A study of path measures based on second-order Hamilton--Jacobi equations and their applications in stochastic thermodynamics

Jean-Claude Zambrini; Jianyu Hu; Qiao Huang; Yuanfei Huang

arxiv: 2508.02469 · v4 · submitted 2025-08-04 · 🧮 math-ph · math.MP

A study of path measures based on second-order Hamilton--Jacobi equations and their applications in stochastic thermodynamics

Jianyu Hu , Qiao Huang , Yuanfei Huang , Jean-Claude Zambrini This is my paper

Pith reviewed 2026-05-19 00:57 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords path measuressecond-order Hamilton-Jacobi equationslarge deviation principlesentropy productionstochastic thermodynamicsOnsager-Machlup functionalstochastic geometric mechanicsSchrödinger problem

0 comments

The pith

Thermodynamic irreversibility appears as the difference between forward and backward second-order Hamilton-Jacobi equations, with large deviation rate functions equivalent to Onsager-Machlup functionals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines path measures for stochastic differential equations by means of second-order Hamilton-Jacobi equations. These equations supply a single structure that yields large-deviation rate functions, solves entropy-minimization problems including Schrödinger's problem, and decomposes entropy production. The central result states that the rate function coincides with the Onsager-Machlup functional of stochastic gradient systems once the second-order Hamilton-Jacobi structure is imposed. Thermodynamic irreversibility is then read off directly as the mismatch between the forward and backward versions of the same equation. The construction therefore links measure-theoretic properties of paths to concrete thermodynamic quantities without intermediate approximations.

Core claim

Second-order Hamilton-Jacobi equations derived from the probabilistic structure of path measures give the large deviation rate function, which is proved identical to the Onsager-Machlup functional for stochastic gradient systems. Entropy production then decomposes so that its irreversible part equals the difference between the forward and backward second-order Hamilton-Jacobi equations. The same equations also recover the solutions of finite-horizon entropy minimization and Schrödinger's problem, thereby embedding stochastic thermodynamics inside stochastic geometric mechanics.

What carries the argument

Second-order Hamilton-Jacobi equations that convert the probabilistic law of path measures into both a large-deviation rate function and an entropy-production decomposition.

If this is right

Large-deviation principles for path measures follow directly once the second-order Hamilton-Jacobi structure is known.
Entropy-minimization problems on finite time horizons and Schrödinger's problem are recovered as instances of stochastic geometric mechanics.
Entropy production splits into a reversible part and an irreversible part given exactly by the difference of forward and backward second-order Hamilton-Jacobi equations.
The same framework yields a unified treatment of large deviations, entropy minimization, and thermodynamic irreversibility.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The forward-backward difference may supply a practical route to compute entropy production by solving the pair of Hamilton-Jacobi equations numerically.
The same second-order structure could be tested on non-gradient drifts to see whether the equivalence with Onsager-Machlup functionals survives.
Connections between the Hamilton-Jacobi formulation and optimal-control or quantum-dissipation problems become visible once the second-order equations are written explicitly.

Load-bearing premise

Second-order Hamilton-Jacobi equations supply a complete and rigorous link between the law of path measures and the entropy-production decomposition without extra regularity or approximation conditions on the underlying processes.

What would settle it

A concrete stochastic gradient system in which the large-deviation rate function computed from the path-measure law differs from the Onsager-Machlup functional obtained from the associated second-order Hamilton-Jacobi equation.

read the original abstract

This paper provides a systematic investigation of the mathematical structure of path measures and their profound connections to stochastic differential equations (SDEs) through the framework of second-order Hamilton--Jacobi (HJ) equations. This approach establishes a unified methodology for analyzing large deviation principles (LDPs), entropy minimization, and entropy production in stochastic systems. Second-order HJ equations are shown to play a central role in bridging stochastic dynamics and measure theory while forming the foundation of stochastic geometric mechanics and their applications in stochastic thermodynamics. The large deviation rate function is rigorously derived from the probabilistic structure of path measures and proved to be equivalent to the Onsager--Machlup functional of stochastic gradient systems coupled with second-order HJ equations. We revisit entropy minimization problems, including finite time horizon problems and Schr\"{o}dinger's problem, demonstrating the connections with stochastic geometric mechanics. Furthermore, we present a novel decomposition of entropy production for stochastic systems, revealing that thermodynamic irreversibility can be interpreted as the difference of the corresponding forward and backward second-order HJ equations. Together, this work establishes a comprehensive mathematical study of the relations between path measures and stochastic dynamical systems, and their diverse applications in stochastic thermodynamics and beyond.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a PDE decomposition of entropy production via forward and backward second-order HJ equations plus an equivalence to Onsager-Machlup functionals, but the rigor of the SDE-to-HJ step needs close checking.

