pith. sign in

arxiv: 2605.23308 · v1 · pith:WDFHOJKTnew · submitted 2026-05-22 · 🧮 math.PR

Convergence rate estimates for semigroups and heat kernels associated with resistance forms

Koyo Oishi This is my paper

Pith reviewed 2026-05-25 03:57 UTC · model grok-4.3

classification 🧮 math.PR
keywords convergence ratesresistance formsheat kernelssemigroupsGromov-Hausdorff-vague convergenceSierpinski gasketBouchaud trap modelhomogenization
0
0 comments X

The pith

Explicit convergence rates are established for semigroups and heat kernels on resistance forms under measure regularity and lower resistance estimates.

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

The paper seeks to turn previous qualitative statements about convergence of heat kernels and semigroups into quantitative ones. When measured resistance metric spaces converge in the Gromov-Hausdorff-vague sense, the associated semigroups and heat kernels converge at explicit rates provided the spaces obey measure regularity and lower resistance estimates. A new metric is constructed that generates the same topology yet simplifies the task of obtaining the rates. The bounds are applied first to random-walk approximations of Brownian motion on the Sierpinski gasket, yielding the initial rate estimate for that setting. The same bounds are then used on the one-dimensional Bouchaud trap model, extending the range of parameters where homogenization is proved and sharpening the existing rate estimates by at least a quadratic factor.

Core claim

Under the assumptions of measure regularity and lower resistance estimates, explicit convergence rates hold for the semigroups and heat kernels whenever the underlying measured resistance metric spaces converge in the Gromov-Hausdorff-vague topology. A new metric is introduced that induces this topology and is convenient for evaluating the rates. The resulting bounds deliver the first convergence-rate estimate for random-walk approximation of Brownian motion on the Sierpinski gasket and extend the known parameter regime for homogenization in the one-dimensional Bouchaud trap model while improving prior rates by at least a quadratic factor.

What carries the argument

Quantitative bounds obtained from measure regularity together with lower resistance estimates, supported by a newly defined metric that induces the Gromov-Hausdorff-vague topology.

If this is right

  • First explicit rate estimate for random-walk approximation of Brownian motion on the Sierpinski gasket.
  • Homogenization proved for the one-dimensional Bouchaud trap model in every parameter regime where it occurs.
  • Existing convergence-rate estimates for the Bouchaud trap model improved by at least a quadratic factor.
  • A new metric simplifies direct calculation of distances in the Gromov-Hausdorff-vague topology.

Where Pith is reading between the lines

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

  • The rates supply concrete error controls that could be used to certify numerical discretizations of diffusions on fractals.
  • The same quantitative approach may transfer to other classes of Dirichlet forms once analogous regularity conditions are verified.
  • Improved rates in the trap model suggest that similar sharpening is possible for homogenization problems in higher-dimensional or non-one-dimensional settings.
  • The new metric could serve as a computational tool for checking convergence of other stochastic processes whose state spaces are resistance spaces.

Load-bearing premise

The measured resistance metric spaces satisfy measure regularity and lower resistance estimates.

What would settle it

A concrete sequence of spaces obeying measure regularity and lower resistance estimates whose semigroup or heat-kernel convergence is slower than the explicit rate given by the main theorem.

Figures

Figures reproduced from arXiv: 2605.23308 by Koyo Oishi.

Figure 1
Figure 1. Figure 1: Sierpinski Gasket Remark 1.15. We used the open set condition and the fact that the resistance metric and the Euclidean metric are comparable up to a power to verify Assumption 1.1 for the Sierpinski gasket. Note that (A2) or (A3) are satisfied provided that the connectivity of the space or the continuity of the process is established. Consequently, our results apply to a wider class of fractals satisfying… view at source ↗
Figure 2
Figure 2. Figure 2: Graph of the exponent E = E(α) appearing in the quenched estimate supx∈R |P0(X1 ≤ x) − P0(Xn 1 ≤ x)| ≲ n −E+ε . The blue line represents the exponent derived in [3], and the red line represents the one derived in Theorem 8.14. 2 Preliminaries 2.1 Bounded Lipschitz metric In this section, we introduce an appropriate metric on a space of measures. In [21], the Gromov-Hausdorff￾vague metric (or topology) is u… view at source ↗
Figure 3
Figure 3. Figure 3: Let V0 be the set of vertices of a regular triangle. We define Vn as Vn := ∪ 3 i=1fi(Vn−1), where the maps fi , i = 1, 2, 3 are defined at the beginning of this section. Note that µn can be expressed as µn({x}) = degVn (x) X y∈Vn degVn (y) , x ∈ Vn. Here, we denote the degree of a vertex x ∈ Vn in the graph Vn by degVn (x). We denote the log2 3- dimensional probability Hausdorff measure on K by µ. We write… view at source ↗
read the original abstract

