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

Abstract

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

Figures (7)

• Figure 1: $H=0.3,\sigma_0=2,A=1.5,\omega=10$
• Figure 2: $H=0.5,\sigma_0=2,A=1.5,\omega=6\pi$
• Figure 3: $H=0.51,\sigma_0=2,A=0.1,\omega=8\pi$
• Figure 4: The average path and the optimal path for \ref{['equation 2']}
• Figure 5: The average path and the optimal path for \ref{['equation 2']}, with $H=0.3, \sigma_t = 2 + \cos(8\pi t)$ and $\gamma=0.1$

Theorems & Definitions (24)

• Definition 2.1
• Definition 2.2
• Theorem 2.3: Biagini2008
• Lemma 2.4: Biagini2008
• Lemma 2.5: Young1936
• Lemma 2.6: Young1936
• Remark 1
• Remark 2
• Remark 3
• Example 3.1: Pendulum equation