Non-Gaussianity of invariant measures to SPDEs in Da Prato-Debussche regime

TL;DR

This paper introduces an algebraic method to demonstrate the non-Gaussian nature of invariant measures for certain parabolic SPDEs with polynomial nonlinearities, especially in the Da Prato-Debussche regime.

Contribution

It presents a novel, elementary algebraic approach using the generator equation at stationarity to establish non-Gaussianity of invariant measures in specific SPDEs.

Findings

Proves non-Gaussianity of invariant measures for a class of SPDEs.

Covers $\

) in dimensions less than 14/5, including singular measures.

Abstract

We propose an elementary method to show non-Gaussianity of invariant measures of parabolic stochastic partial differential equations with polynomial non-linearities in the Da Prato--Debussche regime. The approach is essentially algebraic and involves using the generator equation of the SPDE at stationarity. Our results in particular cover the $\Phi^4_\delta$ measures in dimensions $\delta<\frac{14}{5}$, which includes cases where the invariant measure is singular with respect to the invariant measure of the linear solution.