Semi-discrete semi-linear parabolic SPDEs

Consider an infinite system .tut(x)=(Lut)(x)+.(ut(x)).tBt(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 .. We will show that, locally in time, the solution behaves as a collection of independent diffusions. We prove also that the kth moment Lyapunov exponent is frequently of sharp order k2, in contrast to the continuous-space stochastic heat equation whose kth moment Lyapunov exponent can be of sharp order k3. 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 .1(Zd).