Renormalised Models for Variable Coefficient Singular SPDEs

TL;DR

This paper extends the theory of regularity structures to variable coefficient singular SPDEs, proving convergence of renormalised models and establishing their dependence on the coefficient field.

Contribution

It introduces renormalisation functions for non-translation invariant SPDEs and proves their local dependence and continuous dependence on coefficients.

Findings

Proves convergence of renormalised models for variable coefficient SPDEs.

Extends results beyond translation invariant settings.

Shows models depend continuously on the coefficient field.

Abstract

In this work we prove convergence of renormalised models in the framework of regularity structures [Hai14] for a wide class of variable coefficient singular SPDEs in their full subcritical regimes. In particular, we provide for the first time an extension of the main results of [CH16, HS24, BH23] beyond the translation invariant setting. In the non-translation invariant setting, it is necessary to introduce renormalisation functions rather than renormalisation constants. We show that under a very general assumption, which we prove covers the case of second order parabolic operators, these renormalisation functions can be chosen to be local in the sense that their space-time dependence enters only through a finite order jet of the coefficient field of the differential operator at the given space-time point. Furthermore we show that the models we construct depend continuously on the coefficient field.