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Pathological Large Deviations of the KMP Process in Dimension $d\ge 2$

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Pith's one-line read Pathological trajectories appear with finite rate in the large deviations of the KMP process in dimensions d ≥ 2.

desk verdict Heydecker recasts the KMP rate function as a linear PDE with rough drift to show pathological trajectories carry finite cost in d≥2 and validates the Bertini-Gabrielli-Lebowitz lower bound. read the letter →

arxiv 2605.26652 v1 pith:4HUIOMDR submitted 2026-05-26 math.PR math.AP

Daniel Heydecker This is my paper · ORCID
classification math.PRmath.AP
keywords largedeviationsKMPprocessratefunctionhydrodynamiclimitsinteractingparticlesystemspathologicaltrajectorieshyperbolic-parabolicequation
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 examines dynamic large deviations for the Kipnis-Marchioro-Presutti process on the discrete torus. It shows that certain non-standard trajectories occur with only finite rate cost under the rate function when the dimension is two or higher. The proof proceeds by rewriting the candidate rate function as the cost associated to solutions of a linear hyperbolic-parabolic equation driven by a rough drift. The same reformulation is used to confirm that an earlier lower bound on the rate function holds in every dimension. A reader would care because the result enlarges the set of admissible paths that can arise with positive probability on long time scales.

What carries the argument

Recasting the candidate rate function in terms of a linear hyperbolic-parabolic equation with rough drift

What would settle it

Direct computation of the rate assigned by the recast equation to one explicit pathological trajectory, such as a density profile that jumps across a discontinuity, to check whether the value is finite.

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Extended reading notes

Core claim

By recasting the candidate rate function in terms of a linear hyperbolic-parabolic equation with rough drift, we show that pathological trajectories appear in the large deviations with finite rate in any dimension d≥2. Along the way, we rigorously validate the lower bound derived by Bertini-Gabrielli-Lebowitz in any dimension.

Load-bearing premise

The candidate rate function can be recast in terms of a linear hyperbolic-parabolic equation with rough drift.

Editorial extensions

If this is right

  • Pathological trajectories carry finite rate in every dimension d≥2.
  • The lower bound on the large-deviation rate function holds without dimension restriction.
  • The set of trajectories with finite rate includes solutions to the hyperbolic-parabolic equation that would be excluded by standard hydrodynamic regularity.
  • The KMP process on the torus therefore admits more low-cost fluctuation paths than expected from parabolic theory alone.

Reading between the lines

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

  • The enlarged set of finite-rate paths may alter which initial conditions are selected by the large-deviation principle in hydrodynamic scaling limits.
  • The rough-drift reformulation technique could be tested on other conservative lattice gases to see whether pathological finite-rate paths appear there as well.
  • Proving a matching upper bound on the rate function may now require controlling the roughness of the drift term in dimensions greater than one.
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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.

Referee Report

1 major / 0 minor

Summary. The paper studies dynamic large deviations for the Kipnis--Marchioro--Presutti (KMP) process on the discrete torus T_N^d. By recasting the candidate rate function in terms of a linear hyperbolic-parabolic equation with rough drift, the authors establish that pathological trajectories appear in the large deviations with finite rate for any dimension d≥2. They also rigorously validate the lower bound previously obtained by Bertini-Gabrielli-Lebowitz in any dimension.

Significance. If the recasting and the explicit construction of the pathological trajectories hold, the result would be significant for large-deviation theory in interacting particle systems: it shows that the rate function for the KMP process admits non-regular paths of finite cost in dimensions two and higher, extending the picture beyond the one-dimensional case and confirming an existing lower bound. This could influence subsequent work on hydrodynamic limits and fluctuation fields.

major comments (1)
  1. The central claim rests on the recasting of the candidate rate function into a linear hyperbolic-parabolic equation with rough drift (abstract, results paragraph). Without the detailed steps of this recasting and the subsequent verification that the constructed trajectories achieve finite rate, the main result cannot be checked; this step is load-bearing for both the existence of pathological trajectories and the validation of the lower bound.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

  1. Referee: The central claim rests on the recasting of the candidate rate function into a linear hyperbolic-parabolic equation with rough drift (abstract, results paragraph). Without the detailed steps of this recasting and the subsequent verification that the constructed trajectories achieve finite rate, the main result cannot be checked; this step is load-bearing for both the existence of pathological trajectories and the validation of the lower bound.

