Stochastic PDES, regularity structures, and interacting particle systems
\n

Abstract

These lecture notes grew out of a series of lectures given by the
\nsecond 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 [26]. This theory gives a way to study wellposednessfor a class of stochastic PDEs that could not be treated previously.Prominent examples include the KPZ equation as well as the dynamic Φ43model.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.
\nResum ´ e. ´ Ces notes sont bas´ees sur trois cours que le deuxi`eme auteur a donn´es `a Toulouse, Matsumoto et Darmstadt. L’objectif principal est d’expliquer certains aspects de la th´eorie des “structures de r´egularit´e” d´evelopp´ee r´ecemment par Hairer [26]. Cette th´eorie permet de montrer que certaines EDP stochastiques,
\nqui ne pouvaient ˆetre trait´ees auparavant, sont bien pos´ees. Parmi les exemples se trouvent l’´equation KPZ et le mod`ele Φ4
\n2 dynamique. De telles ´equations peuvent ˆetre d´evelopp´ees en s´eries perturbatives formelles. La th´eorie des structures de r´egularit´e permet de tronquer ce d´eveloppement apr´es un nombre fini de termes, et de r´esoudre un probl`eme de point fixe pour le reste. L’id´ee principale est une nouvelle notion de r´egularit´e des distributions,
\nqui d´epend des termes de ce d´eveloppement.