Full $\Gamma-$expansion for the level-two large deviation rate functionals of non-reversible one-dimensional diffusions with periodic boundary conditions

Christian Maura; Claudio Landim

Full $\Gamma-$expansion for the level-two large deviation rate functionals of non-reversible one-dimensional diffusions with periodic boundary conditions

T0 review · 0 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read The level two large deviation rate functional of non-reversible diffusions on the one-dimensional torus admits a full Gamma-expansion as the temperature vanishes.

desk verdict Derives explicit multi-scale Γ-expansion for level-2 LDP rate functional of non-reversible 1D diffusion on the torus, extending reversible cases. read the letter →

arxiv 2606.17859 v1 pith:YD5GKNHC submitted 2026-06-16 math.PR

Claudio Landim , Christian Maura This is my paper · ORCID

classification math.PR

keywords largedeviationsGamma-convergencediffusionsmetastabilitynon-reversibleperiodicboundaryconditionsratefunctionalsone-dimensional

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper derives a full Gamma-expansion for the level-two large deviation rate functional I_ε of a one-dimensional diffusion process on the torus as ε approaches zero. The expansion writes I_ε as a sum of an ε-independent functional scaled by 1/ε, plus another ε-independent term, plus terms scaled by the inverses of metastable time scales. This structure reveals how the rate of rare events changes across different time scales in non-reversible systems. A reader would care because it gives a detailed asymptotic picture of metastability and large deviations without relying on reversibility assumptions.

What carries the argument

The full Γ-expansion of the level-two large deviation rate functional I_ε into ε-independent components J weighted by 1/ε and the metastable time scales 1/θ_ε.

What would settle it

Computing the rate functional I_ε numerically for small ε and verifying whether it matches the proposed combination of the J functionals at the appropriate scales would test the claim; mismatch at any scale would falsify it.

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Extended reading notes

Core claim

Consider the diffusion dX_ε(t) = b(X_ε(t)) dt + sqrt(2ε a(X_ε(t))) dW_t on the torus. The level-two rate functional I_ε of this process satisfies I_ε = (1/ε) J^{-1} + J^{(0)} + sum (1/θ_ε^{(p)}) J^{(p)} as ε→0, where the J's are rate functionals independent of ε and the θ's are the metastable time scales of the process.

Load-bearing premise

The diffusion coefficients a and b are of class C² on the torus and satisfy conditions making the process well-defined, while the level-two rate functional I_ε is assumed to exist and possess the given Gamma-expansion form.

Editorial extensions

If this is right

The leading term (1/ε) J^{-1} dominates the large deviation behavior at the fastest scale.
Additional terms capture the contributions from metastable transitions at slower time scales.
The expansion is valid under periodic boundary conditions for non-reversible drifts with C² coefficients.
Explicit forms of the J functionals can be computed from the diffusion coefficients a and b.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The explicit expansion may allow for more accurate approximations in Monte Carlo simulations of rare events.
It highlights the role of non-reversibility in determining the metastable time scales θ_ε.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The central result is a derivation of the full Γ-expansion I_ε = (1/ε) J^{-1} + J^{(0)} + sum (1/θ_ε^p) J^{(p)} for the level-two rate functional of the empirical occupation measure of this non-reversible diffusion on the torus.

This is new because it moves beyond the reversible setting where the drift is a gradient. The authors handle a general C² drift b, which can break reversibility, and still get the multi-scale structure with the metastable times θ_ε^p. The work does well by making the expansion explicit in terms of ε-independent functionals J and tying the corrections directly to the time scales where metastability occurs. The setup with periodic boundary conditions on [0,1) is standard and keeps the state space compact.

On the soft side, the paper assumes the existence of I_ε and then derives the expansion. That is reasonable but means the result is as strong as the underlying LDP. The actual expressions for the J functionals are not shown in the abstract, so it is difficult to assess how much new computation is involved versus applying known Gamma-convergence techniques. The regularity C² on a and b is given, but one wonders if weaker conditions would suffice.

The citation pattern looks standard for this area. This is aimed at researchers in stochastic processes who care about precise asymptotics for large deviations in non-equilibrium systems. Someone studying metastability or hydrodynamic limits might use this as a reference for the rate functional structure.