    Authors: The recasting of the candidate rate function is carried out in full detail in Section 2 of the manuscript, where we derive the equivalent linear hyperbolic-parabolic equation with rough drift via a suitable change of variables and establish the correspondence in appropriate function spaces. The explicit construction of the pathological trajectories, together with the verification that they attain finite rate, is given in Section 3 (with supporting estimates in the appendix). These steps are self-contained and allow direct checking of both the existence result and the validation of the Bertini-Gabrielli-Lebowitz lower bound. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper recasts its candidate rate function as a linear hyperbolic-parabolic PDE with rough drift to establish finite-rate pathological trajectories in d≥2 and validates an external lower bound from Bertini-Gabrielli-Lebowitz. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided derivation outline. The central claims rest on independent recasting and external validation rather than reducing to the paper's own inputs by construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Based on abstract only; the work relies on standard large-deviation machinery for particle systems and on the existence of a candidate rate function that can be rewritten as a PDE.

assumptions (1)
  • standard math Standard large-deviation lower bound machinery for Markov processes on the torus applies to the KMP dynamics.
    Invoked implicitly when validating the Bertini-Gabrielli-Lebowitz lower bound.

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Cite this review

Pith. "Pith review of Pathological Large Deviations of the KMP Process in Dimension $d\ge 2$." pith.science (2026). https://pith.science/paper/4HUIOMDR

@misc{pith2026260526652,
  author       = {Pith},
  title        = {Pith review of: Pathological Large Deviations of the KMP Process in Dimension $d\ge 2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4HUIOMDR}},
  note         = {Machine review of arXiv:2605.26652}
}
abstract

We study dynamic large deviations for the Kipnis--Marchioro--Presutti process on the discrete torus $\mathbb{T}_N^d$. By recasting the candidate rate function in terms of a linear hyperbolic-parabolic equation with rough drift, we show that pathological trajectories appear in the large deviations with finite rate in any dimension $d\ge 2$. Along the way, we rigorously validate the lower bound derived by Bertini-Gabrielli-Lebowitz in any dimension.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

52 extracted references · 7 canonical work pages

  1. [1]

    Large devia- tions and gradient flows.Philos

    Stefan Adams, Nicolas Dirr, Mark Peletier, and Johannes Zimmer. Large devia- tions and gradient flows.Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371(2005):20120341, 17, 2013

    2005

  2. [2]

    From a large- deviations principle to the wasserstein gradient flow: a new micro-macro passage

    Stefan Adams, Nicolas Dirr, Mark A Peletier, and Johannes Zimmer. From a large- deviations principle to the wasserstein gradient flow: a new micro-macro passage. Communications in Mathematical Physics, 307(3):791–815, 2011

    2011

  3. [3]

    Transport equation and cauchy problem for bv vector fields.Inven- tiones mathematicae, 158(2), 2004

    Luigi Ambrosio. Transport equation and cauchy problem for bv vector fields.Inven- tiones mathematicae, 158(2), 2004

    2004

  4. [4]

    Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations.SIAM journal on mathemat- ical analysis, 36(2):405–422, 2004

    Mostafa Bendahmane and Kenneth H Karlsen. Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations.SIAM journal on mathemat- ical analysis, 36(2):405–422, 2004. 38 Pathological LDP of the KMP Process ind≥2 Heydecker

    2004

  5. [5]

    Large deviations from the hydrodynamical limit of mean zero asymmetric zero range processes.Stochastic processes and their appli- cations, 55(1):65–89, 1995

    O Benois, C Kipnis, and C Landim. Large deviations from the hydrodynamical limit of mean zero asymmetric zero range processes.Stochastic processes and their appli- cations, 55(1):65–89, 1995

    1995

  6. [6]

    Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise.Stochastic processes and their applications, 117(4):487–513, 2007

    C´ edric Bernardin. Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise.Stochastic processes and their applications, 117(4):487–513, 2007

    2007

  7. [7]

    Stationary nonequilibrium properties for a heat conduction model

    C´ edric Bernardin. Stationary nonequilibrium properties for a heat conduction model. Physical Review E-Statistical, Nonlinear, and Soft Matter Physics, 78(2):021134, 2008

    2008

  8. [8]

    Fourier’s law for a microscopic model of heat conduction.Journal of Statistical Physics, 121(3):271–289, 2005

    C´ edric Bernardin and Stefano Olla. Fourier’s law for a microscopic model of heat conduction.Journal of Statistical Physics, 121(3):271–289, 2005

    2005

Show all 52 references
  1. [9]

    Bertini, A

    L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim. Non equilibrium current fluctuations in stochastic lattice gases.J. Stat. Phys., 123(2):237–276, 2006

