This article is concerned with the existence and stability of invariant or periodic probability measures for a wide class of lattice reversible Selkov systems with coupled nonlinear terms of polynomial growth of arbitrary order defined on the entire integer set \(\mathbb {Z}\) driven by locally Lipschitz noise. We first formulate the stochastic lattice equations to an abstract system defined in the space \(\ell ^2\times \ell ^2\) of square-summable sequences, and then prove the global-in-time existence and uniqueness of solutions to the abstract system. When the noise intensity is controllable in a suitable range, we prove that the weak stability of a family probability distribution laws of the solutions is just an invariant measures by using a Krylov–Bogolyubov’s method. When the time-dependent forces are periodic, we also show the existence of periodic measures on \(\ell ^2\times \ell ^2\) by using this Krylov–Bogolyubov’s method. The tightness and stability of the collection of all invariant or periodic measures are also discussed. The classical idea of uniform tail-estimates developed by Wang (Phys D 128:41–52, 1999) is employed to overcome the difficulties caused by the lack of compactness in infinite lattices. Our results are new even for the cubic nonlinearity case.