In this paper, we derive quantitative convergence rates for stochastic processes associated with resistance forms. While the qualitative convergence of heat kernels and semigroups under the Gromov-Hausdorff-vague convergence of underlying measured resistance metric spaces has been investigated previously, their quantitative convergence rates have remained unexplored. We establish explicit convergence rates for the associated semigroups and heat kernels under the assumptions of measure regularity and lower resistance estimates. Furthermore, we introduce a new metric that induces the Gromov-Hausdorff-vague topology, and is convenient for evaluation. As applications of our main results, we present two illustrative examples. First, we derive first estimate on the convergence rate for the random walk approximation of Brownian motion on the Sierpinski gasket. Second, we apply our results to the one-dimensional Bouchaud trap model, successfully extending the previously known parameter regime to all cases where homogenization occurs and improving the convergence rate estimates in the existing regime by at least a quadratic factor.

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 paper derives explicit quantitative convergence rates for semigroups and heat kernels associated with resistance forms on measured resistance metric spaces, under the assumptions of measure regularity and lower resistance estimates. It introduces a new metric inducing the Gromov-Hausdorff-vague topology and applies the results to obtain a first convergence-rate estimate for the random-walk approximation of Brownian motion on the Sierpinski gasket, as well as an extension of the parameter regime (to all cases where homogenization occurs) together with a quadratic-factor improvement in the existing regime for the one-dimensional Bouchaud trap model.

Significance. If the derivations hold, the work supplies the first quantitative rates in a setting where only qualitative convergence under GHV convergence was previously available. The applications are concrete and improve upon existing results in the Bouchaud model while providing the initial rate for the gasket example. The new metric is presented as convenient for evaluation and may facilitate further quantitative work on resistance forms.

minor comments (2)
  1. [Abstract / Introduction] The abstract states that the results hold under measure regularity and lower resistance estimates, but the precise statements of these assumptions (and any uniformity requirements across the sequence of spaces) should be recalled explicitly in the introduction or §2 to make the hypotheses self-contained for readers.
  2. [Application section on Bouchaud model] The claim of a 'quadratic factor' improvement in the Bouchaud trap model should be accompanied by a direct comparison (in a table or remark) with the best previously published rate, citing the specific theorem or proposition being improved.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for acknowledging its potential significance in providing the first quantitative convergence rates under Gromov-Hausdorff-vague convergence for resistance forms, along with the concrete applications to the Sierpinski gasket and Bouchaud trap model. The recommendation is listed as uncertain, but the report contains no specific major comments or points of concern. We would be happy to address any particular questions or requests for clarification if provided.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under stated assumptions

full rationale

The paper derives explicit quantitative convergence rates for semigroups and heat kernels on resistance forms, conditional on the external assumptions of measure regularity and lower resistance estimates. These assumptions are invoked as inputs to obtain the bounds rather than being redefined in terms of the rates themselves. The new metric inducing the GHV topology is introduced as a tool for evaluation, not as a self-referential construct. The two applications (Sierpinski gasket random walk and Bouchaud trap model) are presented as illustrative uses of the main results, with no indication that the claimed rates reduce to fitted parameters or prior self-citations by construction. The derivation chain remains independent of its outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable.

pith-pipeline@v0.9.0 · 5687 in / 972 out tokens · 16048 ms · 2026-05-25T03:57:07.174636+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages · 2 internal anchors

