Onsager--Machlup Functional for Fractional Stochastic Newton Dynamics with Time-Dependent Noise Intensities

In this paper, we derive the Onsager–Machlup functional for a second-order Newton-type stochastic system driven by time-dependent fractional noise, $$X_t^{\mathrm{\prime }\mathrm{\prime }}=f_t(X_t,X_t^{\mathrm{\prime }})+{\sigma }_t{\xi }_t^H,$$ where $H\in (1\mathrm{/}4,1)$. The analysis relies on applying a Girsanov transformation to the non-degenerate components and evaluating the limiting conditional expectation associated with the noise term, for which the stochastic Fubini theorem plays a crucial role. To illustrate the applicability of the result, we study two mechanical systems perturbed by noise and provide supporting numerical simulations.