Conditioned Onsager–Machlup Integral

• The conditioned Onsager–Machlup integral is a functional framework that quantifies the likelihood of entire trajectories in stochastic systems, forming the basis for large deviation and rare-event analysis.
• It extends classical diffusion methods to incorporate jump-diffusion, Lévy noise, and degenerate stochastic differential equations using modified variational and geometric techniques.
• The framework underpins practical applications in optimal control, data assimilation, and transition path sampling through analytical and numerical optimization of most probable paths.

The conditioned Onsager–Machlup (OM) integral is a functional framework central to the probabilistic, variational, and geometric analysis of stochastic processes, quantifying the likelihood of entire paths (trajectories) under noise and drift. In recent decades, the theory has been extensively developed for degenerate stochastic differential equations (SDEs) including non-Gaussian Lévy noise, as well as for analysis in both finite- and infinite-dimensional metric measure spaces. The OM integral under conditioning (for instance, on initial/final state or on small “tubes” around a path) plays a central role in large deviation theory, rare-event analysis, transition path sampling, and optimal control.

1. Foundations of the Onsager–Machlup Integral

The Onsager–Machlup functional was originally motivated by the paper of fluctuations and reversibility in nonequilibrium thermodynamics, as a Lagrangian quantifying the exponential weight for observing a given fluctuation or trajectory of a stochastic system. For diffusions, the functional typically takes the form

$$P[x(\cdot)] \propto \exp\left(-\frac{1}{2D} S[x(\cdot)]\right),$$

where, for a path $x(\cdot)$ governed by an SDE $dx_t = f(x_t)dt + \sigma dB_t$, the OM action is

$$S[x(\cdot)] = \int_0^T \frac{1}{2} \left|\dot{x}(t) - f(x(t))\right|^2 dt + \frac{1}{2}\int_0^T \operatorname{div} f(x(t)) dt,$$

with the divergence term arising due to the volume change in path space. A central object of interest is the probability that the process stays in a small tube about a smooth reference path $\phi(\cdot)$, which, in regimes where large deviation estimates hold, asymptotically behaves as

$$P(\|X-\phi\| < \epsilon) \sim C(\epsilon)\exp(-S[\phi]/2D).$$

The minimizer(s) of $S[\phi]$ (“most probable paths”) dominate transition probabilities.

2. Extensions to Jump–Diffusion and Lévy Systems

In many physical, biological, and engineering contexts, stochastic systems are not purely diffusive but include jump noise, leading to jump–diffusions and Lévy-driven models. These processes are governed by SDEs of the form

$$dX_t = f(X_{t-}) dt + \sigma dB_t + \int_{\mathbb R} x \,\tilde N(dt, dx),$$

where $\tilde N(dt, dx)$ is a compensated Poisson random measure; the corresponding generator includes nonlocal (integro-differential) terms.

Classical path-integral and Girsanov techniques from the diffusive setting are only partially applicable in the presence of jumps (especially with infinite activity), due to the complexity of the path space and the failure of standard small-ball and tube probability arguments. Recent works, e.g. "Probability Flow Approach to the Onsager–Machlup Functional for Jump-Diffusion Processes" (Huang et al., 2 Sep 2024), introduce a probability flow equivalence between jump–diffusions and pure diffusions, enabling a rigorous derivation of a closed-form OM functional. For finite jump activity,

$$S[\phi] = \int_0^T \frac{1}{2\sigma^2} \left( \dot\phi(t) - f(\phi(t)) - \ell_J(\phi(t)) \right)^2 dt + \int_0^T V_J(\phi(t)) dt,$$

where $\ell_J$ and $V_J$ encode drift corrections due to the jump intensity (including an explicit term related to the Lévy measure at the origin). In the infinite activity regime, only a time-discrete (finite-dimensional) OM functional can generally be defined.

3. Degenerate and Distribution-Dependent SDEs

Much contemporary research explores generalizations of OM theory to degenerate SDEs (where noise acts only on a subset of variables), SDEs driven by fractional Brownian motion (fBm), and mean-field (McKean–Vlasov) type models with law-dependent drift.

For distribution-dependent SDEs with fBm noise, e.g. $dX_t = b(X_t, \mathcal{L}_{X_t})dt + dB^H_t$ for Hurst parameter $H\in(1/4,1)$, the OM functional acquires nonlocal and fractional calculus features: $$J(\phi, \dot\phi) = -\frac{1}{2} \int_0^1 \left[\dot\phi(s) - \mathcal{K}b(\phi_s,\mathcal{L}_{\phi_s})\right]^2 ds + d_H \int_0^1 b_x(\phi_s,\mathcal{L}_{\phi_s}) ds,$$ where $\mathcal{K}$ is an integral or derivative operator reflecting the memory in the noise, and $d_H$ is a constant depending on $H$ (Zhu et al., 20 Mar 2025). The minimizers in this context are characterized by fractional Euler–Lagrange equations (Liu et al., 2023).

