  Consider an infinite system \[\partial_tu_t(x)=(\mathscr{L}u_t)(x)+
\sigma\bigl(u_t(x)\bigr)\partial_tB_t(x)\] of interacting It\^{o} diffusions,
started at a nonnegative deterministic bounded initial profile. We study local
and global features of the solution under standard regularity assumptions on
the nonlinearity $\sigma$. We will show that, locally in time, the solution
behaves as a collection of independent diffusions. We prove also that the $k$th
moment Lyapunov exponent is frequently of sharp order $k^2$, in contrast to the
continuous-space stochastic heat equation whose $k$th moment Lyapunov exponent
can be of sharp order $k^3$. When the underlying walk is transient and the
noise level is sufficiently low, we prove also that the solution is a.s.
uniformly dissipative provided that the initial profile is in $\ell^1(\mathbf
{Z}^d)$.