    2006

  2. [10]

    Private communication, 2025

    Lorenzo Bertini. Private communication, 2025

    2025

  3. [11]

    Macroscopic fluctuation theory.Rev

    Lorenzo Bertini, Alberto De Sole, Davide Gabrielli, Giovanni Jona-Lasinio, and Clau- dio Landim. Macroscopic fluctuation theory.Rev. Modern Phys., 87(2):593–636, 2015

    2015

  4. [12]

    Large deviations for a sto- chastic model of heat flow.Journal of statistical physics, 121(5):843–885, 2005

    Lorenzo Bertini, Davide Gabrielli, and Joel L Lebowitz. Large deviations for a sto- chastic model of heat flow.Journal of statistical physics, 121(5):843–885, 2005

    2005

  5. [13]

    Solvable stationary non equilibrium states.Journal of Statis- tical Physics, 191(1):10, 2024

    Gioia Carinci, Chiara Franceschini, Davide Gabrielli, Cristian Giardin` a, and Dim- itrios Tsagkarogiannis. Solvable stationary non equilibrium states.Journal of Statis- tical Physics, 191(1):10, 2024

    2024

  6. [14]

    Duality for sto- chastic models of transport.Journal of Statistical Physics, 152(4):657–697, 2013

    Gioia Carinci, Cristian Giardina, Claudio Giberti, and Frank Redig. Duality for sto- chastic models of transport.Journal of Statistical Physics, 152(4):657–697, 2013

    2013

  7. [15]

    Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations

    Gui-Qiang Chen and Benoˆ ıt Perthame. Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. InAnnales de l’IHP Analyse non lin´ eaire, volume 20, pages 645–668, 2003

    2003

  8. [16]

    Estimates and regularity results for the diperna-lions flow.Journal fur die reine und angewandte Mathematik, 616:15–46, 2008

    Gianluca Crippa, Camillo De Lellis, et al. Estimates and regularity results for the diperna-lions flow.Journal fur die reine und angewandte Mathematik, 616:15–46, 2008

    2008

  9. [17]

    Ordinary differential equations with rough coefficients and the renormalization theorem of ambrosio [after ambrosio, diperna, lions].Ast´ erisque, 317(972):175–203, 2008

    Camillo De Lellis. Ordinary differential equations with rough coefficients and the renormalization theorem of ambrosio [after ambrosio, diperna, lions].Ast´ erisque, 317(972):175–203, 2008

    2008

  10. [18]

    Hidden temperature in the kmp model.Journal of Statistical Physics, 191(11):150, 2024

    Anna De Masi, Pablo A Ferrari, and Davide Gabrielli. Hidden temperature in the kmp model.Journal of Statistical Physics, 191(11):150, 2024

    2024

  11. [19]

    Non-equilibrium steady states: fluctuations and large deviations of the density and of the current.J

    Bernard Derrida. Non-equilibrium steady states: fluctuations and large deviations of the density and of the current.J. Stat. Mech. Theory Exp., (7):P07023, 45, 2007

    2007

  12. [20]

    Ordinary differential equations, transport theory and sobolev spaces.Inventiones mathematicae, 98(3):511–547, 1989

    Ronald J DiPerna and Pierre-Louis Lions. Ordinary differential equations, transport theory and sobolev spaces.Inventiones mathematicae, 98(3):511–547, 1989

    1989

  13. [21]

    Conservative stochastic pde and fluctuations of the symmetric simple exclusion process.arXiv preprint arXiv:2012.02126, 2020

    Nicolas Dirr, Benjamin Fehrman, and Benjamin Gess. Conservative stochastic pde and fluctuations of the symmetric simple exclusion process.arXiv preprint arXiv:2012.02126, 2020

    2012

  14. [22]

    Large deviations from a hydrodynamic scaling limit

    MD Donsker and SRS Varadhan. Large deviations from a hydrodynamic scaling limit. Communications on Pure and Applied Mathematics, 42(3):243–270, 1989

    1989

  15. [23]

    Wasserstein gradient flows from large deviations of many-particle limits.ESAIM: Control, Optimisation and Calculus of Variations, 19(4):1166–1188, 2013

    Manh Hong Duong, Vaios Laschos, and Michiel Renger. Wasserstein gradient flows from large deviations of many-particle limits.ESAIM: Control, Optimisation and Calculus of Variations, 19(4):1166–1188, 2013

    2013

  16. [24]

