Semi-discrete, semi-linear SPDEs

Consider an infinite system of interacting Ito 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 s. 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 quadratic order k^2, in contrast to the continuous-space stochastic heat equation whose k-th moment Lyapunov exponent can be of sharp cubic order. 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 regular enough.