Asymptotic behavior of non-autonomous stochastic Boussinesq lattice system

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.