  We study discrete nonlinear parabolic stochastic heat equations of the form,
$u_{n+1}(x)-u_n(x)=(\mathcal {L}u_n)(x)+\sigma(u_n(x))\xi_n(x)$, for $n\in
{\mathbf{Z}}_+$ and $x\in {\mathbf{Z}}^d$, where $\boldsymbol
\xi:=\{\xi_n(x)\}_{n\ge 0,x\in {\mathbf{Z}}^d}$ denotes random forcing and
$\mathcal {L}$ the generator of a random walk on ${\mathbf{Z}}^d$. Under mild
conditions, we prove that the preceding stochastic PDE has a unique solution
that grows at most exponentially in time. And that, under natural conditions,
it is "weakly intermittent." Along the way, we establish a comparison principle
as well as a finite support property.
