Onsager–Machlup Integral in Stochastic Systems

• The Onsager–Machlup Integral (OMI) is a variational framework that defines path probabilities in stochastic systems by extending the concept of density to entire trajectories.
• It incorporates effects of drift, diffusion, curvature, and jump statistics, and adapts to transformations in measure and geometry for both diffusive and non-diffusive processes.
• OMI underpins applications in large-deviation theory, non-equilibrium thermodynamics, optimal control, and Bayesian inference, linking stochastic calculus with geometric analysis.

The Onsager–Machlup Integral (OMI) provides a rigorous variational and probabilistic framework for the characterization of path probabilities in stochastic dynamical systems, bridging stochastic calculus, statistical mechanics, and geometric analysis. The OMI generalizes the notion of probability density to path spaces, quantifying the likelihood of entire trajectories in terms of nonlinear functionals that capture the contributions of drift, diffusion, geometry, and, for non-diffusive processes, jump statistics. The functional has critical roles in large-deviation theory, non-equilibrium thermodynamics, inference, and optimal control, with deep connections to stochastic differential equations (SDEs), kinetic theory, and Riemannian geometry.

1. Foundational Definition: Metric Measure Spaces and Transformation Laws

The Onsager–Machlup (OM) functional is defined for a metric measure space $(X,d,\mu)$ via the small-ball ratio limit: $$\lim_{r\to 0+} \frac{\mu( B(r,x) )}{\mu( B(r,y) )} = \exp( OM(y) - OM(x) ),$$ for a suitable subset $Z \subset X$ and Borel measure $\mu$, with $B(r,z)$ the open metric ball around $z$ (Selk, 12 Oct 2025). The OM functional generalizes the logarithm of density: in Euclidean space with $\mu = e^{-f}dy$, one recovers $OM(x) = f(x)$, so that $e^{-OM(x)}$ is the reference density.

When the metric and measure are jointly transformed,

$\mu = e^{-V} \mu_0,$

$$d(x,y) = \inf_\gamma \int_0^1 e^{-U(\gamma(t))} \lvert \gamma'(t) \rvert_{d_0} dt,$$

and $\mu_0$ satisfies a small-ball power law with parameter $p$ (e.g., the dimension), the OM functional transforms as

$OM = OM_0 - p U + V,$

up to an additive constant, emphasizing the intertwined influences of measure and geometry on small-ball asymptotics (Selk, 12 Oct 2025).

In infinite-dimensional spaces, such as Gaussian measures on function spaces, OM functionals generally fail to exist under nontrivial metric reweighting, underlining their sensitivity to the underlying geometric structure. In finite-dimensional settings, conformal geometry and measure rescaling are entirely equivalent from the OM perspective, a fact underpinning applications in geometric statistics and population cartogram construction.

2. Classical Forms and Extensions: Riemannian, Non-Equilibrium, and Large Deviations

For SDEs on Riemannian manifolds ($M,g$), the OM functional quantifies the small-tube asymptotics for diffusions: $$L(x,v) = \frac{1}{2} \| v - f(x) \|_g^2 + \frac{1}{2} \mathrm{div}_g f(x) + \frac{1}{12} R(x),$$ where $f$ is the drift, $\mathrm{div}_g f$ the Riemannian divergence, and $R(x)$ the scalar curvature (Hara et al., 2016). The associated action

$$S_{OM}[\gamma] = \int_0^T L(\gamma(t),\dot{\gamma}(t)) dt$$

determines path probabilities and most probable (instanton) trajectories, incorporating both dissipative (quadratic) and geometric (curvature) corrections.

In non-equilibrium and non-stationary settings, the OMI adapts by incorporating time- and state-dependent metrics and drift,

$$L(a, \dot{a}, t) = \frac{1}{4}( \dot a - F )^T D^{-1} ( \dot a - F ) + \frac{1}{2} \nabla\cdot F,$$

accommodating nontrivial noise covariances and deterministic flows (Peredo-Ortiz et al., 2023). The structure persists for vector-valued nonstationary processes and piecewise-stationary Markov approximations essential for the analysis of time-dependent irreversible phenomena.

In large-deviation theory for Markov jump processes (e.g., chemical reactions in detailed balance), the pathwise rate functional generalizes to

$$\mathcal{I}_{[0,T]}(\rho) = \int_0^T \hat\Psi(\rho,\dot\rho) dt + \int_0^T \hat\Psi^*(\rho,-\tfrac{1}{2}\nabla V(\rho)) dt + \frac{1}{2}( V(\rho(T)) - V(\rho(0)) ),$$

where $\hat\Psi$ and its dual $\hat\Psi^*$ encode the dissipation structure and $V(\rho)$ is the quasipotential of the invariant measure (Renger, 2021).

3. Onsager–Machlup Integral for General Stochastic Systems

For the standard overdamped SDE

$dx(t) = a(x) dt + \sqrt{2D} dW(t),$

the OM Lagrangian in the Stratonovich convention is

$$L(x,\dot x) = \frac{1}{4D} ( \dot x - a(x) )^2 + \frac{1}{2} a'(x),$$

leading to transition-weighted path probabilities $P[x] \propto \exp(-A[x]/2k_BT)$ (Yasuda et al., 29 Apr 2024, Cugliandolo et al., 2017).

