Strong law of large numbers and central limit theorem for stochastic lattice differential equations

In this paper, we establish the strong law of large numbers (SLLN) and central limit theorem (CLT) for the time-inhomogeneous stochastic lattice differential equations in different spaces. These equations are driven by a deterministic time periodic force and a white noise. First, under some suitable conditions on the coefficients, we discuss the existence of $ \tau $-periodic solutions in distribution. We further investigate some version of Feynman-Kac formula, which follows a Feynman-Kac evolution operator, as well as a $ \tau $-periodic measure. We demonstrate two main limit theorems for the continuous time-inhomogeneous periodic solution processes, and apply them to two models: a dynamical viral network model and a three-compartment model.