Stochastic transport equation with Lévy noise

Abstract

We study the stochastic transport equation with globally $\beta$-Hölder continuous and bounded vector field driven by a non-degenerate pure-jump Lévy 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 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.

128 pages