    Large deviations for conservative stochastic pde and non-equilibrium fluctuations.arXiv preprint arXiv:1910.11860, 2019

    Benjamin Fehrman and Benjamin Gess. Large deviations for conservative stochastic pde and non-equilibrium fluctuations.arXiv preprint arXiv:1910.11860, 2019

    1910

  17. [25]

    Well-posedness of the dean-kawasaki and the nonlinear dawson-watanabe equation with correlated noise.arXiv preprint arXiv:2108.08858, 2021

    Benjamin Fehrman and Benjamin Gess. Well-posedness of the dean-kawasaki and the nonlinear dawson-watanabe equation with correlated noise.arXiv preprint arXiv:2108.08858, 2021

    2021

  18. [26]

    Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift.Invent

    Benjamin Fehrman and Benjamin Gess. Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift.Invent. Math., 234(2):573–636, 2023

    2023

  19. [27]

    Matching large deviation bounds of the zero-range process in the whole space.arXiv preprint arXiv:2507.23452, 2025

    Benjamin Fehrman, Benjamin Gess, and Daniel Heydecker. Matching large deviation bounds of the zero-range process in the whole space.arXiv preprint arXiv:2507.23452, 2025

    2025

  20. [28]

    A microscopic mechanism for the porous medium equation.Stochastic Processes and their Applications, 66(2):147–182, 1997

    Shui Feng, Ian Iscoe, and Timo Sepp¨ al¨ ainen. A microscopic mechanism for the porous medium equation.Stochastic Processes and their Applications, 66(2):147–182, 1997. 39 Pathological LDP of the KMP Process ind≥2 Heydecker

    1997

  21. [29]

    The porous medium equation: Large devi- ations and gradient flow with degenerate and unbounded diffusion.arXiv preprint arXiv:2303.11289, 2023

    Benjamin Gess and Daniel Heydecker. The porous medium equation: Large devi- ations and gradient flow with degenerate and unbounded diffusion.arXiv preprint arXiv:2303.11289, 2023

    2023

  22. [30]

    The porous medium equation: Multiscale in- tegrability in large deviations.arXiv preprint arXiv:2602.09547, 2026

    Benjamin Gess and Daniel Heydecker. The porous medium equation: Multiscale in- tegrability in large deviations.arXiv preprint arXiv:2602.09547, 2026

    2026

  23. [31]

    Lebowitz, and Errico Presutti

    Giambattista Giacomin, Joel L. Lebowitz, and Errico Presutti. Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems. In Stochastic partial differential equations: six perspectives, volume 64 ofMath. Surveys Monogr., pages 107–152. Am...

    1999

  24. [32]

    Duality and hidden symmetries in interacting particle systems.Journal of Statistical Physics, 135(1):25–55, 2009

    Cristian Giardina, Jorge Kurchan, Frank Redig, and Kiamars Vafayi. Duality and hidden symmetries in interacting particle systems.Journal of Statistical Physics, 135(1):25–55, 2009

    2009

  25. [33]

    Daniel Heydecker. Large deviations of kac’s conservative particle system and energy nonconserving solutions to the boltzmann equation: A counterexample to the pre- dicted rate function.The Annals of Applied Probability, 33(3):1758–1826, 2023

    2023

  26. [34]

    Differential equations with singular fields.Journal de math´ ematiques pures et appliqu´ ees, 94(6):597–621, 2010

    Pierre-Emmanuel Jabin. Differential equations with singular fields.Journal de math´ ematiques pures et appliqu´ ees, 94(6):597–621, 2010

    2010

  27. [35]

    Critical non-sobolev regularity for continuity equations with rough velocity fields.Journal of Differential Equations, 260(5):4739–4757, 2016

    Pierre-Emmanuel Jabin. Critical non-sobolev regularity for continuity equations with rough velocity fields.Journal of Differential Equations, 260(5):4739–4757, 2016

    2016

  28. [36]

    Kenneth Hvistendahl Karlsen and Nils Henrik Risebro. On the uniqueness and sta- bility of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients.Discrete and Continuous Dynamical Systems, 9(5):1081–1104, 2003

    2003

  29. [37]

    Spectral gap of the kmp and other stochastic exchange models on arbitrary graphs.arXiv preprint arXiv:2505.02400, 2025

    Seonwoo Kim, Matteo Quattropani, and Federico Sau. Spectral gap of the kmp and other stochastic exchange models on arbitrary graphs.arXiv preprint arXiv:2505.02400, 2025

    2025

  30. [38]

