  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
$x\in\mathbb{R}^d$, $\sigma:\mathbb{R}\to\mathbb{R}$ 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
function $f$. 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 of such discussions. The proof rests on novel
facts about functions of positive type, and on strong localization bounds for
comparison of SPDEs.
