Stochastic transport equation with L\'evy noise

TL;DR

This paper establishes existence and uniqueness of solutions for a stochastic transport equation driven by pure jump Lévy noise, extending previous results from Brownian to jump noise and covering the supercritical regime.

Contribution

It generalizes the well-posedness results of stochastic transport equations from Brownian to pure jump Lévy noise, including the supercritical case.

Findings

Proved existence and pathwise uniqueness of weak solutions.

Developed sharp regularity results for stochastic flows.

Extended the theory to the entire Lévy stability range  .

Abstract

We study the stochastic transport equation with globally $\beta$-H\"older continuous and bounded vector field driven by a non-degenerate pure-jump L\'evy noise of $\alpha$-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 $\frac{\alpha}{2}+\beta>1$. Notably, our results cover the entire range $\alpha \in (0,2)$, including the supercritical regime $\alpha\in(0,1)$ where the driving noise exhibits notoriously weak regularization. A key step of our strategy is the development of a \emph{sharp} $C^{1+\delta}$-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 L\'evy 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.