  Consider a parabolic stochastic PDE of the form $\partial_t
u=\frac{1}{2}\Delta u + \sigma(u)\eta$, where $u=u(t\,,x)$ for $t\ge0$ and
$x\in\mathbb{R}^d$, $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz
continuous and non random, and $\eta$ is a centered Gaussian noise that is
white in time and colored in space, with a possibly-signed homogeneous spatial
correlation $f$. If, in addition, $u(0)\equiv1$, then we prove that, under a
mild decay condition on $f$, the process $x\mapsto u(t\,,x)$ is stationary and
ergodic at all times $t>0$. It has been argued that, when coupled with moment
estimates, spatial ergodicity of $u$ teaches us about the intermittent nature
of the solution to such SPDEs \cite{BertiniCancrini1995,KhCBMS}.
  Our results provide rigorous justification of such discussions.
  Our methods hinge on novel facts from harmonic analysis and functions of
positive type, as well as from Malliavin calculus and Poincar\'e inequalities.
  We further showcase the utility of these Poincar\'e inequalities by: (a)
describing conditions that ensure that the random field $u(t)$ is mixing for
every $t>0$; and by (b) giving a quick proof of a conjecture of Conus et al
\cite{CJK12} about the "size" of the intermittency islands of $u$.
  The ergodicity and the mixing results of this paper are sharp, as they
include the classical theory of Maruyama \cite{Maruyama} (see also Dym and
McKean \cite{DymMcKean}) in the simple setting where the nonlinear term
$\sigma$ is a constant function.
