Random exponential attractor for second-order nonautonomous stochastic lattice systems with multiplicative white noise

We studied the existence of a random exponential attractor in the weighted space of infinite sequences for second-order nonautonomous stochastic lattice system with linear multiplicative white noise. Firstly, we present some sufficient conditions for the existence of a random exponential attractor for a continuous cocycle defined on a weighted space of infinite sequences. Secondly, we transferred the second-order stochastic lattice system with multiplicative white noise into a random lattice system without noise through the Ornstein–Uhlenbeck process, whose solutions generate a continuous cocycle on a weighted space of infinite sequences. Thirdly, we estimated the bound and tail of solutions for the random system. Fourthly, we verified the Lipschitz continuity of the continuous cocycle and decomposed the difference between two solutions into a sum of two parts, and carefully estimated the bound of the norm of each part and the expectations of some random variables. Finally, we obtained the existence of a random exponential attractor for the considered system.