to \mathbb{R}$, the stochastic proc

Theorems & Definitions (114)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof : Proof of Lemma is simple and hence omitted.
  • Definition 2.3
  • Proposition 2.6
  • ...and 104 more