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

TL;DR

This paper derives the Onsager--Machlup functional for second-order stochastic Newton systems driven by fractional noise with time-dependent intensities, using Girsanov transformation and stochastic Fubini theorem, with applications to mechanical systems.

Contribution

It introduces a novel Onsager--Machlup functional for fractional stochastic Newton dynamics with time-dependent noise, expanding theoretical understanding and computational tools.

Findings

Derived the Onsager--Machlup functional for fractional Newton systems.

Applied Girsanov transformation and stochastic Fubini theorem in the derivation.

Validated results with numerical simulations on mechanical systems.

Abstract

In this paper, we derive the Onsager--Machlup functional for a second-order Newton-type stochastic system driven by time-dependent fractional noise, \[ X_t'' = f_t(X_t, X_t') + \sigma_t \,\xi_t^{H}, \] where \( H \in (1/4,1) \). 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.