The Onsager-Machlup functional for distribution dependent SDEs driven by fractional Brownian motion

TL;DR

The paper addresses the problem of characterizing the most probable transition paths for distribution-dependent SDEs driven by fractional Brownian motion by deriving the Onsager-Machlup functional in both singular ($\tfrac{1}{4}<H<\tfrac{1}{2}$) and regular ($H>\tfrac{1}{2}$) regimes. The authors employ a fractional Girsanov transform combined with a distribution-fixed Taylor expansion to obtain a closed-form OM functional that involves the inverse fractional operator $ (K^H)^{-1} $, with the explicit form depending on $H$. They extend the results to finite dimensions and provide numerical validation via Euler-Lagrange equations, including a stochastic pendulum example, revealing how the most probable paths depend on the Hurst parameter. This work advances mean-field SDEs with memory by delivering a rigorous variational framework for pathwise likelihoods under fractional noise, enabling analysis of transitions in systems with distribution dependence and long-range dependence.

Abstract

In this paper, we compute the Onsager-Machlup functional for distribution dependent SDEs driven by fractional Brownian motions with Hurst parameter $H\in (\frac{1}{4},1)$. In the case $ \frac{1}{4} < H < \frac{1}{2} $, the norm can be either the supremum norm or Hölder norms of order $ β$ with $ 0 < β< H - \frac{1}{4} $. In the case $\frac{1}{2} < H < 1 $, the norms can be a Hölder norm of order $ β$ with $ H - \frac{1}{2} < β< H - \frac{1}{4} $. As an example, we compute the Onsager-Machlup functional for the stochastic pendulum equation

Figures (7)

• Figure 1: The stochastic process $X_t$ follows equation \ref{['exam']}, transitioning from the initial state $x = \pi$ to the final state $x = 2$. The most probable path, $\phi_t$ (depicted by the red line), is illustrated for $H = \frac{3}{10}$, $H = \frac{1}{2}$, and $H = \frac{7}{10}$.
• Figure 2: Phase portrait of the deterministic pendulum system \ref{['exam22']}
• Figure 3: The stochastic process $X_t$ follows equation \ref{['ex21']}, transitioning from the initial state $(-\frac{\pi}{2},0)$ to the final state $(\frac{\pi}{2},0)$. The most probable path, $\phi_t$ (depicted by the red line), is illustrated for $H = \frac{3}{10}$, $H = \frac{1}{2}$, and $H = \frac{7}{10}$.
• Figure :
• Figure :

Theorems & Definitions (31)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5: Nualart
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9
• Remark 1