Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise

Abstract

In this paper we investigate the long term behavior of a stochastic lattice dynamical system with a diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and a different multiplicative white noise at each node. We prove that this stochastic lattice equation generates a random dynamical system that possesses a global random attractor. In particular, we first establish an existence theorem for weak solutions to general random evolution equations, which is later applied to the specific stochastic lattice system to show that it has weak solutions and the solutions generate a random dynamical system. We then prove the existence of a random attractor of the underlying random dynamical system by constructing a random compact absorbing set and using an embedding theorem. The major novelty of this work is that we consider a different multiplicative white noise term at each different node, which significantly improves the previous results in the literature where the same multiplicative noise was considered at all the nodes. As a consequence, the techniques used in the existing literature are not applicable here and a new methodology has to be developed to study such systems.