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arxiv: 2604.00315 · v2 · pith:XIWLDYHKnew · submitted 2026-03-31 · 🧮 math.AP · math.OC· math.PR

Quantification of ergodicity for Hamilton--Jacobi equations in a dynamic random environment

Xiaoqin Guo , Wenjia Jing , Hung Vinh Tran , Yuming Paul Zhang This is my paper

Pith reviewed 2026-05-22 10:19 UTC · model grok-4.3

classification 🧮 math.AP math.OCmath.PR
keywords Hamilton-Jacobi equationsergodicityrandom environmentconvergence ratesmetric problemKPZ equationlarge-time averagesstochastic growth
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The pith

Solutions to Hamilton-Jacobi equations in dynamic random environments converge to ergodic limits at rate 1/2 up to slowly varying factors.

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

The paper aims to quantify the speed at which solutions to Hamilton-Jacobi equations driven by a stationary ergodic random environment settle into their long-run average behavior. This rate of convergence, shown to be 1/2 with possible slow variations, matters for models of stochastic growth such as those linked to the inviscid KPZ equation, where understanding fluctuations over time is key. The work achieves this by developing a new almost-Lipschitz regularity result for the related metric problem, which controls how solutions behave and enables the rate estimates. A sympathetic reader would see this as providing concrete bounds on ergodicity in random media, helping predict large-time limits in physical systems modeled by these equations.

Core claim

We establish, up to slowly varying factors, convergence rates with exponent 1/2 for the large-time averages of both the solutions and the associated metric problem toward their ergodic limits. Our proof relies crucially on a new almost-Lipschitz regularity theory for the metric problem, which is of independent interest.

What carries the argument

The new almost-Lipschitz regularity theory for the metric problem, which supplies the estimates needed to derive the 1/2 convergence rates for large-time averages of solutions and the metric problem itself.

If this is right

  • Large-time averages of the solutions approach their ergodic limits at rate 1/2 up to slowly varying factors.
  • The associated metric problem admits almost-Lipschitz regularity estimates that control its behavior.
  • These quantitative rates apply directly to stochastic growth models formulated as Hamilton-Jacobi equations with random forcing.
  • The ergodic limits for both the solutions and the metric problem become accessible with explicit error bounds.

Where Pith is reading between the lines

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

  • The regularity theory might extend to related equations with different types of random forcing beyond unit-range time dependence.
  • Such rates could improve error bounds when approximating solutions numerically over long times in growth models.
  • The results suggest a pathway to quantify fluctuations in other interface evolution problems that reduce to Hamilton-Jacobi equations.
  • Connections to scaling in physical growth processes imply possible use in predicting statistical properties of interfaces.

Load-bearing premise

The dynamic random environment is stationary ergodic and has unit-range dependence in time.

What would settle it

A numerical simulation or explicit example in which the difference between the large-time average and the ergodic limit fails to decay like t to the power of -1/2 times any slowly varying function.

Figures

Figures reproduced from arXiv: 2604.00315 by Hung Vinh Tran, Wenjia Jing, Xiaoqin Guo, Yuming Paul Zhang.

Figure 1.1
Figure 1.1. Figure 1.1: Some admissible curves connecting (x1, t1) to (x2, t2) We define, for (x, t) ∈ R d+1 , m(x, t; x, t, ω) ≡ 0. For (x, t) ∈ R d × [0, ∞), we write m(x, t, ω) = m(0, 0; x, t, ω) for simplicity. Thanks to [25, 33, 22, 23, 35], the effective Lagrangian is defined as, for v ∈ R d , L(v) := m(v, 1) = limt→∞ Em(tv, t) t = lim t→∞ m(tv, t, ω) t , (1.6) [PITH_FULL_IMAGE:figures/full_fig_p005_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Curves γ and β 2.3. Improved regularity via iterations. For a ∈ (0, 1], set f(a) := a (1 − a)(q − 1)q + q + q − 1 q , g(a) := a (1 − a)(q − 1) + 1. Notice that f(a) = 1 q g(a) + 1 q ′ and 1 − f(a) = 1 q (1 − g(a)) [PITH_FULL_IMAGE:figures/full_fig_p011_2_1.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: An example of Qx,t and Fx,t Proposition 4.2. Let M, a > 1. Assume that the following statement is true: For each (x, t) ∈ R d × [M, ∞), there exist n ≥ 1 and a Qx,t-skeleton with k + 1 vertices from (0, 0) to (nx, nt) such that k ≤ an. ) (4.9) Then, inequality (4.3) holds for all t ≥ M, with C = C(M, a) > 0. Observe that, for any Qx,t-skeleton {(yi , ti)} k i=0 from (0, 0) to (nx, nt), with n ≥ 1, we mus… view at source ↗
read the original abstract

