Stochastic transport equation with Levy noise

主讲人：翟建梁 中国科学技术大学副教授

时间：2026年1月24日14：00

地点：徐汇校区三号楼332室

举办单位：数理学院

主讲人介绍：翟建梁，中国科学技术大学副教授。主要研究方向是Levy过程驱动的随机偏微分方程，主要成果有：Levy过程驱动的随机偏微分方程解的存在唯一性、遍历性、大偏差原理、平稳测度支撑的渐近行为。已发表论文50余篇, 包括“JEMS”“J. Math. Pures Appl.”“J. Funct. Anal.”等国际重要杂志。

内容介绍：We study the stochastic transport equation with globally β-Holder continuous and bounded vector field driven by a non-degenerate pure-jump Levy 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 α/2+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 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 Levy 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 pure-jump noise.