  1. [1]

    Abraham, J.-F

    R. Abraham, J.-F. Delmas, and P. Hoscheit. A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces.Electron. J. Probab., 18:Paper No. 14, 21, 2013

    work page 2013

  2. [2]

    D. Aldous. The continuum random tree. III.Ann. Probab., 21(1):248–289, 1993

    work page 1993

  3. [3]

    Andres, D

    S. Andres, D. A. Croydon, and T. Kumagai. Heat kernel fluctuations and quantitative homogeniza- tion for the one-dimensional Bouchaud trap model.Stochastic Process. Appl., 172:Paper No. 104336, 20, 2024

    work page 2024

  4. [4]

    The scaling limit of random walk and the intrinsic metric on planar critical percolation

    I. Ðanković, M. Markering, J. Miller, and Y. Yuan. The scaling limit of random walk and the intrinsic metric on planar critical percolation, 2026. Preprint available at arXiv:2604.14122v1

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  5. [5]

    Athreya, W

    S. Athreya, W. Löhr, and A. Winter. The gap between Gromov-vague and Gromov-Hausdorff-vague topology.Stochastic Process. Appl., 126(9):2527–2553, 2016

    work page 2016

  6. [6]

    Athreya, W

    S. Athreya, W. Löhr, and A. Winter. Invariance principle for variable speed random walks on trees. Ann. Probab., 45(2):625–667, 2017

    work page 2017

  7. [7]

    M. T. Barlow. Diffusions on fractals. InLectures on probability theory and statistics (Saint-Flour, 1995), volume 1690 ofLecture Notes in Math., pages 1–121. Springer, Berlin, 1998

    work page 1995

  8. [8]

    M. T. Barlow.Random walks and heat kernels on graphs, volume 438 ofLondon Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2017

    work page 2017

  9. [9]

    M. T. Barlow and R. F. Bass. The construction of Brownian motion on the Sierpiński carpet.Ann. Inst. Henri Poincaré Probab. Stat., 25(3):225–257, 1989

    work page 1989

  10. [10]

    M. T. Barlow and E. A. Perkins. Brownian motion on the Sierpiński gasket.Probab. Theory Related Fields, 79(4):543–623, 1988

    work page 1988

  11. [11]

    Ben Arous and J

    G. Ben Arous and J. Černý. Dynamics of trap models. InMathematical statistical physics, pages 331–394. Elsevier B. V., Amsterdam, 2006

    work page 2006

  12. [12]

    A. C. Berry. The accuracy of the Gaussian approximation to the sum of independent variates.Trans. Amer. Math. Soc., 49:122–136, 1941

    work page 1941

  13. [13]

    J. P. Bouchaud. Weak ergodicity breaking and aging in disordered systems.J. Phys. I, 2(9):1705– 1713, 1992

    work page 1992

  14. [14]

    Boucheron, G

    S. Boucheron, G. Lugosi, and P. Massart.Concentration inequalities: A nonasymptotic theory of independence. Oxford University Press, Oxford, 2013. With a foreword by Michel Ledoux

    work page 2013

  15. [15]

    Bousquet

    O. Bousquet. A Bennett concentration inequality and its application to suprema of empirical pro- cesses.C. R. Math. Acad. Sci. Paris, 334(6):495–500, 2002

    work page 2002

  16. [16]

    Burago, Y

    D. Burago, Y. Burago, and S. Ivanov.A course in metric geometry, volume 33 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001

    work page 2001

  17. [17]

    Chen and M

    Z.-Q. Chen and M. Fukushima.Symmetric Markov processes, time change, and boundary theory, vol- ume 35 ofLondon Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ„ 2012. 90 Convergence rate estimates for semigroups and heat kernels associated with resistance forms

    work page 2012

  18. [18]

    D. Croydon. Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree.Ann. Inst. Henri Poincaré Probab. Stat., 44(6):987–1019, 2008

    work page 2008

  19. [19]

    Croydon, B

    D. Croydon, B. Hambly, and T. Kumagai. Time-changes of stochastic processes associated with resistance forms.Electron. J. Probab., 22:Paper No. 82, 41, 2017

    work page 2017

  20. [20]

