Random models for singular SPDEs

TL;DR

This paper proves the convergence of a renormalized model for the generalized KPZ equation using chaos decomposition and a generalized convergence theorem, simplifying previous renormalization approaches.

Contribution

It introduces a new proof of model convergence that avoids full BPHZ renormalization, broadening the applicability to generalized KPZ equations.

Findings

Established convergence of the BHZ renormalized model

Developed a chaos decomposition approach for SPDEs

Extended Hairer-Quastel convergence theorem

Abstract

We give a proof of the convergence of the BHZ renormalized model associated with the generalized (KPZ) equation that does not require the full strength of the BPHZ renormalisation. Our approach is based on a convenient form of chaos decomposition. The other key ingredient is a generalisation of the Hairer-Quastel convergence theorem for Feynman diagrams with certain decorations encoding Taylor remainders. With these ideas we are able to construct the model for the generalised KPZ equation.