Stochastic transport equation with Lévy noise
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| author | Brzeźniak, Zdzisław Priola, Enrico Zhai, Jianliang Zhu, Jiahui |
| author_facet | Brzeźniak, Zdzisław Priola, Enrico Zhai, Jianliang Zhu, Jiahui |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17727 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic transport equation with Lévy noise Brzeźniak, Zdzisław Priola, Enrico Zhai, Jianliang Zhu, Jiahui Probability 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. |
| title | Stochastic transport equation with Lévy noise |
| topic | Probability |
| url | https://arxiv.org/abs/2512.17727 |