read the letter

The main thing to know is that this work claims to derive the large deviation rate function directly from path measures and show it equals the Onsager-Machlup functional for stochastic gradient systems, while also writing entropy production as the difference of the corresponding forward and backward second-order Hamilton-Jacobi equations. That decomposition is presented as a fresh way to read thermodynamic irreversibility. The paper also revisits entropy minimization on finite horizons and Schrödinger's problem, tying them into stochastic geometric mechanics. If the equivalences are solid, the framework could let people move between probabilistic path measures and PDE calculations more cleanly than before. It does a reasonable job laying out the connections in one place and showing how the HJ equations sit at the center of the story. The abstract is clear about starting from the probabilistic structure rather than assuming the thermodynamic quantities upfront. That avoids obvious circularity. The synthesis feels useful for organizing results that are already scattered across large-deviation papers and stochastic thermodynamics. The soft spot is the passage from the underlying SDE to the second-order HJ equation. The equivalence and the irreversibility interpretation rest on that step. If the derivation uses Ito's formula or a variational principle without stating or verifying the needed regularity (C^2 coefficients, uniform ellipticity, linear growth), then the claims about general path measures could be formal rather than fully rigorous. Boundary conditions for the HJ equations also look like they could use more explicit treatment. Without the full proofs it is hard to tell how much is assumed versus proved. This paper is for people already working in stochastic thermodynamics who know both large deviations and Hamilton-Jacobi methods. A reader who wants a PDE route to entropy production calculations would find the synthesis worth looking at. It is coherent on its own terms and engages the literature enough to deserve a serious referee. I would send it to peer review with instructions to focus on the regularity conditions and the measure-theoretic details in the derivations.

Referee Report

1 major / 2 minor

Summary. The paper provides a systematic investigation of the mathematical structure of path measures connected to SDEs via second-order Hamilton-Jacobi equations. It claims to rigorously derive large deviation rate functions from the probabilistic structure of path measures and prove their equivalence to the Onsager-Machlup functional for stochastic gradient systems, revisit entropy minimization problems including finite-horizon and Schrödinger problems, and present a novel decomposition of entropy production as the difference between corresponding forward and backward second-order HJ equations, thereby interpreting thermodynamic irreversibility.

Significance. If the central derivations hold with the necessary rigor, the work establishes a unified framework bridging path measures, second-order HJ equations, and stochastic thermodynamics. The claimed equivalence of the LDP rate function to the Onsager-Machlup functional and the HJ-based decomposition of entropy production could offer new analytical tools for irreversibility in stochastic systems and strengthen connections to stochastic geometric mechanics.

major comments (1)

[Derivation of LDP rate function and equivalence to Onsager-Machlup functional (main theorems section)] The abstract and main claims assert a rigorous derivation of the large deviation rate function from path measures and its equivalence to the Onsager-Machlup functional of stochastic gradient systems coupled with second-order HJ equations. This equivalence is load-bearing for both the LDP and the entropy production decomposition. However, the passage to the second-order HJ equations (likely via Itô's formula or variational principles) requires unstated regularity conditions on the SDE coefficients, such as C² smoothness, uniform ellipticity, and linear growth bounds. These are not explicitly stated or verified for the general path measures considered, creating a potential gap that would render the thermodynamic interpretations non-rigorous.

minor comments (2)

[Abstract] The abstract uses the term 'rigorously derived' without a brief parenthetical on the key regularity assumptions; adding this would improve clarity for readers.
[Introduction and notation sections] Notation for forward and backward second-order HJ equations should be introduced with explicit definitions and any domain/boundary conditions in the early sections to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment below and will incorporate the suggested clarifications to strengthen the rigor of the derivations.