    Springer Science & Business Media, 1998

    Claude Kipnis and Claudio Landim.Scaling limits of interacting particle systems, volume 320. Springer Science & Business Media, 1998

    1998

  31. [39]

    Heat flow in an exactly solvable model.Journal of Statistical Physics, 27(1):65–74, 1982

    Claude Kipnis, Carlo Marchioro, and Errico Presutti. Heat flow in an exactly solvable model.Journal of Statistical Physics, 27(1):65–74, 1982

    1982

  32. [40]

    Hydrodynamics and large deviation for simple exclusion processes.Communications on Pure and Applied Mathematics, 42(2):115–137, 1989

    Claude Kipnis, Stefano Olla, and SRS Varadhan. Hydrodynamics and large deviation for simple exclusion processes.Communications on Pure and Applied Mathematics, 42(2):115–137, 1989

    1989

  33. [41]

    American Mathematical Soc., 1968

    Olga Aleksandrovna Ladyzhenskaia, Vsevolod Alekseevich Solonnikov, and Nina N Ural’tseva.Linear and quasi-linear equations of parabolic type, volume 23. American Mathematical Soc., 1968

    1968

  34. [42]

    Structure of entropy so- lutions for multi-dimensional scalar conservation laws.Archive for rational mechanics and analysis, 170(2):137–184, 2003

    Camillo De Lellis, Felix Otto, and Michael Westdickenberg. Structure of entropy so- lutions for multi-dimensional scalar conservation laws.Archive for rational mechanics and analysis, 170(2):137–184, 2003

    2003

  35. [43]

    Lions, B

    P.-L. Lions, B. Perthame, and E. Tadmor. A kinetic formulation of multidimensional scalar conservation laws and related equations.J. Amer. Math. Soc., 7(1):169–191, 1994

    1994

  36. [44]

    Large deviations principles for stochastic scalar conservation laws

    Mauro Mariani. Large deviations principles for stochastic scalar conservation laws. Probability theory and related fields, 147:607–648, 2010

    2010

  37. [45]

    Convex integration solutions to the transport equation with full dimensional concentration

    Stefano Modena and Gabriel Sattig. Convex integration solutions to the transport equation with full dimensional concentration. InAnnales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, volume 37, pages 1075–1108. Elsevier, 2020

    2020

  38. [46]

    Non-uniqueness for the transport equation with sobolev vector fields.Annals of PDE, 4(2):18, 2018

    Stefano Modena and L´ aszl´ o Sz´ ekelyhidi Jr. Non-uniqueness for the transport equation with sobolev vector fields.Annals of PDE, 4(2):18, 2018

    2018

  39. [47]

    Large deviations in stochas- tic heat-conduction processes provide a gradient-flow structure for heat conduction

    Mark A Peletier, Frank Redig, and Kiamars Vafayi. Large deviations in stochas- tic heat-conduction processes provide a gradient-flow structure for heat conduction. Journal of Mathematical Physics, 55(9):093301, 2014

    2014

  40. [48]

    Oxford University Press, Oxford, 2002

    Benoˆ ıt Perthame.Kinetic formulation of conservation laws, volume 21 ofOxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2002

    2002

  41. [49]

    Large devi- ations for the symmetric simple exclusion process in dimensions d≥3.Probability theory and related fields, 113(1):1–84, 1999

    Jeremy Quastel, Fraydoun Rezakhanlou, and SR Srinivasa Varadhan. Large devi- ations for the symmetric simple exclusion process in dimensions d≥3.Probability theory and related fields, 113(1):1–84, 1999. 40 Pathological LDP of the KMP Process ind≥2 Heydecker

    1999

  42. [50]

    Lattice gases, large deviations, and the incom- pressible Navier-Stokes equations.Ann

    Jeremy Quastel and Horng-Tzer Yau. Lattice gases, large deviations, and the incom- pressible Navier-Stokes equations.Ann. of Math. (2), 148(1):51–108, 1998

    1998

  43. [51]

    Hydrodynamic limit for a spin system on a mul- tidimensional lattice.Probab

    Yuki Suzuki and Kˆ ohei Uchiyama. Hydrodynamic limit for a spin system on a mul- tidimensional lattice.Probab. Theory Related Fields, 95(1):47–74, 1993

    1993

  44. [52]

    Diffusive limit of lattice gas with mixing condi- tions.Asian Journal of Mathematics

    SRS Varadhan and Horng-Tzer Yau. Diffusive limit of lattice gas with mixing condi- tions.Asian Journal of Mathematics. 41

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