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arxiv: 2508.04647 · v3 · submitted 2025-08-06 · ❄️ cond-mat.stat-mech · math.PR

Stochastic Calculus for Pathwise Observables of Markov-Jump Processes: Unification of Diffusion and Jump Dynamics

Lars Torbj{\o}rn Stutzer , Cai Dieball , Alja\v{z} Godec This is my paper

Pith reviewed 2026-05-18 23:57 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math.PR
keywords stochastic calculusMarkov jump processespathwise observablesthermodynamic inequalitiesLangevin equationdiffusion processesresponse functionsunification
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The pith

Markov-jump processes admit a stochastic calculus for pathwise observables in exact parallel to diffusion processes.

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

The paper develops a complete stochastic calculus for path-wise observables of Markov-jump processes that mirrors the existing framework for continuous diffusion. It formulates a Langevin equation for jumps, defines general observables, establishes their covariation structure while fully accounting for transients and time-inhomogeneous dynamics, proves thermodynamic inequalities in general form, introduces a response-function formalism, and demonstrates unification via the continuum limit. A sympathetic reader would care because path-wise observables are central to time-average statistical mechanics and enable thermodynamic inference when not all dissipative degrees of freedom are accessible. The unification treats jump and diffusion dynamics under one structural theory rather than separate patchwork approaches.

Core claim

We develop, in an exact parallelism with continuous-space diffusion, a complete stochastic calculus for path-wise observables of Markov-jump processes. We formulate a Langevin equation for jump processes, define general path-wise observables, establish their covariation structure whereby we fully account for transients and time-inhomogeneous dynamics, prove the known kinds of thermodynamic inequalities in their most general form and discuss saturation conditions, determine the response of path-wise observables to general perturbations and introduce a corresponding response-function formalism, carry out the continuum limit to achieve the complete unification of diffusion and jump dynamics, in

What carries the argument

The Langevin equation for Markov-jump processes, which generates stochastic increments and defines the covariation structure for pathwise observables in exact parallel to the diffusion case.

Load-bearing premise

A Langevin equation for jump processes can be formulated such that the covariation structure and thermodynamic inequalities follow in exact structural parallelism with the diffusion case, without additional ad-hoc rules for transients or time-inhomogeneity.

What would settle it

A concrete calculation for a time-inhomogeneous Markov-jump process in which the covariation between two specific pathwise observables deviates from the structure predicted by the Langevin equation, or in which a thermodynamic inequality fails to hold under conditions the framework claims to cover.

Figures

Figures reproduced from arXiv: 2508.04647 by Alja\v{z} Godec, Cai Dieball, Lars Torbj{\o}rn Stutzer.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison of key quantities in continuous and discrete space, namely the equations of motion, noise statistics, funda [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic comparison of discrete and continuous tra [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Example trajectory and examples of an arising cur [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Two densities, [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Statistics of transitions and waiting times in the sta [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Overview of models used to visualize the results: (a) Markov model for secondary active transport (SAT); the states [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Saturation of CTUR in the transient calmodulin fold [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Quality factor as a function of density weighting [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of TUR and transport bound with the periodic extension of the SAT model in Fig. [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Correlation bound applied to the driven ring, see Fig. [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Perturbations of the equilibrium and driven rings. [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Visualization of the intermediate point considered [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Validity and comparison of generalized trans [PITH_FULL_IMAGE:figures/full_fig_p032_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Shown are quality factor of the CTUR in (a-c) using values of [PITH_FULL_IMAGE:figures/full_fig_p036_14.png] view at source ↗
read the original abstract

Path-wise observables--functionals of stochastic trajectories--are at the heart of time-average statistical mechanics and are central to thermodynamic inequalities such as uncertainty relations, speed limits, and correlation-bounds. They provide a means of thermodynamic inference in the typical situation, when not all dissipative degrees of freedom in a system are experimentally accessible. So far, theories focusing on path-wise observables have been developing in two major directions, diffusion processes and Markov-jump dynamics, in a virtually disjoint manner. Moreover, even the respective results for diffusion and jump dynamics were derived with a patchwork of different approaches that are predominantly indirect. Stochastic calculus was recently shown to provide a direct approach to path-wise observables of diffusion processes, while a corresponding framework for jump dynamics remained elusive. In our work we develop, in an exact parallelism with continuous-space diffusion, a complete stochastic calculus for path-wise observables of Markov-jump processes. We formulate a "Langevin equation" for jump processes, define general path-wise observables, and establish their covariation structure, whereby we fully account for transients and time-inhomogeneous dynamics. We prove the known kinds of thermodynamic inequalities in their most general form and discus saturation conditions. We determine the response of path-wise observables to general (incl. thermal) perturbations and introduce a corresponding response-function formalism. We carry out the continuum limit to achieve the complete unification of diffusion and jump dynamics. In addition, we connect the framework to quantum unraveling and the Belavkin equation for open quantum systems, associating quantum and classical descriptions of thermal systems.

