Singularity of solutions to singular SPDEs

TL;DR

This paper establishes conditions under which solutions to certain singular stochastic partial differential equations (SPDEs) have distributions that are singular with respect to Gaussian measures, with applications to quantum field theory models.

Contribution

It provides a new sufficient condition for the singularity of solution distributions of singular SPDEs, extending previous results and applicable to a broad class of such equations.

Findings

Proves the singularity of the $ ext{Φ}^4_3$-measure relative to the Gaussian free field.

Identifies parameter ranges where fractional $ ext{Φ}^4$-measures are singular.

Applies the approach to various singular SPDEs.

Abstract

Building on the notes [Hai17], we give a sufficient condition for the marginal distribution of the solution of singular SPDEs on the $d$-dimensional torus to be singular with respect to the law of the Gaussian measure induced by the linearised equation. As applications we obtain the singularity of the $\Phi^4_3$-measure with respect to the Gaussian free field measure and the border of parameters for the fractional $\Phi^4$-measure to be singular with respect to the Gaussian free field measure. Our approach is applicable to quite a large class of singular SPDEs.