2025/12/19 by Zdzisław Brzeźniak, Brzeźniak, Zdzisław, Enrico Priola +5
Economics, Econometrics and Finance · Physics and Astronomy · Mathematics · #Stochastic processes and financial applications #stochastic dynamics and bifurcation #Advanced Mathematical Physics Problems
paper · doi:10.48550/arxiv.2512.17727
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 \fracα2+β>1. Notably, our results cover the entire range α∈ (0,2), including the supercritical regime α∈(0,1) where the driving noise exhibits notoriously weak regularization. A key step of our strategy is the development of a sharp C1+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.