We study quantitative large-time averages for Hamilton--Jacobi equations in a dynamic random environment that is stationary ergodic and has unit-range dependence in time. Our motivation comes from stochastic growth models related to the tensionless (inviscid) KPZ equation, which can be formulated as Hamilton--Jacobi equations with random forcing. Understanding the large-time behavior of solutions is closely connected to fundamental questions concerning fluctuations and scaling in such growth processes. In this article, we establish, up to slowly varying factors, convergence rates with exponent $1/2$ for the large-time averages of both the solutions and the associated metric problem toward their ergodic limits. Our proof relies crucially on a new almost-Lipschitz regularity theory for the metric problem, which is of independent interest.

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

0 major / 2 minor

Summary. The manuscript studies quantitative large-time averages for Hamilton--Jacobi equations in a stationary ergodic dynamic random environment with unit-range time dependence. It claims to establish convergence rates of order 1/2 (up to slowly varying factors) for the large-time averages of both the solutions and the associated metric problem to their ergodic limits. The argument relies on a new almost-Lipschitz regularity theory for the metric problem, which is presented as being of independent interest and motivated by connections to stochastic growth models and the tensionless KPZ equation.

Significance. If the claimed rates and regularity hold, the work would be significant for providing explicit quantitative ergodicity results in random media, advancing understanding of fluctuations and scaling in related growth processes. The new almost-Lipschitz regularity theory for the metric problem under unit-range time dependence is a clear strength that could have applications beyond this setting; the internal consistency of the argument (unit-range independence supplying decorrelation for the diffusive scaling) is a positive feature.

minor comments (2)
  1. [Abstract] The abstract and introduction state the 1/2-rate result 'up to slowly varying factors' without specifying the form of these factors or their dependence on parameters; adding a brief clarification would improve readability.
  2. [Introduction] A short outline of the key steps in the proof of the regularity theory (e.g., how unit-range dependence is used to control oscillations) would help readers assess the argument structure without needing to consult later sections immediately.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. The summary accurately reflects our main results on quantitative large-time averages with 1/2-rate convergence (up to slowly varying factors) for both the solutions and the metric problem, as well as the new almost-Lipschitz regularity theory for the metric problem under unit-range time dependence. We appreciate the recognition of the potential independent interest of this regularity result and its connections to stochastic growth models and the tensionless KPZ equation. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new regularity theory

full rationale

The paper claims 1/2-rate convergence (up to slowly varying factors) for large-time averages of solutions and the metric problem, relying on a newly developed almost-Lipschitz regularity theory for the metric problem under stationary ergodic dynamics with unit-range time dependence. This regularity is explicitly positioned as independent and of separate interest. No load-bearing step reduces the claimed rates or ergodic limits to a fitted parameter, self-citation chain, or definitional equivalence; the unit-range independence directly supplies decorrelation for the variance bounds yielding the diffusive scaling, without hidden assumptions that loop back to the target result. The argument structure is internally consistent and externally falsifiable via the stated mixing properties.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the random environment satisfying stationarity, ergodicity, and unit-range time dependence; these are domain assumptions standard in the field but required for the quantitative rates to hold. No free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption The dynamic random environment is stationary ergodic and has unit-range dependence in time.
    Explicitly stated as the setting in which the Hamilton-Jacobi equations and metric problem are studied.

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Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    K. S. Alexander. Approximation of subadditive functions and convergence rates in limiting- shape results.Ann. Probab., 25(1):30–55, 1997

    work page 1997

  2. [2]

    Armstrong and P

    S. Armstrong and P. Cardaliaguet. Stochastic homogenization of quasilinear Hamilton–Jacobi equations and geometric motions.Journal of the European Mathematical Society, 20(4):797– 864, 2018. QUANTIFICATION OF ERGODICITY FOR HJ EQUATIONS 49

    work page 2018

  3. [3]

    Armstrong, P

    S. Armstrong, P. Cardaliaguet, and P. Souganidis. Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations.Journal of the American Mathematical Society, 27(2):479–540, 2014

    work page 2014

  4. [4]

    Armstrong and H

    S. Armstrong and H. Tran. Stochastic homogenization of viscous Hamilton–Jacobi equations and applications.Analysis & PDE, 7(8):1969–2007, 2015

    work page 1969

  5. [5]

    S. N. Armstrong, H. V. Tran, and Y. Yu. Stochastic homogenization of a nonconvex Hamilton– Jacobi equation.Calculus of Variations and Partial Differential Equations, 54(2):1507–1524, 2015

    work page 2015

  6. [6]

    S. N. Armstrong, H. V. Tran, and Y. Yu. Stochastic homogenization of nonconvex Hamilton– Jacobi equations in one space dimension.Journal of Differential Equations, 261(5):2702–2737, 2016

    work page 2016

  7. [7]

