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

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.