Invariant measures of stochastic Klein–Gordon–Schrödinger equations on infinite lattices

We study the long-time dynamics of stochastic Klein–Gordon–Schrödinger equations driven by infinite-dimensional nonlinear noise defined on integer set. Firstly, we formulate the stochastic lattice equations as an abstract system defined in an appropriated space of square-summable sequences, and then prove the existence and uniqueness of global solutions to the abstract system. To such solutions, we establish the uniform boundedness and uniform estimates on the tails of solutions, which are necessary to ensure the tightness of a family of probability distributions. Finally, we prove the existence of invariant measures for the stochastic lattice equations using the Krylov–Bogolyubov’s method.