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.

Referee Report

1 major / 2 minor

Summary. The paper develops a stochastic calculus for path-wise observables of Markov-jump processes in exact structural parallelism with the diffusion case. It formulates a Langevin equation for jumps, defines general path-wise observables, establishes their covariation structure while fully accounting for transients and time-inhomogeneous dynamics, proves thermodynamic inequalities in general form with saturation conditions, introduces a response-function formalism for general (including thermal) perturbations, performs the continuum limit to unify diffusion and jump dynamics, and connects the framework to quantum unraveling and the Belavkin equation for open quantum systems.

Significance. If the derivations are rigorous and the claimed exact parallelism holds without hidden ad-hoc rules, the work would provide a valuable unification of two previously disjoint lines of research on path-wise observables in stochastic thermodynamics. This could streamline proofs of uncertainty relations, speed limits, and correlation bounds for jump processes common in chemical and biological systems, while the explicit treatment of time-inhomogeneity and the quantum-classical link add breadth. The direct stochastic-calculus approach, as opposed to patchwork indirect methods, is a methodological strength if substantiated by explicit equations.

major comments (1)
  1. [Covariation structure (following Langevin formulation for jumps)] The central claim of exact covariation parallelism with diffusion requires that, for a time-dependent jump rate λ(t), the quadratic covariation of an integrated path-wise observable acquires no extra integral term involving dλ/dt that lacks a counterpart in the diffusion Itô table. Please provide the explicit derivation (likely in the section establishing the covariation structure following the Langevin equation) and show how any such term is either absent or absorbed without redefining the noise or introducing case-specific compensators; otherwise the structural identity and the subsequent continuum limit unification are at risk.
minor comments (2)
  1. [Abstract] The abstract contains the typo 'discus saturation conditions' (should be 'discuss').
  2. [Introduction or Langevin equation section] The quotation marks around 'Langevin equation' for jump processes are appropriate but would benefit from an early explicit statement of how this equation differs in interpretation and noise properties from the standard diffusion Langevin equation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment regarding the covariation structure. We address this point in detail below and will incorporate clarifications into the revised version.

read point-by-point responses

  1. Referee: [Covariation structure (following Langevin formulation for jumps)] The central claim of exact covariation parallelism with diffusion requires that, for a time-dependent jump rate λ(t), the quadratic covariation of an integrated path-wise observable acquires no extra integral term involving dλ/dt that lacks a counterpart in the diffusion Itô table. Please provide the explicit derivation (likely in the section establishing the covariation structure following the Langevin equation) and show how any such term is either absent or absorbed without redefining the noise or introducing case-specific compensators; otherwise the structural identity and the subsequent continuum limit unification are at risk.

    Authors: We agree that an explicit derivation is needed for full clarity. In the section establishing the covariation structure (immediately following the Langevin equation for jumps), the quadratic covariation is obtained directly from the definition for semimartingales: for a path-wise observable O_t = ∫_0^t f(s) dN_s where N is the counting process with intensity λ(t), the quadratic variation process [O,O]_t equals the sum of squared jumps ∑_{s≤t} (ΔO_s)^2. Because jumps are of finite size and occur at random times, this sum contains no continuous contribution and therefore no term proportional to dλ/dt. The intensity λ(t) enters only the compensator (predictable quadratic variation) ⟨O,O⟩_t = ∫_0^t λ(s) f(s)^2 ds, which is the expectation of [O,O]_t but does not alter the pathwise quadratic covariation itself. Consequently, no extra integral involving dλ/dt appears in the covariation table, and no redefinition of the noise or case-specific compensators is required. This construction is identical in structure to the diffusion case, where the quadratic variation likewise arises solely from the continuous martingale part without derivative terms on the drift. We will add a step-by-step expansion of this derivation, including the explicit Itô product rule for two such observables, to the revised manuscript. The continuum limit to diffusion then proceeds unchanged, as the jump size vanishes while the rate diverges such that the compensator converges to the diffusion quadratic variation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation develops independent jump-process framework

full rationale

The paper formulates a Langevin equation for Markov-jump processes, defines path-wise observables, derives their covariation structure while explicitly handling transients and time-inhomogeneous rates, proves thermodynamic inequalities, introduces a response-function formalism, and performs the continuum limit to recover diffusion results. These steps are presented as direct constructions and proofs within the manuscript rather than reductions to prior fitted inputs or self-citations. The unification claim rests on showing that the newly defined jump calculus limits to the known diffusion case, which is a consistency check rather than a definitional equivalence. No load-bearing premise is justified solely by overlapping-author citations or by renaming an input as a prediction. The framework is therefore self-contained against external benchmarks for the jump case.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Framework rests on standard stochastic-process axioms (Markov property, existence of jump rates, well-defined quadratic variation) plus the modeling choice that a Langevin equation for jumps exists in exact structural analogy to diffusion. No free parameters or new invented entities are indicated in the abstract.

axioms (1)
  • domain assumption Markov-jump processes admit a well-defined stochastic calculus with covariation rules that parallel those of Itô calculus for diffusions
    Invoked when defining the Langevin equation for jumps and establishing covariation structure.

