Synchronization of stochastic lattice equations and upper semicontinuity of attractors

Abstract

We consider a system of two coupled stochastic lattice equations driven by additive white noise processes, where the strength of the coupling is given by a parameter $\kappa \ge 0.$ We show that these equations generate a random dynamical system which has a random pullback attractor. This attractor naturally depends on the parameter κ. When the intensity of the coupling becomes large, we observe that the components of the given system synchronize. To describe this phenomenon, we prove the upper semicontinuity of the family of attractors with respect to the attractor of a specific limiting system.