Extended Onsager–Machlup Theory

• Extended Onsager–Machlup theory is a generalization of the classical variational principle that determines the most probable stochastic trajectory by minimizing an action functional.
• It broadens the framework by incorporating diverse elements such as Lévy noise, infinite-dimensional settings, fractional dynamics, and nonlocal memory kernels.
• The theory underpins both analytic methods and computational applications in data assimilation, generalized Langevin dynamics, and nonequilibrium as well as equilibrium processes.

Extended Onsager–Machlup theory denotes a family of generalizations of the classical Onsager–Machlup variational principle beyond finite-dimensional Gaussian diffusion. In the classical setting, the probability that a stochastic trajectory remains in a small tube around a smooth reference curve is exponentially governed by an action functional, and the most probable path is obtained by minimizing that action. Current extensions preserve this basic variational logic while enlarging the admissible state spaces, noise classes, and thermodynamic settings: jump-diffusions with Lévy noise, stochastic partial differential equations and stochastic lattice systems, fractional and degenerate dynamics, non-Markovian memory kernels, active and non-reciprocal systems, and distribution-dependent McKean–Vlasov equations all admit Onsager–Machlup-type formulations under additional structural assumptions (Chao et al., 2018, Hu et al., 2020, Kim, 2 Jul 2026, Chao et al., 25 Jun 2026).

1. Classical variational structure and what is being extended

A recurring core structure is the small-tube asymptotic

$$\mathbb P\big(\|X-\phi\|<\varepsilon\big)\propto C(\varepsilon,T)\exp\{-I[\phi]\},$$

or an equivalent path-probability representation. In diffusion form, one standard Onsager–Machlup functional is

$$J_{\mathrm{OM}}(\phi)=\frac12\int_0^T \left(\left|\dot{\phi}(t)-f(\phi(t))\right|^2+\mathrm{div}\,f(\phi(t))\right)\,dt,$$

while Rayleighian-based approaches write a modified Rayleighian $R-R_{\min}$ and define an Onsager–Machlup integral over the full trajectory rather than only an instantaneous variational step (Sugiura, 2017, Doi et al., 2019).

The most probable path is then the minimizer of the action over admissible trajectories with fixed endpoints. This role of the Onsager–Machlup function as a Lagrangian is retained in many later extensions, although the integrand may acquire jump corrections, geometric terms, trace terms, fractional operators, or nonlocal memory kernels (Chao et al., 2018, Chao et al., 25 Jun 2026).

A central interpretive clarification is that a discrete sample-path density is not identical to the asymptotic probability of a smooth tube. In data assimilation, this distinction leads to different discrete priors for weak-constraint $4$D-Var and for MCMC with the Euler scheme: the divergence of the drift is essential in the former but not necessary in the latter (Sugiura, 2017).

Direction of extension	Characteristic modification	Representative papers
Lévy and jump dynamics	Small-jump drift corrections; Lévy-intensity terms; discrete OM for infinite activity	(Chao et al., 2018, Huang et al., 2024, Chao et al., 9 Jan 2025)
Infinite-dimensional and nonlocal dynamics	Hilbert-space trace terms, time-varying diffusion operators, memory kernels	(Hu et al., 2020, Zhang et al., 2024, Kim, 2 Jul 2026)
Fractional, mean-field, active, inertial settings	Fractional operators, law-dependent geometry, odd elasticity, covariant acceleration	(Maayan, 2017, Chao et al., 25 Jun 2026, Yasuda et al., 2021, Yasuda et al., 24 Apr 2026)

2. Lévy noise, jump asymmetry, and degenerate jump systems

A major extension replaces purely Brownian forcing by Brownian motion plus Lévy noise. For the scalar constant-diffusion jump-diffusion

$$dX_t=f(X_{t-})\,dt+c\,dB_t+\int_{|x|<1}x\,\tilde N(dt,dx),$$

the Onsager–Machlup function is, up to an additive constant,

$$OM(\dot z,z)=\left(\frac{\dot z-f(z)}{c}\right)^2+f'(z)+2\frac{\dot z-f(z)}{c^2}\int_{|\xi|<1}\xi\,\nu(d\xi).$$

