February 2024 Stochastic transport equation with singular drift
Damir Kinzebulatov, Yuliy A. Semënov, Renming Song
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 731-752 (February 2024). DOI: 10.1214/22-AIHP1353


We prove existence, uniqueness and Sobolev regularity of weak solution of the Cauchy problem of the stochastic transport equation with drift in a large class of singular vector fields containing, in particular, the Ld class, the weak Ld class, as well as some vector fields that are not even in Lloc2+ε for any ε>0.

On obtient des résultats sur l’existence, l’unicité et la régularité en sens de Sobolev de la solution faible au problème de Cauchy pour l’équation de transport stochastique avec dérive dans une large classe de champs vectoriels singuliers contenant, en particulier, la classe Ld, la classe Ld faible, ainsi que certains champs vectoriels qui n’appartiennent même pas à Lloc2+ε pour ε>0 arbitraire.

Funding Statement

The research of D. K. is supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada (RGPIN-2017-05567).
The research of R. S. is supported in part by a grant from the Simons Foundation (#429343).


We thank the referee for very helpful comments that helped to improve the presentation.

Part of the research for this paper was done while the first-named author was supported by a grant of the Fonds de recherche du Québec – Nature et technologies.

Part of the research for this paper was done while the third-named author was visiting Jiangsu Normal University, where he was partially supported by a grant from the National Natural Science Foundation of China (11931004, Yingchao Xie).


Download Citation

Damir Kinzebulatov. Yuliy A. Semënov. Renming Song. "Stochastic transport equation with singular drift." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 731 - 752, February 2024. https://doi.org/10.1214/22-AIHP1353


Received: 23 February 2021; Revised: 20 October 2022; Accepted: 22 November 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718397
Digital Object Identifier: 10.1214/22-AIHP1353

Primary: 35R60
Secondary: 35A21

Keywords: Gradient estimates , Singular drift , Stochastic differential equations , stochastic transport equation

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré


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Vol.60 • No. 1 • February 2024
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