Characterization of the support for Wick powers of the additive stochastic heat equation

TL;DR

This paper characterizes the support of Wick powers of the stationary solution to the additive stochastic heat equation on a 2D torus, providing new insights into the support of related stochastic processes and equations.

Contribution

It offers an elementary proof of a support theorem for the dynamic P(Φ)_2 equation and extends the approach to Gaussian multiplicative chaos.

Findings

Support of Wick powers characterized

Elementary proof of support theorem for P(Φ)_2

Support of Gaussian multiplicative chaos law determined

Abstract

Let $Z$ be the stationary solution of the additive stochastic heat equation $\partial_t Z = (\Delta - 1) Z + \xi$ on the two-dimensional torus, where $\xi$ is the space-time white noise. The aim of this paper is to determine the support of Wick powers $\{Z^{:k:}\}_{k=1}^{\infty}$. This leads to an elementary proof of a support theorem for the dynamic $P(\Phi)_2$ equation. In addition, we show that the approach can be used to determine the support of the law of the Gaussian multiplicative chaos in the $L^2$-phase.