    K. Azuma. Weighted sums of certain dependent random variables.Tohoku Mathematical Journal, Second Series, 19(3):357–367, 1967

    work page 1967

  8. [8]

    Bahraminasab, S

    A. Bahraminasab, S. M. A. Tabei, A. A. Masoudi, F. Shahbazi, and M. R. Rahimi Tabar. Zero tension Kardar-Parisi-Zhang equation in (d+ 1)-dimensions.J. Statist. Phys., 116(5- 6):1521–1544, 2004

    work page 2004

  9. [9]

    Bakhtin and D

    Y. Bakhtin and D. Dow. Differentiability of the effective lagrangian for hamilton–jacobi– bellman equations in dynamic random environments.Stochastics and Partial Differential Equations: Analysis and Computations, 2025

    work page 2025

  10. [10]

    Bakhtin and K

    Y. Bakhtin and K. Khanin. On global solutions of the random Hamilton–Jacobi equations and the KPZ problem.Nonlinearity, 31:R93, 2018

    work page 2018

  11. [11]

    Burago, S

    D. Burago, S. Ivanov, and A. Novikov. Feeble fish in time-dependent waters and homogeniza- tion of the G-equation.Communications on Pure and Applied Mathematics, 73(7):1453–1489, 2020

    work page 2020

  12. [12]

    Cannarsa and P

    P. Cannarsa and P. Cardaliaguet. H¨ older estimates in space-time for viscosity solutions of Hamilton-Jacobi equations.Comm. Pure Appl. Math., 63(5):590–629, 2010

    work page 2010

  13. [13]

    Cardaliaguet, J

    P. Cardaliaguet, J. Nolen, and P. E. Souganidis. Homogenization and enhancement for the G-equation.Arch. Ration. Mech. Anal., 199(2):527–561, 2011

    work page 2011

  14. [14]

    Cardaliaguet and B

    P. Cardaliaguet and B. Seeger. H¨ older regularity of Hamilton-Jacobi equations with stochastic forcing.Transactions of the American Mathematical Society, 374(10):7197–7233, 2021

    work page 2021

  15. [15]

    Cardaliaguet and L

    P. Cardaliaguet and L. Silvestre. H¨ older continuity to Hamilton-Jacobi equations with su- perquadratic growth in the gradient and unbounded right-hand side.Comm. Partial Differ- ential Equations, 37(9):1668–1688, 2012

    work page 2012

  16. [16]

    Cardaliaguet and P

    P. Cardaliaguet and P. E. Souganidis. Homogenization and enhancement of the G-equation in random environments.Communications on Pure and Applied Mathematics, 66(10):1582–1628, 2013

    work page 2013

  17. [17]

    Y. S. Chow and H. Teicher.Probability theory. Springer Texts in Statistics. Springer-Verlag, New York, third edition, 1997. Independence, interchangeability, martingales

    work page 1997

  18. [18]

    A. Davini. Stochastic homogenization of nondegenerate viscous hj equations in 1d, 2024

    work page 2024

  19. [19]

    W. M. Feldman and P. E. Souganidis. Homogenization and non-homogenization of cer- tain non-convex Hamilton–Jacobi equations.Journal de Math´ ematiques Pures et Appliqu´ ees, 108(5):751–782, 2017

    work page 2017

  20. [20]

    H. Gao. Random homogenization of coercive Hamilton-Jacobi equations in 1d.Calc. Var. Partial Differential Equations, 55(2):Art. 30, 39, 2016

    work page 2016

  21. [21]

    Gassiat, B

    P. Gassiat, B. Gess, P.-L. Lions, and P. E. Souganidis. Long-time behavior of stochastic Hamilton-Jacobi equations.Journal of Functional Analysis, 286(4):110269, 2024

    work page 2024

  22. [22]

    W. Jing, P. E. Souganidis, and H. V. Tran. Stochastic homogenization of viscous su- perquadratic Hamilton–Jacobi equations in dynamic random environment.Res. Math. Sci., 4:4:6, 2017. 50 X. GUO, W. JING, H. V. TRAN, Y. P. ZHANG

    work page 2017

  23. [23]

    W. Jing, P. E. Souganidis, and H. V. Tran. Large time average of reachable sets and ap- plications to homogenization of interfaces moving with oscillatory spatio-temporal velocity. Discrete Contin. Dyn. Syst. Ser. S, 11(5):915–939, 2018

    work page 2018

  24. [24]

    Kosygina, F

    E. Kosygina, F. Rezakhanlou, and S. R. S. Varadhan. Stochastic homogenization of Hamilton- Jacobi-Bellman equations.Comm. Pure Appl. Math., 59(10):1489–1521, 2006

    work page 2006

  25. [25]

