February 2024 Martingale solutions to the stochastic thin-film equation in two dimensions
Max Sauerbrey
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 373-412 (February 2024). DOI: 10.1214/22-AIHP1328


We construct solutions to the stochastic thin-film equation with quadratic mobility and Stratonovich gradient noise in the physically relevant dimension d=2 and allow in particular for solutions with non-full support. The construction relies on a Trotter–Kato time-splitting scheme, which was recently employed in d=1. The additional analytical challenges due to the higher spatial dimension are overcome using α-entropy estimates and corresponding tightness arguments.

Nous construisons des solutions de l’équation aux dérivées partielles stochastique des couches minces avec une mobilité quadratique et un forçage stochastique gradient de type Stratonovich en dimension d=2, physiquement pertinente. Les solutions à support non plein sont autorisées. La construction repose sur une méthode de Trotter–Kato en fractionnant l’intervalle de temps, récemment utilisée dans le cas d=1. Les difficultés supplémentaires, dues à la dimension spatiale supérieure, sont surmontées à l’aide d’estimations de l’α-entropie et d’arguments de tension correspondants.


The author thanks his doctoral advisor Manuel Gnann for many insightful discussions on this subject as well as the careful reading of this document. He thanks Mark Veraar for pointing out the possibility to relax integrability assumptions on the initial value. He also thanks the anonymous referees for the careful reading of this document and their valuable suggestions.


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Max Sauerbrey. "Martingale solutions to the stochastic thin-film equation in two dimensions." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 373 - 412, February 2024. https://doi.org/10.1214/22-AIHP1328


Received: 12 August 2021; Revised: 13 September 2022; Accepted: 26 September 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718385
Digital Object Identifier: 10.1214/22-AIHP1328

Primary: 35R60 , 76A20

Keywords: noise , Stochastic compactness method , thin-film equation , α-Entropy estimates

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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