  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.