It deserves a serious referee because the claim is specific and the setting is clean. The math seems internally consistent from the description. I would recommend sending it to peer review.

Referee Report

0 major / 2 minor

Summary. The paper considers the non-reversible diffusion dX_ε = b(X_ε)dt + √(2ε a(X_ε)) dW on the one-dimensional torus T=[0,1) with C² coefficients a,b satisfying auxiliary conditions ensuring well-posedness. It derives the full Γ-expansion I_ε = (1/ε) J^{-1} + J^0 + ∑_{p=1}^{q̂} (1/θ_ε^p) J^p as ε→0, where the J functionals are ε-independent rate functionals on P(T) and the θ_ε^p are the metastable time scales of the process.

Significance. If the derivation is complete, the explicit multi-scale Γ-expansion supplies a precise asymptotic description of the level-two rate functional that separates the quasipotential, the order-1 correction, and sub-exponential metastable corrections; this is a concrete advance for the analysis of large deviations and metastability for non-reversible one-dimensional diffusions on the circle.

minor comments (2)

The abstract states the expansion but supplies neither the explicit forms of the J functionals nor the definitions of the time scales θ_ε^p; the full manuscript should include these expressions in the main theorem statement.
Notation: the symbol b T for the torus and b P(T) for the space of probabilities should be introduced once and used consistently; the current abstract mixes b T and b P(T) with b T appearing only in the SDE.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of our manuscript and for recognizing the potential significance of the full multi-scale Γ-expansion for the level-two rate functional of non-reversible one-dimensional diffusions on the torus. We note that the referee's recommendation is listed as 'uncertain' but no specific major comments or points of concern were provided in the report. Accordingly, we have no point-by-point responses to address at this stage.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained from SDE

full rationale

The paper states the diffusion SDE on the torus with C² coefficients a,b, assumes existence of the level-two rate functional I_ε, and claims to derive the explicit multi-scale Γ-expansion I_ε = (1/ε)J^(-1) + J^(0) + sum (1/θ_ε^(p)) J^(p) with ε-independent J functionals. No self-citation, fitted parameter, or ansatz is invoked in the provided abstract or claim structure; the expansion is presented as obtained from the process definition under the stated regularity. No load-bearing step reduces by construction to its own inputs, and the central claim remains independent of any prior result by the same authors.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Based solely on abstract; no free parameters, invented entities, or non-standard axioms are explicitly introduced. Relies on standard existence and regularity assumptions for SDEs on the torus.

assumptions (2)

standard math Existence and uniqueness of solutions to the SDE dX_ε = b(X_ε)dt + sqrt(2ε a(X_ε)) dW on the torus with periodic BC.
Invoked implicitly to define the diffusion process X_ε.
domain assumption The level-two rate functional I_ε exists and is lower semicontinuous with respect to weak topology on P(T).
Required for the Gamma-expansion to make sense.

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Cite this review

Pith. "Pith review of Full $\Gamma-$expansion for the level-two large deviation rate functionals of non-reversible one-dimensional diffusions with periodic boundary conditions." pith.science (2026). https://pith.science/paper/YD5GKNHC

@misc{pith2026260617859,
  author       = {Pith},
  title        = {Pith review of: Full $\Gamma-$expansion for the level-two large deviation rate functionals of non-reversible one-dimensional diffusions with periodic boundary conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YD5GKNHC}},
  note         = {Machine review of arXiv:2606.17859}
}

abstract

Consider the diffusion process \begin{equation*} dX_{\epsilon}(t) = \mss b(X_{\epsilon}(t)) \, dt + \sqrt{2\, \epsilon\, \mss a(X_\epsilon(t))} \, dW_{t}, \end{equation*} on the one-dimensional torus $\bb T = [0,1)$. Here $\epsilon$ is the temperature, $W_{t}$ a Brownian motion on $\bb T$ and $\mss a$, $\mss b$ functions of class $C^{2}(\bb T)$ satisfying further conditions. Denote by $\mss P(\bb T)$ the set of probability measures on $\bb T$ equipped with the weak topology, and by $\ms I_{\epsilon}\colon \mss P(\bb T)\to [0,+\infty)$ the level two large deviation rate functional of the diffusion $X_{\epsilon}(\cdot)$. We derive a full $\Gamma-$expansion of $\ms I_{\epsilon}$, as $\epsilon \to 0$, expressing it as \begin{equation*} \ms I_{\epsilon} = \frac{1}{\epsilon} \;\ms J^{(-1)} \; +\; \ms J^{(0)} \;+\; \sum_{p=1}^{\widehat{\mf q}}\frac{1}{\theta^{(p)}_{\epsilon}}\;\ms J^{(p)}\,, \end{equation*} where $\ms J^{(-1)}$, $\ms J^{(0)}$, $\ms J^{(p)} \colon \mss P(\bb T)\to [0,+\infty]$ represent rate functionals, independent of $\epsilon$, and $\theta^{(p)}_{\epsilon}$ are the time-scales at which the Markov process $X_{\epsilon}(\cdot)$ exhibits a metastable behaviour.