    D. A. Croydon. Heat kernel fluctuations for a resistance form with non-uniform volume growth. Proc. Lond. Math. Soc. (3), 94(3):672–694, 2007

    work page 2007

  21. [21]

    D. A. Croydon. Scaling limits of stochastic processes associated with resistance forms.Ann. Inst. Henri Poincaré Probab. Stat., 54(4):1939–1968, 2018

    work page 1939

  22. [22]

    D. A. Croydon, N. Feldheim, and O. Feldheim. Zeroes of fractional Gaussian fields on fractals. In preparation

    work page

  23. [23]

    R. M. Dudley.Real analysis and probability, volume 74 ofCambridge Studies in Advanced Mathe- matics. Cambridge University Press, Cambridge, 2002. Revised reprint of the 1989 original

    work page 2002

  24. [24]

    Duquesne

    T. Duquesne. A limit theorem for the contour process of conditioned Galton-Watson trees.Ann. Probab., 31(2):996–1027, 2003

    work page 2003

  25. [25]

    Erhard, T

    D. Erhard, T. Franco, and D. S. da Silva. The slow bond random walk and the snapping out Brownian motion.Ann. Appl. Probab., 31(1):99–127, 2021

    work page 2021

  26. [26]

    C.-G. Esseen. On the Liapounoff limit of error in the theory of probability.Ark. Mat. Astr. Fys., 28A(9):19, 1942

    work page 1942

  27. [27]

    S. N. Ethier and T. G. Kurtz.Markov processes: Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986

    work page 1986

  28. [28]

    S. N. Evans, J. Pitman, and A. Winter. Rayleigh processes, real trees, and root growth with re- grafting.Probab. Theory Related Fields, 134(1):81–126, 2006

    work page 2006

  29. [29]

    P. J. Fitzsimmons, B. M. Hambly, and T. Kumagai. Transition density estimates for Brownian motion on affine nested fractals.Comm. Math. Phys., 165(3):595–620, 1994

    work page 1994

  30. [30]

    L. R. G. Fontes, M. Isopi, and C. M. Newman. Chaotic time dependence in a disordered spin system. Probab. Theory Related Fields, 115(3):417–443, 1999

    work page 1999

  31. [31]

    Fukushima, Y

    M. Fukushima, Y. ¯Oshima, and M. Takeda.Dirichlet forms and symmetric Markov processes, volume 19 ofDe Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1994

    work page 1994

  32. [32]

    Goldstein

    S. Goldstein. Random walks and diffusions on fractals. InPercolation theory and ergodic theory of infinite particle systems (Minneapolis, Minn., 1984–1985), volume 8 ofIMA Vol. Math. Appl., pages 121–129. Springer, New York, 1987

    work page 1984

  33. [33]

    Gut.Probability: a graduate course, volume 200

    A. Gut.Probability: a graduate course, volume 200. Springer, 2006

    work page 2006

  34. [34]

    J. E. Hutchinson. Fractals and self-similarity.Indiana Univ. Math. J., 30(5):713–747, 1981

    work page 1981

  35. [35]

    A. S. Kechris.Classical descriptive set theory, volume 156 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1995

    work page 1995

  36. [36]

    A. Khezeli. A unified framework for generalizing the Gromov-Hausdorff metric.Probab. Surv., 20:837–896, 2023

    work page 2023

  37. [37]

    J. Kigami. Harmonic calculus on limits of networks and its application to dendrites.J. Funct. Anal., 128(1):48–86, 1995

    work page 1995

  38. [38]

    Kigami.Analysis on fractals, volume 143 ofCambridge Tracts in Mathematics

    J. Kigami.Analysis on fractals, volume 143 ofCambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2001. 91 K.Oishi

    work page 2001

  39. [39]

    J. Kigami. Harmonic analysis for resistance forms.J. Funct. Anal., 204(2):399–444, 2003

    work page 2003

  40. [40]

    J. Kigami. Resistance forms, quasisymmetric maps and heat kernel estimates.Mem. Amer. Math. Soc., 216(1015):vi+132, 2012

    work page 2012

  41. [41]

