Periodic measures for the stochastic delay modified Swift–Hohenberg lattice systems

In this paper, the existence and the limiting behavior of periodic measures for the periodic stochastic modified Swift–Hohenberg lattice systems with variable delays are analyzed. We first prove the existence and uniqueness of global solution when the nonlinear T-periodic drift and diffusion terms are locally Lipchitz continuous and linearly growing. Then we show the existence of periodic measures of the system under some assumptions. Finally, by strengthening the assumptions, we prove that the set of all periodic measures is weakly compact, and we also show that every limit point of a sequence of periodic measures of the original system must be a periodic measure of the limiting system when the noise intensity tends to zero.