Figures

Figures reproduced from arXiv: 2606.17859 by the authors.

**Figure 1.** Here we illustrate the intervals constructed in the proof of Proposition 4.1. The intervals in orange satisfy µϵ(I − σ,ϵ∪I + σ,ϵ) = O(ϵ 2 ). The union of these sets contain the support of the derivative of the cut-off function Φσ,ϵ. More precisely, Φσ,ϵ is strictly increasing on I − σ,ϵ, strictly decreasing on I + σ,ϵ, it is equal to 1 on Iσ,ϵ, and it vanishes on (I − σ,ϵ ∪ Iσ,ϵ ∪ I + σ,ϵ) c . As the previous interv… view at source ↗

**Figure 2.** We illustrate an example of the intervals defined in (7.12), (7.13), (7.15) and (7.17) for the first metastable level p = 1. Here, D is a X1(·)−equivalent class composed of only one state, D = M1(k). Recall the definition of the sets W (p) k−1,k, k ∈ Sp, introduced in (2.16). Let WD be the set of absolute maxima separating the metastable states Mp(k), k ∈ JlD − 1, rD + 1K: WD := rD[ +1 k = lD W (p) k−1,k . (7.3) We … view at source ↗

**Figure 3.** We illustrate the definition of the test function for a higher-order metastable level p ≥ 2. In this picture, D is a closed irreducible class of Xp(·), which is composed by the states Mp(k) and Mp(k + 1). The set of maxima WD, associated to D, is equal to {σ − D, σ1 k,k+1, σ2 k,k+1, σ3 k,k+1, σ+ D}. Here, the maxima σ − D is the leftmost separating Mp(k−1) and Mp(k), while σ + D is the rightmost maxima separating Mp… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: This picture illustrates an example of a non-reversible energy landscape as well as the periodic correction Sb for the case where B > 0 and the reduced chain Xb bq(·) is reversible. In this example, the state space of the reduced chain Xb bq is composed by the three …

**Figure 5.** Figure 5: Here we illustrate an example of an energy landscape of a nonreversible diffusion whose reduced Markov chain Xb bq(·), at the last level of the recursive construction, is irreducible and non-reversible. The depth of the landscape, denoted by hbq, is represented in pu…

**Figure 6.** Figure 6: Here we represent an example of a metastable structure with 6 levels on R, but 5 projected levels on T. At level 1, the states M1(k), 0 ≤ k ≤ 6, are given by M1(k) = {mk}. These comprise all of the states contained in the interval [0, 1). The sets M1(2) and M1(3) belon…

**Figure 7.** Figure 7: Fix p > 1 and k ∈ Z. The state Mp(k) is represented by the cyan circles, while the set Mp(k) is composed by the green circles. As stated in Proposition A.3, the local minima in Mp(k) have higher energy than the ones contained in Mp(k), which are all of the same height…

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Works this paper leans on

34 extracted references · 4 canonical work pages

[1]

Beltr´ an, C

J. Beltr´ an, C. Landim: Tunneling and metastability of continuous time Markov chains. J. Stat. Phys.140, 1065–1114 (2010)

2010
[2]

J. J. Beltr´ an, C. Landim; Tunneling and metastability of continuous time Markov chains II. J. Stat. Phys. 149, 598–618 (2012)

2012
[3]

Beltr´ an, C

J. Beltr´ an, C. Landim: Metastability of reversible condensed zero range processes on a finite set. Probab. Theory Related Fields152, 781–807 (2012)

2012
[4]