The additional term is proportional to

$\int_{|\xi|<1}\xi\,\nu(d\xi),$

so it measures the net effect of asymmetric small jumps. If the Lévy measure is symmetric, this integral vanishes in principal value and the classical diffusion Onsager–Machlup function is recovered. The most probable path remains the minimizer of

$I[z]=\int_s^u OM(\dot z,z)\,dt,$

and smooth minimizers satisfy the Euler–Lagrange equation

$$\ddot z_m=\frac{c^2}{2}f''(z_m)+f'(z_m)f(z_m)-f'(z_m)\int_{|\xi|<1}\xi\,\nu(d\xi)$$

with fixed endpoints (Chao et al., 2018).

In the double-well example

$$dX_t=(X_{t-}-X_{t-}^3)\,dt+dB_t+\int_{|x|<1}x\,\tilde N(dt,dx),$$

the numerical results show that the path shape depends strongly on the time horizon $$J_{\mathrm{OM}}(\phi)=\frac12\int_0^T \left(\left|\dot{\phi}(t)-f(\phi(t))\right|^2+\mathrm{div}\,f(\phi(t))\right)\,dt,$$0, that asymmetric Lévy noise changes the transition geometry, that the Gaussian-noise path is recovered when the asymmetry parameter vanishes, and that for some parameter ranges the boundary value problem has no solution, so no most probable path exists in the sense of the theory (Chao et al., 2018).

A later finite-activity theory derives a closed-form jump-diffusion Onsager–Machlup functional by introducing an equivalent probabilistic flow between the jump-diffusion and a diffusion with modified drift. In that formulation the action contains a mean jump drift correction $$J_{\mathrm{OM}}(\phi)=\frac12\int_0^T \left(\left|\dot{\phi}(t)-f(\phi(t))\right|^2+\mathrm{div}\,f(\phi(t))\right)\,dt,$$1, an extra jump correction $$J_{\mathrm{OM}}(\phi)=\frac12\int_0^T \left(\left|\dot{\phi}(t)-f(\phi(t))\right|^2+\mathrm{div}\,f(\phi(t))\right)\,dt,$$2, and a term involving the Lévy intensity at the origin $$J_{\mathrm{OM}}(\phi)=\frac12\int_0^T \left(\left|\dot{\phi}(t)-f(\phi(t))\right|^2+\mathrm{div}\,f(\phi(t))\right)\,dt,$$3. For infinite jump activity, the same framework yields a time-discrete Onsager–Machlup functional rather than a continuous-time one (Huang et al., 2024).

Degenerate jump systems introduce an additional structural constraint because one component is noise-free. For

$$J_{\mathrm{OM}}(\phi)=\frac12\int_0^T \left(\left|\dot{\phi}(t)-f(\phi(t))\right|^2+\mathrm{div}\,f(\phi(t))\right)\,dt,$$4

the Onsager–Machlup function depends only on the noisy component velocity $$J_{\mathrm{OM}}(\phi)=\frac12\int_0^T \left(\left|\dot{\phi}(t)-f(\phi(t))\right|^2+\mathrm{div}\,f(\phi(t))\right)\,dt,$$5 together with the deterministic constraint $$J_{\mathrm{OM}}(\phi)=\frac12\int_0^T \left(\left|\dot{\phi}(t)-f(\phi(t))\right|^2+\mathrm{div}\,f(\phi(t))\right)\,dt,$$6. The corresponding variational problem is handled by a Hamilton–Pontryagin principle rather than a standard unconstrained Euler–Lagrange equation, and the kinetic Langevin example leads to a higher-order boundary value problem for the position variable (Chao et al., 9 Jan 2025).

3. Infinite-dimensional, lattice, and nonlocal path-measure extensions

In infinite-dimensional Hilbert-space settings, the Onsager–Machlup functional acquires operator-theoretic terms. For the SPDE

$$J_{\mathrm{OM}}(\phi)=\frac12\int_0^T \left(\left|\dot{\phi}(t)-f(\phi(t))\right|^2+\mathrm{div}\,f(\phi(t))\right)\,dt,$$7

driven by Brownian and Lévy noise, the action derived via Girsanov transformation and path representation is

$$J_{\mathrm{OM}}(\phi)=\frac12\int_0^T \left(\left|\dot{\phi}(t)-f(\phi(t))\right|^2+\mathrm{div}\,f(\phi(t))\right)\,dt,$$8

with

$$J_{\mathrm{OM}}(\phi)=\frac12\int_0^T \left(\left|\dot{\phi}(t)-f(\phi(t))\right|^2+\mathrm{div}\,f(\phi(t))\right)\,dt,$$9

This extension covers both small and large jumps and turns the most probable transition problem for an SPDE into an optimization problem in Hilbert space (Hu et al., 2020).