    V. N. Kolokoltsov. The rates of convergence for functional limit theorems with stable subordinators and for CTRW approximations to fractional evolutions.Fractal Fract., 7(4):335, 2023

    work page 2023

  42. [42]

    Komlós, P

    J. Komlós, P. Major, and G. Tusnády. An approximation of partial sums of independent RV’s and the sample DF. I.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32:111–131, 1975

    work page 1975

  43. [43]

    Komlós, P

    J. Komlós, P. Major, and G. Tusnády. An approximation of partial sums of independent RV’s, and the sample DF. II.Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 34(1):33–58, 1976

    work page 1976

  44. [44]

    T. Kumagai. Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms.Publ. Res. Inst. Math. Sci., 40(3):793–818, 2004

    work page 2004

  45. [45]

    S. Kusuoka. A diffusion process on a fractal. InProbabilistic methods in mathematical physics (Katata/Kyoto, 1985), pages 251–274. Academic Press, Boston, MA, 1987

    work page 1985

  46. [46]

    R. Latała. Estimation of moments of sums of independent real random variables.Ann. Probab., 25(3):1502–1513, 1997

    work page 1997

  47. [47]

    J.-F. Le Gall. Random real trees.Ann. Fac. Sci. Toulouse Math. (6), 15(1):35–62, 2006

    work page 2006

  48. [48]

    D. A. Levin, Y. Peres, and E. L. Wilmer.Markov chains and mixing times. American Mathematical Society, Providence, RI, 2009. With a chapter by James G. Propp and David B. Wilson

    work page 2009

  49. [49]

    M. B. Marcus and J. Rosen.Markov processes, Gaussian processes, and local times, volume 100 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006

    work page 2006

  50. [50]

    E. J. McShane. Extension of range of functions.Bull. Amer. Math. Soc., 40(12):837–842, 1934

    work page 1934

  51. [51]

    Miermont

    G. Miermont. Tessellations of random maps of arbitrary genus.Ann. Sci. Éc. Norm. Supér. (4), 42(5):725–781, 2009

    work page 2009

  52. [52]

    J. Miller. Existence and uniqueness of the canonical Brownian motion in non-simple conformal loop ensemble gaskets, 2025. Preprint available at arXiv:2512.04807v1

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  53. [53]

    M. Murugan. Heat kernel for reflected diffusion and extension property on uniform domains.Probab. Theory Related Fields, 190(1-2):543–599, 2024

    work page 2024

  54. [54]

    R. Noda. Metrization of Gromov-Hausdorff-type topologies on boundedly-compact metric spaces,

    work page

  55. [55]

    Preprint available at arXiv:2404.19681v2

    work page arXiv

  56. [56]

    R. Noda. Scaling limits of discrete-time Markov chains and their local times on electrical networks,

    work page

  57. [57]

    Preprint available at arXiv:2405.01871v2

    work page arXiv

  58. [58]

    R. Noda. Convergence of local times of stochastic processes associated with resistance forms.Ann. Inst. Henri Poincaré Probab. Stat., To appear

    work page

  59. [59]

    Post and J

    O. Post and J. Simmer. Approximation of fractals by discrete graphs: norm resolvent and spectral convergence.Integral Equations Operator Theory, 90(6):68, 2018

    work page 2018

  60. [60]

    A. W. van der Vaart and J. A. Wellner.Weak convergence and empirical processes: With applications to statistics. Springer Series in Statistics. Springer-Verlag, New York, 1996

    work page 1996

  61. [61]

    Vershynin.High-dimensional probability: An introduction with applications in data science, vol- ume 47 ofCambridge Series in Statistical and Probabilistic Mathematics

    R. Vershynin.High-dimensional probability: An introduction with applications in data science, vol- ume 47 ofCambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2018. With a foreword by Sara van de Geer

    work page 2018

  62. [62]

    F. G. Viens and A. B. Vizcarra. Supremum concentration inequality and modulus of continuity for sub-nth chaos processes.J. Funct. Anal., 248(1):1–26, 2007. 92

    work page 2007