Renormalising SPDEs in regularity structures

TL;DR

This paper develops a framework for renormalising singular stochastic PDEs within regularity structures, establishing a link between the renormalisation group actions and solutions, leading to a general stability theorem.

Contribution

It constructs an action of the renormalisation group on stochastic PDEs, connecting it with model actions and solutions, extending previous theoretical results.

Findings

Solutions from BPHZ lift match classical solutions of modified PDEs

Provides a general local existence and stability theorem for singular SPDEs

Establishes a robust renormalisation procedure for a wide class of noises

Abstract

The formalism recently introduced in arXiv:1610.08468 allows one to assign a regularity structure, as well as a corresponding "renormalisation group", to any subcritical system of semilinear stochastic PDEs. Under very mild additional assumptions, it was then shown in arXiv:1612.08138 that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to such a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT. The present work completes this programme by constructing an action of the renormalisation group onto a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models. This shows in particular that solutions constructed from the BPHZ lift of a smooth driving noise coincide with the classical solutions of a modified PDE. This yields a very general black box type local existence and stability theorem for a wide class of singular nonlinear SPDEs.