A related infinite-dimensional extension treats stochastic lattice dynamical systems in the weighted sequence space

$R-R_{\min}$0

with time-varying diffusion operator $R-R_{\min}$1. The Onsager–Machlup functional is

$R-R_{\min}$2

Its derivation combines well-posedness in $R-R_{\min}$3, an infinite-dimensional Girsanov transform, Karhunen–Loève expansion, and probability estimates for Brownian motion balls. The numerical example is a lattice disease-transmission model with time-varying noise intensity (Zhang et al., 2024).

A conceptually different extension treats non-Markovian latent dynamics generated by attention-induced memory. There the predictive object is no longer a local action

$R-R_{\min}$4

but a nonlocal path measure

$R-R_{\min}$5

The memory kernel is generated by eliminating a hidden linear Markov augmentation, and the kernel class is characterized as completely monotone. The local Onsager–Machlup theory is recovered as the short-memory limit $R-R_{\min}$6; the paper states that locality is an effective approximation rather than the primitive description in the memory-carrying regime (Kim, 2 Jul 2026).

4. Fractional noise, degeneracy, and distribution dependence

Fractional Brownian motion replaces semimartingale structure by long-range dependence and a Cameron–Martin geometry tied to the Hurst index $R-R_{\min}$7. For the additive-noise SDE

$R-R_{\min}$8

with multidimensional fractional Brownian motion and $R-R_{\min}$9, the Onsager–Machlup functional with respect to the driving fractional Brownian motion is

$4$0

for $4$1 under the stated norm restrictions, with a more general conditional formulation for all $4$2 and $4$3 (Maayan, 2017).

Degenerate fractional systems alter both the admissible reference paths and the correction term. For the two-dimensional system

$4$4

the small-ball event reduces to the $4$5-component, the admissible path must satisfy

$4$6

and the Onsager–Machlup action involves a fractional integral for $4$7 or a fractional derivative for $4$8, together with the correction

$4$9

The resulting most probable path satisfies a fractional Euler–Lagrange equation (Liu et al., 2023).

Mean-field dependence produces a different enlargement of the theory. For the McKean–Vlasov SDE

$$dX_t=f(X_{t-})\,dt+c\,dB_t+\int_{|x|<1}x\,\tilde N(dt,dx),$$0

an Euler-type approximation freezes the measure argument on a time grid, applies classical Onsager–Machlup theory to the resulting distribution-free SDEs, and passes to the limit. The limiting Onsager–Machlup integrand is

$$dX_t=f(X_{t-})\,dt+c\,dB_t+\int_{|x|<1}x\,\tilde N(dt,dx),$$1

where $$dX_t=f(X_{t-})\,dt+c\,dB_t+\int_{|x|<1}x\,\tilde N(dt,dx),$$2, $$dX_t=f(X_{t-})\,dt+c\,dB_t+\int_{|x|<1}x\,\tilde N(dt,dx),$$3, $$dX_t=f(X_{t-})\,dt+c\,dB_t+\int_{|x|<1}x\,\tilde N(dt,dx),$$4 is the modified drift, and $$dX_t=f(X_{t-})\,dt+c\,dB_t+\int_{|x|<1}x\,\tilde N(dt,dx),$$5 is the scalar curvature, all evaluated at $$dX_t=f(X_{t-})\,dt+c\,dB_t+\int_{|x|<1}x\,\tilde N(dt,dx),$$6. The Euler–Lagrange equation includes a Lions-derivative term

$$dX_t=f(X_{t-})\,dt+c\,dB_t+\int_{|x|<1}x\,\tilde N(dt,dx),$$7

so the variational problem is explicitly law-dependent (Chao et al., 25 Jun 2026).

