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.

• The paper derives an explicit Onsager–Machlup functional for degenerate SDEs with time-dependent fractional noise, addressing coupling and memory effects in complex systems.
• It employs degenerate Girsanov transformations and infinite-dimensional Gaussian correlation inequalities to manage multidimensional noise and implicit history-dependent interactions.
• Numerical experiments on systems like the stochastic pendulum and Duffing oscillator validate that most probable paths often coincide with deterministic noise-free solutions.

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

Introduction and Problem Formulation

The paper "Onsager--Machlup Functionals for Generalized Newtonian Equations of Motion with Time-Varying Fractional Noise" (2607.04193) addresses the derivation and analysis of Onsager–Machlup (OM) functionals for a large class of degenerate stochastic differential equations (SDEs) on $\mathbb{R}^{m+n}$, subject to $n$-dimensional fractional Brownian motion (fBm) with time-dependent diffusion coefficients, for Hurst parameter $H \in (1/4,1)$. The primary motivation is modeling complex systems—such as viscoelastic media and protein dynamics—where stochastic forcing acts only on a subset of variables (degeneracy) and exhibits both temporal correlations and time-varying intensity.

The OM functional provides a path-space action that characterizes the relative likelihood of rare noise-driven transitions between metastable states. In the context of SDEs perturbed by fBm, especially with time inhomogeneity and degeneracy, such quantitative characterizations are analytically challenging due to strong coupling and memory effects.

Main Results: Explicit Formulation of the OM Functional

The main contribution is a rigorous derivation of an explicit OM functional for degenerate SDEs of the form

$$\begin{cases} dX_t = a_t(X_t, Y_t) \, dt, \ dY_t = b_t(X_t, Y_t) \, dt + \sigma_t \, dB^H_t, \ \end{cases}$$

where $\sigma_t$ is a time-dependent, non-degenerate diagonal diffusion matrix and $B^H_t$ is $n$-dimensional fBm with $H \in (1/4,1)$. The coupling between position $X_t$ and velocity $Y_t$ generates nontrivial dependencies, as noise acts only on $n$0 but propagates throughout the system via the deterministic flow.

The OM functional $n$1 associated to a reference path $n$2 in $n$3 and the corresponding deterministic flow $n$4 in $n$5 is given in terms of the inverse of a $n$6-weighted fractional Volterra operator $n$7:

$n$8

with explicit formulas for $n$9 in regimes $H \in (1/4,1)$0, $H \in (1/4,1)$1, and $H \in (1/4,1)$2. Here $H \in (1/4,1)$3 is a universal constant depending on $H \in (1/4,1)$4.

Analytical Techniques and Decoupling Mechanisms

Two core technical difficulties are overcome:

• Degeneracy-induced coupling: The drift $H \in (1/4,1)$5 couples $H \in (1/4,1)$6 into $H \in (1/4,1)$7, and $H \in (1/4,1)$8 itself evolves via a deterministic equation driven by $H \in (1/4,1)$9. This leads to implicit, history-dependent interactions that cannot be treated by standard (non-degenerate) Girsanov transformations.
• Multidimensional noise coupling: The multidimensionality of fBm yields nontrivial cross-correlation terms not representable by deterministic kernels or reducible to independent components.

To manage these, the analysis deploys:

1. Degenerate Girsanov transformation: Shifts only the noise-affected ($$\begin{cases} dX_t = a_t(X_t, Y_t) \, dt, \ dY_t = b_t(X_t, Y_t) \, dt + \sigma_t \, dB^H_t, \ \end{cases}$$0) subspace, reconstructing $$\begin{cases} dX_t = a_t(X_t, Y_t) \, dt, \ dY_t = b_t(X_t, Y_t) \, dt + \sigma_t \, dB^H_t, \ \end{cases}$$1 via deterministic coupling.
2. Gaussian Correlation Inequality (GCI): Extends the classical GCI from finite to infinite-dimensional Gaussian measures, permitting reduction of pathwise conditional exponential moments to unconditional Gaussian expectations under Hölder small-ball conditioning.
3. Sharp regularity and Gronwall estimates: Establishes that under small-noise constraints, the deviations of the degenerate components scale optimally, allowing vanishing of higher-order coupling in the OM functional.

Explicit Form of the OM Functional

The OM functional is computed in full generality for $$\begin{cases} dX_t = a_t(X_t, Y_t) \, dt, \ dY_t = b_t(X_t, Y_t) \, dt + \sigma_t \, dB^H_t, \ \end{cases}$$2 (fractional) and $$\begin{cases} dX_t = a_t(X_t, Y_t) \, dt, \ dY_t = b_t(X_t, Y_t) \, dt + \sigma_t \, dB^H_t, \ \end{cases}$$3 (classical Brownian) regimes:

$$\begin{cases} dX_t = a_t(X_t, Y_t) \, dt, \ dY_t = b_t(X_t, Y_t) \, dt + \sigma_t \, dB^H_t, \ \end{cases}$$4

where $$\begin{cases} dX_t = a_t(X_t, Y_t) \, dt, \ dY_t = b_t(X_t, Y_t) \, dt + \sigma_t \, dB^H_t, \ \end{cases}$$5, and fractional integrals/derivatives are defined in the Riemann–Liouville sense.

