Malliavin Calculus for regularity structures: the case of gPAM

TL;DR

This paper extends Malliavin calculus within regularity structures to analyze the generalized parabolic Anderson model in 2D, establishing the existence of a density at positive times for this singular SPDE.

Contribution

It introduces a new extension of regularity structures that includes a deterministic translation operator, enabling Malliavin calculus for certain singular SPDEs.

Findings

Proves the existence of a density for the solution at positive times.

Develops a novel deterministic translation operator within regularity structures.

Applies the framework to the 2D generalized parabolic Anderson model.

Abstract

Malliavin calculus is implemented in the context of [M. Hairer, A theory of regularity structures, Invent. Math. 2014]. This involves some constructions of independent interest, notably an extension of the structure which accomodates a robust, and purely deterministic, translation operator, in $L^2$-directions, between "models". In the concrete context of the generalized parabolic Anderson model in 2D - one of the singular SPDEs discussed in the afore-mentioned article - we establish existence of a density at positive times.