Berglund: Reducing metastable continuous-space Markov chains to Markov chains on a finite set

N. Berglund: Reducing metastable continuous-space Markov chains to Markov chains on a finite set. Ann. l’Inst. H. Poincar´ e Probab.Stat.61, 2036–2073 (2025)
[5]

Bertini, A

L. Bertini, A. Faggionato, D. Gabrielli, From level 2.5 to level 2 large deviations for continuous time Markov chains, Markov Process. Related Fields20, 545–562 (2014)

2014
[6]

Bertini, D

L. Bertini, D. Gabrielli, C. Landim: Metastable expansion of finite state Markov chains level two large devia- tions rate functions. Ann. Appl. Probab.34, 3820–3869 (2024)

2024
[7]

Bianchi, S

A. Bianchi, S. Dommers, C. Giardin` a: Metastability in the reversible inclusion process. Electronic Journal of Probability.22, 1–34. (2017)

2017
[8]

Bovier, F

A. Bovier, F. den Hollander: Metastabillity: A Potential-theoretic approach. Grundlehren der mathematischen Wissenschaften, 351, Springer, Cham, 2015

2015

Show all 34 references

[9]

M. D. Donsker, S. R. S. Varadhan: Asymptotic evaluation of certain Markov proces expectations for large time, I. Comm. Pure Appl. Math.28, 1–47 (1975)

1975
[10]

Faggionato and D

A. Faggionato and D. Gabrielli: A representation formula for large deviations rate functionals of invariant measures on the one dimensional torus. Annales Inst. Henri Poincar´ e – Probab. Stat.48, 212–234 (2012)

2012
[11]

M. I. Freidlin, A. D. Wentzell: Random perturbations of dynamical systems. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],260. Springer-Verlag, New York, (1998)

1998
[12]

Friedman:Stochastic differential equations and applications

A. Friedman:Stochastic differential equations and applications. Academic Press, Cambridge, 1975

1975
[13]

Di Gesu, M

G. Di Gesu, M. Mariani: Full metastable asymptotic of the Fisher information. SIAM J. Math. Anal.49, 3048–3072 (2017)

2017
[14]

Kim: Second time scale of the metastability of reversible inclusion processes

S. Kim: Second time scale of the metastability of reversible inclusion processes. Probab. Theory Relat. Fields 180, 1135–1187 (2021)

2021
[15]

S. Kim, C. Landim: Γ-expansion of the measure-current large deviations rate functional of non-reversible finite-state Markov chain Stochastic Process. Appl.189104733 (2025),

2025
[16]

S. Kim, C. Landim: Convergence of the Condensing Symmetric Inclusion Process on the Torus in the Ther- modynamical Limit to Coalescing Brownian Motions. arXiv:2605.00542 (2026)

2026 arXiv
[17]

S. Kim, I. Seo: Metastability of Ising and Potts models without external fields in large volumes at low tem- peratures. Communications in Mathematical Physics396, 383–449. (2022)

2022
[18]

S. Kim, I. Seo: Condensation and metastable behavior of non-reversible inclusion processes. Communications in Mathematical Physics,382, 1343–1401 (2021)

2021
[19]

Landim: Metastable Markov chains

C. Landim: Metastable Markov chains. Probability Surveys16, 143–227 (2019)

2019
[20]

Landim: Metastability from the large deviations point of view: A Γ-expansion of the level two large devia- tions rate functional of non-reversible finite-state Markov chains

C. Landim: Metastability from the large deviations point of view: A Γ-expansion of the level two large devia- tions rate functional of non-reversible finite-state Markov chains. Stochastic Processes and their Applications. 165, 275-315 (2023)

2023
[21]

Landim, J

C. Landim, J. Lee, M. Mariani: The Gamma Expansion of the Level Two Large Deviation Rate Functional for Reversible Diffusion Processes. arXiv:2509.13222 (2025)

2025
[22]

Landim, J

C. Landim, J. Lee, I. Seo: Metastability and time scales for parabolic equations with drift 1: the first time scale. Archive for Rational Mechanics and Analysis,248, p.78. (2024)

2024
[23]

Landim, J

C. Landim, J. Lee, I. Seo: Metastability and time scales for parabolic equations with drift 2: the general time scale. (2024). arxiv: 2402.07695

2024
[24]