Most Probable Paths and Euler–Lagrange Theory

Minimizers of $$\begin{cases} dX_t = a_t(X_t, Y_t) \, dt, \ dY_t = b_t(X_t, Y_t) \, dt + \sigma_t \, dB^H_t, \ \end{cases}$$6 under fixed endpoints yield the most probable transition paths. The paper obtains the necessary Euler–Lagrange equations for these minimizers, both in the classical and fractional setting, under the additional constraint that $$\begin{cases} dX_t = a_t(X_t, Y_t) \, dt, \ dY_t = b_t(X_t, Y_t) \, dt + \sigma_t \, dB^H_t, \ \end{cases}$$7 is coupled to $$\begin{cases} dX_t = a_t(X_t, Y_t) \, dt, \ dY_t = b_t(X_t, Y_t) \, dt + \sigma_t \, dB^H_t, \ \end{cases}$$8:

• The system yields a set of coupled forward-backward ordinary/integro-differential equations for $$\begin{cases} dX_t = a_t(X_t, Y_t) \, dt, \ dY_t = b_t(X_t, Y_t) \, dt + \sigma_t \, dB^H_t, \ \end{cases}$$9 (with $\sigma_t$0 the adjoint Lagrange multiplier), sharply characterizing optimal paths in the degenerate fBm-driven setting.
• For certain systems (e.g., classical mechanics and Hamiltonian flows with constant divergence in velocity), the most probable paths coincide with deterministic noise-free solutions—a phenomenon termed path-preservation. This is demonstrated to arise when the divergence term in the OM functional is path-independent.

Numerical Experiments

The theoretical predictions are substantiated with numerical integration of representative degenerate SDEs, specifically:

• Stochastic pendulum equation: Simulations demonstrate that the deterministic solution coincides with the most probable path and empirical mean, invariant under changes in $\sigma_t$1 and time-modulated noise amplitude.

Figure 1: Simulation of the most probable path for $\sigma_t$2, $\sigma_t$3, $\sigma_t$4, and $\sigma_t$5.

• Stochastic Duffing oscillator: For both autonomous and time-inhomogeneous noise, OM functional minimization yields transition paths consistent with averaged sample paths. For $\sigma_t$6, nonlocal effects of fractional noise manifest in altered path morphology, directly influenced by the noise history and time-dependent signal.

  Figure 2: The average path and the optimal path for the Duffing system (Equation 2) in the classical case.

  

  Figure 3: The average path and the optimal path for Equation 2 with $\sigma_t$7 and rapidly time-varying $\sigma_t$8.

  

  Figure 4: The average path for Equation 2 with $\sigma_t$9 and periodic variation in $B^H_t$0.

Implications and Future Outlook

The derivation of explicit Onsager–Machlup action for degenerate SDEs with time-dependent, multidimensional fractional Brownian forcing fills a significant theoretical gap. From a practical standpoint, this greatly expands the range of physical, biological, and engineered systems—especially those with subspace-restricted noise and history dependence—for which transition statistics and rare-event pathways can be quantitatively evaluated.

The decoupling arguments, especially those leveraging infinite-dimensional Gaussian correlation, should see further application in the study of stochastic PDEs, nonequilibrium statistical mechanics, and inference for rough-path–driven models. The analytical techniques developed here may provide a blueprint for future work on path-space large deviation theory and optimal control under degenerate fractional noise.

Exciting directions include extending these results to:

• Non-diagonal, possibly singular diffusion matrices,
• Non-Gaussian (e.g., Lévy or jump) driving noise,
• Stochastic partial differential equations with spatially degenerate multiplicative fractional noise,
• Analysis of adaptive importance sampling for rare event simulation based on the derived OM functionals.

Conclusion

This work rigorously establishes the Onsager–Machlup functional for a broad class of degenerate stochastic systems with time-varying, multidimensional fractional noise, resolving core analytical obstacles posed by coupling and memory effects. It yields explicit analytical and computational tools for evaluating most probable noise-induced transitions and characterizes the geometry of optimal paths in such strongly non-Markovian systems, with both theoretical and practical ramifications for stochastic dynamics under complex noise.

Reference:

"Onsager--Machlup Functionals for Generalized Newtonian Equations of Motion with Time-Varying Fractional Noise" (2607.04193)