Well-posedness and invariant measures for second-order stochastic lattice systems with superlinear diffusion terms

In this paper, we investigate both well-posedness and existence of an invariant measures for a class of wave equations defined on higher-dimensional integer set [Formula: see text] driven by a family of infinite-dimensional superlinear. By a space-time cut-off technique we prove the global existence as well as uniqueness of solutions in [Formula: see text] with [Formula: see text] for a family of nonlinear drift terms of polynomial growth of arbitrary order as well as a family of nonlinear diffusion terms with superlinear growth rate. In addition, the existence of invariant measures is established in [Formula: see text] by formulating Krylov–Bogolyubov’s method. The tightness of a family of distribution laws of solutions is derived via the idea of uniform tail-estimates of solutions in order to overcome the lack of compactness on infinite lattices.