Onsager-Machlup functional for stochastic lattice dynamical systems driven by time-varying noise

This paper investigates the Onsager-Machlup functional of stochastic lattice dynamical systems (SLDSs) driven by time-varying noise. We extend the Onsager-Machlup functional from finite-dimensional to infinite-dimensional systems, and from constant to time-varying diffusion coefficients. We first verify the existence and uniqueness of general SLDS solutions in the infinite sequence weighted space $l^2_{\rho}$. Building on this foundation, we employ techniques such as the infinite-dimensional Girsanov transform, Karhunen-Lo\`eve expansion, and probability estimation of Brownian motion balls to derive the Onsager-Machlup functionals for SLDSs in $l^2_{\rho}$ space. Additionally, we use a numerical example to illustrate our theoretical findings, based on the Euler Lagrange equation corresponding to the Onsage Machup functional.