Onsager--Machlup Functionals for Generalized Newtonian Equations of Motion with Time-Varying Fractional Noise

Abstract

In this paper, we derive the Onsager--Machlup functional for a class of degenerate stochastic differential equations on $\mathbb{R}^{m+n}$ driven by $n$-dimensional fractional Brownian motion with time-dependent diffusion coefficients, where the Hurst parameter satisfies $H\in(1/4,1)$. The main difficulty arises from the interaction between the degenerate structure and the multidimensional fractional noise, which produces nontrivial coupling terms under small-ball conditioning. By combining the Gaussian correlation inequality with an approximation argument for infinite-dimensional convex sets, we establish a decoupling mechanism that allows these coupling effects to be controlled by unconditional Gaussian expectations. Furthermore, through regularity estimates for the degenerate components and sharp Hölder norm estimates, we eliminate the remaining coupling contributions and obtain an explicit expression for the Onsager--Machlup functional. As applications of the derived functional, we obtain the corresponding constrained Euler--Lagrange equations characterizing the most probable transition paths of the system, and establish a sufficient condition under which the stochastic differential equation preserves the most probable path.