The central limit theorem for the Langevin equation with strong damping

Abstract In this paper, we study the functional central limit theorem (FCLT) for a strongly damped stochastic Langevin equation with Lipschitz continuous coefficients. The theorem states the convergence of independent random processes toward Brownian motion or more general diffusion processes. This convergence provides a fundamental framework for analyzing limiting behaviors in randomly driven systems. We establish the FCLT for the Langevin equation by investigating the mean square convergence of the asymptotic behavior of its solutions. The major difficulty in our proof is that there exists a component in the solution of the equation which is not an Itô integral, preventing the direct application of the Itô isometry. We overcome this difficulty by adopting the method of integration by parts. Finally, to validate our theoretical framework, we apply the theorem to two distinct physical models, utilizing the classical harmonic oscillator as a basic model and a two-dimensional generalized Langevin system featuring nonlinear, state-dependent damping and multiplicative noise. The numerical simulations confirm the theoretical convergence rates and demonstrate the universality of using the linearized diffusion limit to analyze thermodynamic fluctuations in complex macroscopic scenarios.