Semi-discrete semi-linear parabolic SPDEs

Abstract

Consider an infinite system

\[\partial_{t}u_{t}(x)=(\mathscr{L}u_{t})(x)+\sigma(u_{t}(x))\partial_{t}B_{t}(x)\] 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 $\sigma$. 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 $\ell^{1}(\mathbf{Z}^{d})$.