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Mutual Linearity in and out of Stationarity for Markov Jump Processes: A Trajectory-Based Approach

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    A trajectory-level derivation shows mutual linearity holds for non-stationary Markov jump processes and generalizes to other systems.

Reference graph

Works this paper leans on

179 extracted references · 179 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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    A 6), so that the large t quality factor be- comes Q t→∞ − − − →1 log ϵ (ϵ + 1)2 + (ϵ − 1)2 (ϵ2 − 1) − ϵ + 1 ϵ − 1 1 pvarFps(V A) = (ϵ − 1) (ϵ + 1) logϵ 1 pvarFps(V A)

    (see App. A 6), so that the large t quality factor be- comes Q t→∞ − − − →1 log ϵ (ϵ + 1)2 + (ϵ − 1)2 (ϵ2 − 1) − ϵ + 1 ϵ − 1 1 pvarFps(V A) = (ϵ − 1) (ϵ + 1) logϵ 1 pvarFps(V A) . (130) Recall that the choice of F (τ) is free for stationary sys- tems, hence we can choose F (τ) = ⟨V A x + V A y ⟩ps/2 s.t. pvarF ps(V A) = var ps(V A). Using Popoviciu’s ineq...

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    (see App. A 8 for details). However, Eq. (139) al- lows for the same result to be obtained using correlations of stochastic differentials without invoking perturbative calculations. B. Equilibrium Response to Temperature Perturbations For MJP in equilibrium we havepeq i ∝ e−Ei/T in terms of free energies Ei. In addition, we have detailed balance, which, h...

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    Proof of Noise-Time Correlation Lemma To prove how the noise Eq. (37) and time spent in a state in Eq. (36) correlate, i.e., ⟨dεxy(τ)dτi(τ ′)⟩, we consider three cases: the transition from x to y happens (i) before τ ′, (ii) at τ ′, or (iii) after τ ′, corresponding to τ < τ ′, τ = τ ′, and τ > τ ′, respectively. Specifically, we can write ⟨dεxy(τ)dτi(τ ′...

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    Derivation of Increment Correlations a. Jump-Time Correlations We see for an explicit expression for ⟨dnxy(τ)dτi(τ ′)⟩. We can decompose this expectation using Eq. (38) as ⟨dnxy(τ)dτi(τ ′)⟩ = ⟨dεxy(τ)dτi(τ ′)⟩ + rxy⟨dτx(τ)τi(τ ′)⟩. (A7) Using the last two equations in Eqs. (41), we get ⟨dnxy(τ)dτi(τ ′)⟩ rxydτdτ ′ =1τ <τ′P (i, τ ′|y, τ)px(τ) + 1τ ≥τ ′P (x,...

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    Derivation of Eq. (76) Starting from Eq. (75), we can simplify the sum over x, y by using the symmetry of Zxy(τ)[P (i, τ ′|y, τ) − P (i, τ ′|x, τ)] to get ⟨AtJII t ⟩ =1 2 Z t 0 dτ ′ Z t 0 dτ X i,j κij(τ ′)rij1τ <τ′ × X x,y [P (i, τ ′|y, τ) − P (i, τ ′|x, τ)] [px(τ)rxy − py(τ)ryx] = − Z t 0 dτ ′ Z t 0 dτ X i,j κij(τ ′)rij1τ <τ′ (A19) × X x P (i, τ ′|x, τ) ...

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    Indispensability of the Modified Current Consider a three-state system with generator L =   −4 2 1 1 −3 1 3 1 −2   , (A22) with eigenvalues 0, −4, −5. Let the current defining tran- sition weights be κij(τ) = e4τ (δi1δj2 − δi2δj1). Figure 13 shows various quality factors of the transient system with initial distribution pi(0) = (δi1 + 2δi2)/3. Expli...

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    Long-time limit of driven ring Integrated Covariance Here we present details on the large- t limit of the sta- tionary correlation bound in Eq. (123) applied to the driven ring Sec. IV A with the assumption that ϵ > 1. We start by realizing that ∆ V ≡ V A − ⟨ V A⟩1 = 1 2(1, 1, −1, −1)T . Hence, we can write Z t 0 dτcovs[V A τ , V A 0 ] = 1 4 Z t 0 dτ∆V T ...

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    Physically Motivated Rates - Pseudo Potential Form We specifically assume the transition rates to be of the form [164] rxy = Dxye−Vxy/T = ˜DxyT e−Vxy/T , (A37) where Dxy is an edge specific “diffusion coefficient” and Vxy is a pseudo-potential on the edge. This pseudo- potential allows for general nonequilibrium systems to be implemented and therefore all...

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