Weak mean attractors, ergodicity and large deviation principle of stochastic Klein–Gordon–Schrödinger lattice systems

This paper is devoted to the study of the existence, uniqueness and statistical properties of solutions for stochastic Klein–Gordon–Schrödinger (KGS) lattice systems driven by nonlinear noise. First of all, we establish the well-posedness of the stochastic KGS lattice systems with Lipschitz continuous nonlinear diffusion and drift terms, and then define a mean random dynamical system. Next, we demonstrate the existence and uniqueness of weak pullback mean random attractors in a Bochner space. Furthermore, we not only prove the existence of invariant probability measures to autonomous stochastic KGS lattice systems, but also obtain the ergodicity and exponential mixing property of invariant measures. Besides, we investigate the limiting behavior of invariant measures of autonomous stochastic KGS lattice systems as the noise intensity [Formula: see text] approaches to [Formula: see text]. Finally, we establish large deviations of stochastic KGS lattice systems via weak convergence method. It is worth mentioning that we utilize the idea of uniform tail-estimates for verifying the tightness of distributions of solutions and the set of invariant measure to autonomous stochastic KGS lattice systems.