Renormalisation of parabolic stochastic PDEs

TL;DR

This paper surveys recent advances in the analysis of singular stochastic PDEs, focusing on their renormalisation and the role of regularity structures in understanding universality and scaling limits in statistical mechanics.

Contribution

It provides an overview of the development and application of regularity structures for renormalising and analyzing singular stochastic PDEs in statistical mechanics.

Findings

Regularity structures enable rigorous construction of singular SPDEs.

Renormalisation procedures are essential for defining these equations.

Universal behavior in cross-over regimes is described by these SPDEs.

Abstract

We give a survey of recent result regarding scaling limits of systems from statistical mechanics, as well as the universality of the behaviour of such systems in so-called cross-over regimes. It transpires that some of these universal objects are described by singular stochastic PDEs. We then give a survey of the recently developed theory of regularity structures which allows to build these objects and to describe some of their properties. We place particular emphasis on the renormalisation procedure required to give meaning to these equations. These are expanded notes of the 20th Takagi lectures held at Tokyo University on November 4, 2017.