Semi-discrete semi-linear parabolic SPDEs

Consider an infinite system $\partial_tu_t(x)=(\mathscr{L}u_t)(x)+ σ\bigl(u_t(x)\bigr)\partial_tB_t(x)$ of interacting Itô 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 $σ$. 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 $\ell^1(\mathbf {Z}^d)$.

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