Stochastic PDEs, Regularity structures, and interacting particle systems

Ajay Chandra; Hendrik Weber

doi:10.5802/afst.1555

Ajay Chandra ¹ ; Hendrik Weber ¹

¹ University of Warwick

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 4, pp. 847-909.

Résumé
Abstract

Ces notes sont basées sur trois cours que le deuxième auteur a donnés à Toulouse, Matsumoto et Darmstadt. L’objectif principal est d’expliquer certains aspects de la théorie des « structures de régularité » développée récemment par Hairer [27]. Cette théorie permet de montrer que certaines EDP stochastiques, qui ne pouvaient pas être traitées auparavant, sont bien posées. Parmi les exemples se trouvent l’équation KPZ et le modèle $Φ_{3}^{4}$ dynamique.

Telles équations peuvent être développées en séries perturbatives formelles. La théorie des structures de régularité permet de tronquer ce développement aprés un nombre fini de termes, et de résoudre un problème de point fixe pour le reste. L’idée principale est une nouvelle notion de régularité des distributions, qui dépend des termes de ce développement.

These lecture notes grew out of a series of lectures given by the second named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main aim is to explain some aspects of the theory of “Regularity structures” developed recently by Hairer in [27]. This theory gives a way to study well-posedness for a class of stochastic PDEs that could not be treated previously. Prominent examples include the KPZ equation as well as the dynamic $Φ_{3}^{4}$ model.

Such equations can be expanded into formal perturbative expansions. Roughly speaking the theory of regularity structures provides a way to truncate this expansion after finitely many terms and to solve a fixed point problem for the “remainder”. The key ingredient is a new notion of “regularity” which is based on the terms of this expansion.

Publié le : 2017-12-12

DOI : 10.5802/afst.1555

Affiliations des auteurs :

Ajay Chandra ¹ ; Hendrik Weber ¹

¹ University of Warwick

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Ajay Chandra; Hendrik Weber. Stochastic PDEs, Regularity structures, and interacting particle systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 4, pp. 847-909. doi : 10.5802/afst.1555. https://afst.centre-mersenne.org/articles/10.5802/afst.1555/

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