Singularity of solutions to singular SPDEs

TL;DR

This work provides a general criterion for when the marginal law of a nonlinear singular SPDE solution on ${\mathbb T}^d$ is singular relative to the Gaussian law of the linearized dynamics, via a Cameron–Martin space test on the Da Prato–Debussche decomposition. The authors construct a renormalized expansion using $Z$, $Y^N$, and $v$, and prove singularity under a subcritical regime and in the borderline case by carefully analyzing the divergences of key constants. The framework is then applied to the dynamical Φ^4_3 model and to fractional Φ^4 models to establish singularity boundaries, and it is extended to a KPZ-like equation to illustrate the reach and limitations of the method. Overall, the paper extends singular SPDE techniques to a broad class of non-gradient-type equations and provides rigorous results about the singularity of important quantum field theory measures relative to Gaussian references.

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 $Φ^4_3$-measure with respect to the Gaussian free field measure and the border of parameters for the fractional $Φ^4$-measure to be singular with respect to the Gaussian free field measure. Our approach is applicable to quite a large class of singular SPDEs.

Theorems & Definitions (42)

• Remark 1.1
• Lemma 3.1
• Lemma 3.4
• Remark 3.6
• Remark 3.7
• Remark 3.8
• Theorem 3.9
• Remark 3.10
• Lemma 3.11
• Remark 3.12