5. Nonequilibrium, active, and inertial reformulations

Several extensions move away from equilibrium reciprocity while retaining Onsager–Machlup structure in altered form. In nonlinear transport theory, the minimum dissipation principle is extended to stationary nonequilibrium states that remain locally in equilibrium by decomposing the macroscopic variables into a boundary-driven sector $$dX_t=f(X_{t-})\,dt+c\,dB_t+\int_{|x|<1}x\,\tilde N(dt,dx),$$8 and an internal sector $$dX_t=f(X_{t-})\,dt+c\,dB_t+\int_{|x|<1}x\,\tilde N(dt,dx),$$9 that is orthogonal with respect to the transport metric $$OM(\dot z,z)=\left(\frac{\dot z-f(z)}{c}\right)^2+f'(z)+2\frac{\dot z-f(z)}{c^2}\int_{|\xi|<1}\xi\,\nu(d\xi).$$0. Onsager–Machlup theory is then applied only to the $$OM(\dot z,z)=\left(\frac{\dot z-f(z)}{c}\right)^2+f'(z)+2\frac{\dot z-f(z)}{c^2}\int_{|\xi|<1}\xi\,\nu(d\xi).$$1-sector, yielding an approximate projected variational principle whose error is controlled by off-diagonal couplings (Sonnino et al., 2015).

A different nonequilibrium extension treats a globally non-stationary process as a piecewise-stationary process. The mean obeys

$$OM(\dot z,z)=\left(\frac{\dot z-f(z)}{c}\right)^2+f'(z)+2\frac{\dot z-f(z)}{c^2}\int_{|\xi|<1}\xi\,\nu(d\xi).$$2

while fluctuations over a short fluctuation time $$OM(\dot z,z)=\left(\frac{\dot z-f(z)}{c}\right)^2+f'(z)+2\frac{\dot z-f(z)}{c^2}\int_{|\xi|<1}\xi\,\nu(d\xi).$$3 obey a locally stationary non-Markov generalized Langevin equation with waiting-time-dependent kernels. In this construction the local covariance satisfies

$$OM(\dot z,z)=\left(\frac{\dot z-f(z)}{c}\right)^2+f'(z)+2\frac{\dot z-f(z)}{c^2}\int_{|\xi|<1}\xi\,\nu(d\xi).$$4

and the framework is used to derive the non-equilibrium self-consistent generalized Langevin equation theory of irreversible processes in liquids (Peredo-Ortiz et al., 2023).

Modern Onsager–Machlup theory for anisothermal chemical reactions is formulated through large deviations. Under detailed balance, the concentration-level Lagrangian satisfies

$$OM(\dot z,z)=\left(\frac{\dot z-f(z)}{c}\right)^2+f'(z)+2\frac{\dot z-f(z)}{c^2}\int_{|\xi|<1}\xi\,\nu(d\xi).$$5

with quasipotential

$$OM(\dot z,z)=\left(\frac{\dot z-f(z)}{c}\right)^2+f'(z)+2\frac{\dot z-f(z)}{c^2}\int_{|\xi|<1}\xi\,\nu(d\xi).$$6

Beyond detailed balance, the flux action is decomposed within macroscopic fluctuation theory into symmetric and antisymmetric forces with generalized orthogonality (Renger, 2021).

Active and non-reciprocal systems introduce antisymmetric force sectors directly into the Onsager–Machlup equations. For odd elasticity,

$$OM(\dot z,z)=\left(\frac{\dot z-f(z)}{c}\right)^2+f'(z)+2\frac{\dot z-f(z)}{c^2}\int_{|\xi|<1}\xi\,\nu(d\xi).$$7

the Euler–Lagrange equation contains the term

$$OM(\dot z,z)=\left(\frac{\dot z-f(z)}{c}\right)^2+f'(z)+2\frac{\dot z-f(z)}{c^2}\int_{|\xi|<1}\xi\,\nu(d\xi).$$8

so the optimal forward and backward paths differ and the entropy production for a closed loop is

$$OM(\dot z,z)=\left(\frac{\dot z-f(z)}{c}\right)^2+f'(z)+2\frac{\dot z-f(z)}{c^2}\int_{|\xi|<1}\xi\,\nu(d\xi).$$9

The non-reciprocity is therefore tied to the odd elastic part of the force law (Yasuda et al., 2021).