For degenerate SDEs and degenerate McKean–Vlasov models, the structure of the OM functional is modified to reflect that small tube probabilities for unforced coordinates depend on the coupling to noisy components via the drift. The derivation uses fractional Girsanov transformations, Itô calculus for law-dependent functionals, and conditioned exponential inequalities (Liu et al., 2023).

4. Geometric and Infinite-dimensional Aspects

The OM functional also plays a role as a generalized density in metric measure spaces, especially in infinite dimensions where standard densities do not exist. For a metric measure space $(X,d,\mu)$, the OM functional is defined via the small-ball scaling

$$\lim_{r\to 0_+} \frac{\mu(B(r,x))}{\mu(B(r,y))} = \exp(\mathrm{OM}(y) - \mathrm{OM}(x)).$$

Upon reweighting both the measure by $e^{-V(x)}$ and the metric as $$d(x,y) \mapsto \inf_{\gamma}\int e^{-U(\gamma(t))}|\dot\gamma(t)|dt$$, the OM functional transforms as

$\mathrm{OM} = \mathrm{OM}_0 - p U + V,$

where $p$ is the local exponent for small-ball scaling (e.g., the manifold dimension in finite dimensions) (Selk, 12 Oct 2025). The result underscores the geometric sensitivity of probability densities in path space: metric reweightings cause nontrivial changes in the likelihood structure of conditioned OM integrals and, in general, the variational problem for most probable paths.

5. Variational Formulation and Most Probable Transition Paths

The OM functional acts as a Lagrangian in variational problems for the most probable path (instantons, transition events) between prescribed endpoints or states. For jump-diffusion systems, the OM Lagrangian generically acquires new terms reflecting the drift correction from jump noise,

$$\mathrm{OM}(\dot\phi,\phi) = \frac{1}{c^2}\left( \dot\phi - f(\phi) + \int_{|x|<1} x\,\nu(dx) \right)^2 + f'(\phi),$$

leading to Euler–Lagrange equations with explicit jump-induced force terms (Chao et al., 2018, Chao et al., 9 Jan 2025).

In degenerate systems, the variational problem requires the Hamilton–Pontryagin formalism: minimization is subject to constraints (e.g., $\dot\phi_1 = g(\phi_1, \phi_2)$), leading to implicit Euler–Lagrange equations involving auxiliary multipliers (Chao et al., 9 Jan 2025). These systems often admit analytical solutions when the drift and noise structure are quadratic; in generic cases, numerical optimization is required.

6. Connections to Data Assimilation and Large Deviation Theory

In data assimilation (e.g., smoothing, weak-constraint 4D-Var), the OM functional underlies the prior over paths; its explicit form informs both MCMC samplers and MAP estimators. The divergence (Jacobian) correction is essential for correct weighting in variational (MAP) estimation, while for sampling, certain discretizations allow its omission (Sugiura, 2017). In large deviation theory, the OM functional appears as the rate function for pathwise deviations—e.g., in macroscopic fluctuation theory for reaction networks, it splits into dissipative, energetic, and fluctuation-induced components, with extensions to nonequilibrium and anisothermal settings (Renger, 2021).

7. Implications and Future Directions

The conditioned OM integral formalism is presently extended in several directions:

• Non-equilibrium dynamical systems, particularly piecewise-stationary models and non-Markovian noise, via extended OM theory (Peredo-Ortiz et al., 2023).
• Statistical and optimal control problems in infinite-dimensional or geometric settings, where the transformation of the OM functional under changes of metric or measure is nontrivial (Selk, 12 Oct 2025).
• Rare event analysis, optimal transition path computation, and statistical inference in systems driven by non-Gaussian (jump or heavy-tailed) noise, with closed-form and numerically tractable formulas for the OM action (Huang et al., 2 Sep 2024, Chao et al., 2018).

Recent developments leverage probability flow equivalence and variational calculus (Hamilton–Pontryagin and Euler–Lagrange formalisms) to handle degeneracy, infinite activity, and memory effects, significantly broadening the range of stochastic systems where OM-based transition path analysis is possible. Analytical and computational approaches, incorporating fractional calculus and high-dimensional geometry, continue to be actively developed.