Onsager-Machlup functional for stochastic differential equations with time-varying noise

This paper is devoted to studying the Onsager-Machlup functional for stochastic differential equations with time-varying noise of the $ \alpha $-Hölder, $ {0 < \alpha < \frac{1}{4}} $,$ \begin{equation} \begin{aligned} \mathrm{d} X_t = f(t, X_t)\mathrm{d} t + g(t)\mathrm{d}W_t. \end{aligned}\notag \end{equation} $Our study focuses on scenarios where the diffusion coefficient $g(t)$ exhibits temporal variability, starkly contrasting the conventional assumption of a constant diffusion coefficient in the existing literature. This variance brings additional complexity to the analysis. Through this investigation, we derive the Onsager-Machlup functional, which serves as a Lagrangian. Its minimum characterizes the most probable transition path between metastable states in diffusion processes with time‑dependent noise. This is done by introducing new measurable norms and applying an appropriate version of the Girsanov transformation. To illustrate our theoretical advancements, we provide numerical simulations, including cases of a one-dimensional stochastic differential equation and a fast-slow system, which demonstrate the application to multiscale stochastic volatility models, thereby highlighting the significant impact of time-varying diffusion coefficients.