Semi-discrete semi-linear parabolic SPDEs

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 $\ell^1(\mathbf {Z}^d)$.