Semi-discrete semi-linear parabolic SPDEs

Abstract

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

Published at http://dx.doi.org/10.1214/14-AAP1065 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)