A covariant formulation of Onsager and Onsager–Machlup principles for active systems rewrites the Rayleighian as a scalar under point transformations and defines path probabilities through

$\int_{|\xi|<1}\xi\,\nu(d\xi),$0

The same paper extends the framework to inertia by introducing the covariant acceleration

$\int_{|\xi|<1}\xi\,\nu(d\xi),$1

and requiring variation at fixed $\int_{|\xi|<1}\xi\,\nu(d\xi),$2, which yields

$\int_{|\xi|<1}\xi\,\nu(d\xi),$3

The path probability is constrained to obey the detailed fluctuation theorem and thereby matches the stochastic-thermodynamic entropy production formula (Yasuda et al., 24 Apr 2026).

A separate inertial line of work argues that the older second-order Onsager–Machlup ansatz for Newtonian stochastic dynamics is ruled out by Ostrogradsky’s theorem because a non-degenerate higher-derivative Lagrangian leads to a Hamiltonian linear in at least one canonical momentum. It replaces the second-order variational ansatz by a phase-space canonical formalism with coupled first-order equations for position and momentum, so that stochastic forcing enters as an effective force rather than as a higher-derivative modification of the Lagrangian (Jurisch, 2020). Taken together, these papers show that inertia is not treated as a trivial appendage to overdamped Onsager–Machlup theory.

6. Variational equations, computational uses, and interpretive issues

Extended Onsager–Machlup theory is used both as an analytic principle for most probable paths and as a computational criterion for approximate dynamics. In data assimilation, the weak-constraint $\int_{|\xi|<1}\xi\,\nu(d\xi),$4D-Var objective for a smooth tube center includes the divergence term,

$\int_{|\xi|<1}\xi\,\nu(d\xi),$5

whereas MCMC with the Euler scheme uses the path prior without the divergence term. The same work proposes Hutchinson’s trace estimator to approximate the divergence and its derivative in large systems (Sugiura, 2017).

A more explicitly variational application uses the Onsager–Machlup integral as a global score for candidate approximate trajectories: $\int_{|\xi|<1}\xi\,\nu(d\xi),$6 Because $\int_{|\xi|<1}\xi\,\nu(d\xi),$7 and vanishes only on the exact kinetic path, the best approximate trajectory is defined as the trial path with the smallest Onsager–Machlup value. In the free-boundary liquid-coating example this variational procedure predicts the steady film thickness scaling

$\int_{|\xi|<1}\xi\,\nu(d\xi),$8

without integrating the full time-dependent fourth-order PDE (Doi et al., 2019).

Another reformulation, the statistical Onsager–Machlup variational principle, biases the path weight by an observable $\int_{|\xi|<1}\xi\,\nu(d\xi),$9 and maximizes

$I[z]=\int_s^u OM(\dot z,z)\,dt,$0

to obtain cumulant-generating functions rather than only the most probable trajectory. In this form the Onsager–Machlup framework yields the Stokes–Einstein diffusion constant

$I[z]=\int_s^u OM(\dot z,z)\,dt,$1

for a Brownian particle in a viscous fluid and the full Gaussian cumulant structure for a Brownian particle in steady shear flow (Yasuda et al., 2024).

At the geometric extreme, the same Onsager–Machlup philosophy has been transplanted to conformal probability. For multiple SLEs, the multi-chordal and multi-radial Loewner potentials govern small-neighborhood asymptotics of weighted SLE measures, so the deviation cost is encoded by Loewner potential differences plus conformal derivative corrections. In this setting the Onsager–Machlup functional is not a drift-mismatch integral but a conformally covariant potential on curve space with Brownian-loop interaction terms (Fan, 9 Aug 2025).

Across these developments, extended Onsager–Machlup theory remains identifiable by three persistent features: a pathwise exponential weighting, a variational characterization of the dominant trajectory or configuration, and a correction structure that records what has changed relative to classical diffusion—jump asymmetry, infinite-dimensional trace effects, fractional memory, hidden-variable nonlocality, law dependence, non-reciprocal activity, or nonequilibrium thermodynamic forcing.