Existence of densities for the dynamic $\Phi^4_3$ model

TL;DR

This paper proves the existence of probability densities for solutions to the $\

Contribution

It introduces a novel application of Malliavin calculus within the regularity structures framework to establish density existence for the $\

Findings

Density exists for solutions under broad noise conditions.

Applicable to various Gaussian space-time noises, including white noise.

Works for degenerate and rough noises on small scales.

Abstract

We apply Malliavin calculus to the $\Phi^4_3$ equation on the torus and prove existence of densities for the solution of the equation evaluated at regular enough test functions. We work in the framework of regularity structures and rely on Besov-type spaces of modelled distributions in order to prove Malliavin differentiability of the solution. Our result applies to a large family of Gaussian space-time noises including white noise, in particular the noise may be degenerate as long as it is sufficiently rough on small scales.