read point-by-point responses

Referee: [Derivation of LDP rate function and equivalence to Onsager-Machlup functional (main theorems section)] The abstract and main claims assert a rigorous derivation of the large deviation rate function from path measures and its equivalence to the Onsager-Machlup functional of stochastic gradient systems coupled with second-order HJ equations. This equivalence is load-bearing for both the LDP and the entropy production decomposition. However, the passage to the second-order HJ equations (likely via Itô's formula or variational principles) requires unstated regularity conditions on the SDE coefficients, such as C² smoothness, uniform ellipticity, and linear growth bounds. These are not explicitly stated or verified for the general path measures considered, creating a potential gap that would render the thermodynamic interpretations non-rigorous.

Authors: We appreciate the referee for identifying this important point to ensure full mathematical rigor. The derivations indeed rely on Itô's formula applied to solutions of the second-order HJ equations and on variational principles connecting path measures to the Onsager-Machlup functional. These steps require the SDE coefficients to be C² smooth, the diffusion matrix to be uniformly elliptic, and linear growth bounds to hold in order to guarantee well-posedness of the SDEs, validity of the large deviation principle, and applicability of the thermodynamic interpretations. While these conditions are standard for the class of stochastic gradient systems considered in the paper and were implicitly used throughout the analysis, we agree they were not explicitly stated or verified in the main theorems section. In the revised manuscript we will add a dedicated paragraph (or short subsection) immediately preceding the main results that explicitly lists these regularity assumptions and confirms their satisfaction for the path measures under study. This clarification will make the scope of the equivalence and the entropy-production decomposition fully rigorous without changing any of the core statements or proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations start from path measures and SDEs

full rationale

The paper derives the large deviation rate function directly from the probabilistic structure of path measures and establishes its equivalence to the Onsager-Machlup functional through coupling with second-order HJ equations. The entropy production decomposition is introduced as an interpretation of the difference between forward and backward HJ equations. These steps are framed as beginning from the underlying stochastic dynamics and measure theory rather than presupposing thermodynamic outputs or reducing via self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. No equations or claims in the abstract or described chain exhibit the enumerated circular patterns, and the methodology is presented as self-contained against external probabilistic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the central claims rest on standard background results in stochastic analysis and large deviation theory rather than newly introduced free parameters or invented entities.

axioms (1)

domain assumption Second-order Hamilton-Jacobi equations can be coupled to the probabilistic structure of path measures to derive large deviation rate functions.
Invoked in the derivation of the equivalence to the Onsager-Machlup functional.

pith-pipeline@v0.9.0 · 5750 in / 1293 out tokens · 30158 ms · 2026-05-19T00:57:24.501553+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArrowOfTime.lean arrow_from_z echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

thermodynamic irreversibility can be interpreted as the difference of the corresponding forward and backward second-order HJ equations... large deviation rate function proved equivalent to the Onsager-Machlup functional
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

second-order Hamilton–Jacobi equation (2.13) ... Φ(ω)=∫V dt + g(ω(T))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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B. D. Anderson. Reverse-time diffusion equation models.Stochastic Processes and their Applications, 12(3):313–326, 1982

work page 1982

[2] [2]

P. Ao. Emerging of stochastic dynamical equalities and steady state thermodynamics from darwinian dynamics.Communications in Theoretical Physics, 49(5):1073, 2008

work page 2008

[3] [3]

V. I. Arnol’d.Mathematical Methods of Classical Mechanics, volume 60. Springer Science & Business Media, 2013

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work page 2022

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Huang and J.-C

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work page 2023

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Huang and J.-C

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work page 2023

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arXiv preprint arXiv:2504.06628 (2025)

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work page arXiv 2025

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Stochastic derivatives and generalized h-transforms of Markov processes

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L´ eonard

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work page 1920

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L´ eonard

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L´ eonard

C. L´ eonard. A survey of the Schr¨ odinger problem and some of its connections with optimal transport. Discrete & Continuous Dynamical Systems, 34(4):1533–1574, 2014

work page 2014

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L´ eonard

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