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

Let $Z$ be the stationary solution of the additive stochastic heat equation ${\mathrm{\partial }}_{t}Z=(\mathrm{\Delta }-1)Z+\xi$ 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 $\{Z^{:k:}{\}}_{k=1}^{\mathrm{\infty }}$. This leads to an elementary proof of a support theorem for the dynamic $P(\mathrm{\Phi }{)}_2$ 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.