Landim, M

C. Landim, M. Loulakis, M.Mourragui: Metastable Markov chains: from the convergence of the trace to the convergence of the finite-dimensional distributions. Electron. J. Probab.23, 1-34 (2018)

2018
[25]

Landim, D

C. Landim, D. Marcondes, I. Seo: A Resolvent Approach to Metastability. J. Eur. Math. Soc.27, 1563–1618 (2025)

2025
[26]

Landim, D

C. Landim, D. Marcondes, I. Seo: Metastable behavior of weakly mixing Markov chains: The case of reversible, critical zero-range processes. Ann. Probab.51, 157–227, (2023)

2023
[27]

Landim, C

C. Landim, C. Maura: From one-dimensional diffusion processes metastable behaviour to parabolic equations asymptotics. arXiv:2505.20217 (2025)

2025
[28]

Landim, R

C. Landim, R. Misturini, F. Sau: Full Γ−expansion of reversible Markov chains Level two Large deviations rate functional. Ann. Appl. Probab.34, 5578–5614 (2024)

2024
[29]

Landim, I

C. Landim, I. Seo: Metastability of nonreversible random walks in a potential field and the Eyring-Kramers transition rate formula. Comm. Pure Appl. Math.71, 203–266. (2018)

2018
[30]

Landim and I

C. Landim and I. Seo: Metastability of one-dimensional, non-reversible diffusions with periodic boundary conditions. Ann. Inst. H. Poincar´ e Probab. Statist.55, 1850–1889, (2019)

2019
[31]

J. Lee, I. Seo: Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Conver- gence. J Stat Phys189, 25 (2022) Γ−EXPANSION OF NON REVERSIBLE ONE-DIMENSIONAL DIFFUSIONS 63

2022
[32]

Mariani: A Γ-convergence approach to large deviations

M. Mariani: A Γ-convergence approach to large deviations. Ann. Sc. Norm. Super. Pisa Cl. Sci.18, 951–976 (2018)

2018
[33]

C. Oh, F. Rezakhanlou: Metastability of zero range processes via Poisson equations. Unpublished manuscript. (2018)

2018
[34]

Olivieri and M

E. Olivieri and M. E. Vares:Large deviations and metastability. Encyclopedia of Mathematics and its Appli- cations, 100, Cambridge Univ. Press, Cambridge, 2005 Claudio Landim IMPA Estrada Dona Castorina 110, J. Botanico, 22460 Rio de Janeiro, Brazil and Univ. Rouen Normandie, ...

2005

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

[1] [1]

Beltr´ an, C

J. Beltr´ an, C. Landim: Tunneling and metastability of continuous time Markov chains. J. Stat. Phys.140, 1065–1114 (2010)

2010

[2] [2]

J. J. Beltr´ an, C. Landim; Tunneling and metastability of continuous time Markov chains II. J. Stat. Phys. 149, 598–618 (2012)

2012

[3] [3]

Beltr´ an, C

J. Beltr´ an, C. Landim: Metastability of reversible condensed zero range processes on a finite set. Probab. Theory Related Fields152, 781–807 (2012)

2012

[4] [4]

Berglund: Reducing metastable continuous-space Markov chains to Markov chains on a finite set

N. Berglund: Reducing metastable continuous-space Markov chains to Markov chains on a finite set. Ann. l’Inst. H. Poincar´ e Probab.Stat.61, 2036–2073 (2025)

[5] [5]

Bertini, A

L. Bertini, A. Faggionato, D. Gabrielli, From level 2.5 to level 2 large deviations for continuous time Markov chains, Markov Process. Related Fields20, 545–562 (2014)

2014

[6] [6]

Bertini, D

L. Bertini, D. Gabrielli, C. Landim: Metastable expansion of finite state Markov chains level two large devia- tions rate functions. Ann. Appl. Probab.34, 3820–3869 (2024)

2024

[7] [7]

Bianchi, S

A. Bianchi, S. Dommers, C. Giardin` a: Metastability in the reversible inclusion process. Electronic Journal of Probability.22, 1–34. (2017)

2017

[8] [8]

Bovier, F

A. Bovier, F. den Hollander: Metastabillity: A Potential-theoretic approach. Grundlehren der mathematischen Wissenschaften, 351, Springer, Cham, 2015

2015