In this paper, we investigate the existence of a random uniform exponential attractor for the non-autonomous stochastic Boussinesq lattice equation with multiplicative white noise and quasi-periodic forces. We first show the existence and uniqueness of the solution of the considered Boussinesq system. Then, we consider the existence of a uniform absorbing random set for a jointly continuous non-autonomous random dynamical system (NRDS) generated by the system, and make an estimate on the tail of solutions. Third, we verify the Lipschitz continuity of the skew-product cocycle defined on the phase space and the symbol space. Finally, we prove the boundedness of the expectation of some random variables and obtain the existence of a random uniform exponential attractor for the considered system.

The purpose of this paper was to discuss the existence of a random exponential attractor for non-autonomous coupled Klein-Gordon-Schrödinger (KGS) lattice equations with multiplicative noise. We employed the method of estimation on the tails of solutions to prove the existence of a random attractor for a continuous cocycle generated by the random KGS lattice equations on an infinite-dimensional sequence space, and used this abstract result to prove the Lipschitz continuity of the continuous cocycle. Then, we verified the existence of a random exponential attractor for the investigated system according to a known criterion.