Current Developments in Mathematics

Volume 2014

Regularity structures and the dynamical $\Phi^4_3$ model

Pages: 1 – 49



Martin Hairer (The University of Warwick, United Kingdom)


We give a concise overview of the theory of regularity structures as first exposed in “A theory of regularity structures” [M. Hairer. Invent. Math. 198, no. 2, (2014), 269–504]. In order to allow to focus on the conceptual aspects of the theory, many proofs are omitted and statements are simplified. In order to provide both motivation and focus, we concentrate on the study of solutions to the stochastic quantisation equations for the Euclidean $\Phi^4_3$ quantum field theory which can be obtained with the help of this theory. In particular, we sketch the proofs of how one can show that this model arises quite naturally as an idealised limiting object for several classes of smooth models.

Published 31 December 2015