    Kosygina and S

    E. Kosygina and S. R. S. Varadhan. Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium.Comm. Pure Appl. Math., 61(6):816–847, 2008

    work page 2008

  26. [26]

    Kosygina and A

    E. Kosygina and A. Yilmaz. Homogenization of nonconvex viscous Hamilton-Jacobi equations in stationary ergodic media in one dimension.Nonlinearity, 38(7):Paper No. 075017, 25, 2025

    work page 2025

  27. [27]

    Lesigne and D

    E. Lesigne and D. Voln´ y. Large deviations for martingales.Stochastic Process. Appl., 96(1):143–159, 2001

    work page 2001

  28. [28]

    Lions and P

    P.-L. Lions and P. E. Souganidis. Homogenization of “viscous” Hamilton–Jacobi equations in stationary ergodic media.Communications in Partial Difference Equations, 30(3):335–375, 2005

    work page 2005

  29. [29]

    Mitake, P

    H. Mitake, P. Ni, and H. V. Tran. Quantitative homogenization of convex Hamilton–Jacobi equations withu/ε-periodic Hamiltonians.arXiv preprint arXiv:2507.00663, 2025

    work page arXiv 2025

  30. [30]

    J. Qian, H. V. Tran, and Y. Yu. Min-max formulas and other properties of certain classes of nonconvex effective Hamiltonians.Math. Ann., 372(1-2):91–123, 2018

    work page 2018

  31. [31]

    Rezakhanlou and J

    F. Rezakhanlou and J. E. Tarver. Homogenization for stochastic Hamilton-Jacobi equations. Archive for rational mechanics and analysis, 151(4):277–309, 2000

    work page 2000

  32. [32]

    Rodr´ ıguez-Fern´ andez, S

    E. Rodr´ ıguez-Fern´ andez, S. N. Santalla, M. Castro, and R. Cuerno. Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation.Phys. Rev. E, 106:024802, Aug 2022

    work page 2022

  33. [33]

    R. W. Schwab. Stochastic homogenization of Hamilton-Jacobi equations in stationary ergodic spatio-temporal media.Indiana Univ. Math. J., 58(2):537–581, 2009

    work page 2009

  34. [34]

    B. Seeger. Homogenization of pathwise Hamilton–Jacobi equations.Journal de Math´ ematiques Pures et Appliqu´ ees, 110:1–31, 2018

    work page 2018

  35. [35]

    B. Seeger. Homogenization of a stochastically forced Hamilton-Jacobi equation.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 38(4):1217–1253, 2021

    work page 2021

  36. [36]

    B. Seeger. Scaling limits and homogenization of mixing Hamilton-Jacobi equations.Commu- nications in Partial Differential Equations, 46(1):165–199, 2021

    work page 2021

  37. [37]

    P. E. Souganidis. Stochastic homogenization of Hamilton–Jacobi equations and some appli- cations.Asymptotic Analysis, 20(1):1–11, 1999

    work page 1999

  38. [38]

    P. E. Souganidis. Pathwise solutions for fully nonlinear first-and second-order partial differ- ential equations with multiplicative rough time dependence. InSingular Random Dynamics: Cetraro, Italy 2016, pages 75–220. Springer, 2019

    work page 2016

  39. [39]

    H. V. Tran.Hamilton–Jacobi Equations: Theory and Applications, volume 213 ofGraduate Studies in Mathematics. American Mathematical Society, 2021

    work page 2021

  40. [40]

    C. Viterbo. Stochastic homogenization for variational solutions of Hamilton–Jacobi equations. Analysis & PDE, 18(4):805–856, 2025

    work page 2025

  41. [41]

    Xin and Y

    J. Xin and Y. Yu. Periodic homogenization of the inviscidG-equation for incompressible flows. Commun. Math. Sci., 8(4):1067–1078, 2010

    work page 2010

  42. [42]

    Y. P. Zhang and A. Zlatoˇ s. Homogenization for space-time-dependent KPP reaction– diffusion equations and G-equations.Calculus of Variations and Partial Differential Equa- tions, 62(9):248, 2023

    work page 2023

  43. [43]

    Ziliotto

    B. Ziliotto. Stochastic homogenization of nonconvex Hamilton-Jacobi equations: A counterex- ample.Communications on Pure & Applied Mathematics, 70(9), 2017. QUANTIFICATION OF ERGODICITY FOR HJ EQUATIONS 51 (X. Guo)Department of Mathematical Sciences, University of Cincinnati, 2815 Commons W ay, Cincinnati OH 45221-0025 Email address:guoxq@ucmail.uc.edu (W...

    work page 2017