In systems with multiplicative noise, the OM functional requires careful prescription of discretization (the "α-scheme") and introduces additional drift and Jacobian terms. The action for the multiplicative-noise Langevin SDE takes the form

$$\mathcal{L}(x,\dot x) = \frac{1}{4D} \left( \frac{ \dot x - f(x) + 2\alpha D g(x) g'(x)}{g(x)} \right)^2 + \alpha f'(x),$$

with the proper stochastic calculus and measure normalization dictated by discretization and variable change (Cugliandolo et al., 2017).

For inhomogeneous or thermally active environments, the OM Lagrangian generalizes to include a coordinate-dependent variance ("Helmholtz multiplier") $b(q)$, leading to

$$\mathcal{L}(q,\dot q) = \frac{1}{2 \sigma_2 b(q)^2} ( \dot q - a(q) + 2\sigma_2 b(q) b'(q) )^2 + \partial_q a(q) - \sigma_2[(b')^2 + b b'']$$

where $b(q)$ encapsulates spatial noise strength and environmental properties. The associated path integral representation connects directly to solutions of the Fokker–Planck equation (Jurisch, 2020).

4. Lévy Noise, Jump-Diffusions, and Degenerate Systems

For SDEs driven by jump noise, such as Lévy processes, the probability of a path in a small tube is determined by an OM function that incorporates both Gaussian and non-Gaussian effects. For 1D jump-diffusions: $$dX_t = f(X_{t-}) dt + c dB_t + \int_{|x|<1} x\, \tilde{N}(dt, dx),$$ the OM function is

$$OM(z, \dot z) = \left( \frac{\dot z - f(z)}{c} \right)^2 + f'(z) + 2\frac{\dot z - f(z)}{c^2} d_{\nu},$$

where $d_\nu = \int_{|x|<1} x \nu(dx)$ measures asymmetry in jump sizes. The most probable path, or instanton, minimizes

$I[z] = \int OM(z(t), \dot z(t)) dt,$

subject to endpoint data (Chao et al., 2018).

For degenerate systems (e.g., kinetic Langevin with underdamped dynamics and Lévy noise),

$$\begin{cases} dX_t = g(X_t, Y_t) dt,\ dY_t = f(X_t, Y_t) dt + c dW_t + dL_t, \end{cases}$$

the OM function becomes

$$OM(\phi, \dot \phi) = \frac{1}{2} \left| \frac{ \dot \phi_2 - f(\phi_1, \phi_2) + \int_{|\xi|<1} \xi \nu(d\xi) }{ c } \right|^2 + \frac{1}{2} \partial_y f(\phi_1, \phi_2),$$

with the corresponding variational problem handled by the Hamilton–Pontryagin principle; for systems where noise acts only on a subset of variables, the constraint structure dictates the admissible class of extremal trajectories (Chao et al., 9 Jan 2025).

The probability-flow approach can be used to rigorously construct the OM functional for multi-dimensional jump-diffusions, even with infinite jump activity. In the finite-activity case, the OM functional obtains new drift and correction terms reflecting jump statistics, while in the infinite-activity case a time-discrete OM functional is necessary (Huang et al., 2 Sep 2024).

5. Applications: Variational Principles, Path Optimization, and Statistical Mechanics

The OMI underpins diverse variational methods, most notably the Onsager–Machlup variational principle (OMVP), which seeks the most probable path by minimizing the time-global functional over admissible trajectories. Extensions to "statistical formulation" (SOMVP) facilitate the calculation of fluctuation properties such as diffusion constants and cumulants by maximizing suitable OM-based functionals subject to observable constraints (Yasuda et al., 29 Apr 2024).

For active particles (e.g., 3D active Brownian particles), the OMI is constructed with explicit inclusion of dissipative and active power contributions, yielding a path integral whose minimization reveals rich geometric transitions (straight, U-shaped, helical MPPs) under varying boundary conditions, realized efficiently via neural-network optimization in high dimension (Zheng et al., 20 Nov 2025).

In field-theoretic contexts, auxiliary-field expansions of the OM path integral provide non-perturbative access to effective actions, renormalization-group flows, and phase structure in stochastic PDEs, with the OM formalism mapping directly to response-function representations (Martin–Siggia–Rose–Janssen–de Dominicis) (Cooper et al., 2014).

In Bayesian inference on infinite-dimensional spaces, the minimizer of the OM functional coincides with the strong/weak mode (MAP estimator) of the posterior distribution, justifying the use of penalized log-likelihood (e.g., Tikhonov–Phillips regularization) in solvers for inverse problems (Kretschmann, 2022).

6. Generalizations and Theoretical Significance

The OMI persists across a wide array of driven, non-equilibrium, and geometrically complex systems, as long as the noise structure and underlying geometry are sufficiently regular to ensure meaningful small-ball asymptotics. Its geometric transformation law under metric and measure reweighting, critical in finite dimension (Selk, 12 Oct 2025), breaks down in infinite dimensions due to the collapse of small-ball asymptotics, demarcating the boundaries of its applicability.

Recent advances extend OM functionals to SDEs with fractional noise ($B^H$), tracking how the Hurst parameter $H$ modifies both the functional structure and admissible path spaces (Hölder/Sobolev), with robust small-ball estimates underpinning these generalizations (Zhu et al., 12 Nov 2025).

The OMI remains a foundational construct in stochastic variational modeling, large deviation theory, statistical mechanics, and geometric analysis, serving as both a practical computational tool and a theoretical lens through which the subtle interplay of noise, geometry, and non-equilibrium driving can be rigorously quantified.