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Forum Mathematicum

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Volume 30, Issue 5


The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces

Jiaxin Hu / Xuliang Li
Published Online: 2018-02-08 | DOI: https://doi.org/10.1515/forum-2017-0072


We apply the Davies method to prove that for any regular Dirichlet form on a metric measure space, an off-diagonal stable-like upper bound of the heat kernel is equivalent to the conjunction of the on-diagonal upper bound, a cutoff inequality on any two concentric balls, and the jump kernel upper bound, for any walk dimension. If in addition the jump kernel vanishes, that is, if the Dirichlet form is strongly local, we obtain a sub-Gaussian upper bound. This gives a unified approach to obtaining heat kernel upper bounds for both the non-local and the local Dirichlet forms.

Keywords: Heat kernel; Dirichlet form; cutoff inequality on balls; Davies method

MSC 2010: 35K08; 28A80; 60J35


  • [1]

    S. Andres and M. T. Barlow, Energy inequalities for cutoff functions and some applications, J. Reine Angew. Math. 699 (2015), 183–215. Web of ScienceGoogle Scholar

  • [2]

    M. T. Barlow, Diffusions on fractals, Lectures on Probability Theory and Statistics (Saint-Flour 1995), Lecture Notes in Math. 1690, Springer, Berlin (1998), 1–121. Google Scholar

  • [3]

    M. T. Barlow and R. F. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math. 51 (1999), no. 4, 673–744. CrossrefGoogle Scholar

  • [4]

    M. T. Barlow and R. F. Bass, Stability of parabolic Harnack inequalities, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1501–1533. CrossrefGoogle Scholar

  • [5]

    M. T. Barlow, R. F. Bass and T. Kumagai, Stability of parabolic Harnack inequalities on metric measure spaces, J. Math. Soc. Japan 58 (2006), no. 2, 485–519. CrossrefGoogle Scholar

  • [6]

    M. T. Barlow, R. F. Bass, T. Kumagai and A. Teplyaev, Uniquess of Brownian motion on Sierpinski carpets, J. Eur. Math. Soc. (JEMS) 12 (2010), 655–701. Google Scholar

  • [7]

    M. T. Barlow, A. Grigor’yan and T. Kumagai, Heat kernel upper bounds for jump processes and the first exit time, J. Reine Angew. Math. 626 (2009), 135–157. Web of ScienceGoogle Scholar

  • [8]

    M. T. Barlow, A. Grigor’yan and T. Kumagai, On the equivalence of parabolic Harnack inequalities and heat kernel estimates, J. Math. Soc. Japan 64 (2012), no. 4, 1091–1146. CrossrefWeb of ScienceGoogle Scholar

  • [9]

    M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields 79 (1988), no. 4, 543–623. CrossrefGoogle Scholar

  • [10]

    R. F. Bass and D. A. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2933–2953. CrossrefGoogle Scholar

  • [11]

    E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. Henri Poincaré Probab. Stat. 23 (1987), no. 2, 245–287. Google Scholar

  • [12]

    Z.-Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stochastic Process. Appl. 108 (2003), no. 1, 27–62. CrossrefGoogle Scholar

  • [13]

    Z.-Q. Chen and T. Kumagai, Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields 140 (2008), no. 1–2, 277–317. Web of ScienceGoogle Scholar

  • [14]

    Z.-Q. Chen, T. Kumagai and J. Wang, Stability of heat kernel estimates for symmetric jump processes on metric measure spaces, preprint (2016), https://arxiv.org/abs/1604.04035.

  • [15]

    E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math. 109 (1987), no. 2, 319–333. CrossrefGoogle Scholar

  • [16]

    M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, 2nd rev. extended ed., De Gruyter Stud. Math. 19, Walter de Gruyter, Berlin, 2011. Google Scholar

  • [17]

    A. Grigor’yan, Heat kernels and function theory on metric measure spaces, Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris 2002), Contemp. Math. 338, American Mathematical Society, Providence (2003), 143–172. Google Scholar

  • [18]

    A. Grigor’yan, E. Hu and J. Hu, Two sides estimates of heat kernels of non-local Dirichlet forms, preprint (2016), https://www.math.uni-bielefeld.de/~grigor/gcap.pdf.

  • [19]

    A. Grigor’yan, E. Hu and J. Hu, Lower estimates of heat kernels for non-local Dirichlet forms on metric measure spaces, J. Funct. Anal. 272 (2017), no. 8, 3311–3346. CrossrefWeb of ScienceGoogle Scholar

  • [20]

    A. Grigor’yan and J. Hu, Heat kernels and green functions on metric measure spaces, Canad. J. Math. 66 (2014), 641–699. CrossrefGoogle Scholar

  • [21]

    A. Grigor’yan and J. Hu, Upper bounds of heat kernels on doubling spaces, Mosc. Math. J. 14 (2014), no. 3, 505–563, 641–642. Google Scholar

  • [22]

    A. Grigor’yan, J. Hu and K.-S. Lau, Heat kernels on metric measure spaces and an application to semilinear elliptic equations, Trans. Amer. Math. Soc. 355 (2003), no. 5, 2065–2095. CrossrefGoogle Scholar

  • [23]

    A. Grigor’yan, J. Hu and K.-S. Lau, Estimates of heat kernels for non-local regular Dirichlet forms, Trans. Amer. Math. Soc. 366 (2014), no. 12, 6397–6441. CrossrefGoogle Scholar

  • [24]

    A. Grigor’yan, J. Hu and K.-S. Lau, Heat kernels on metric measure spaces, Geometry and Analysis of Fractals, Springer Proc. Math. Stat. 88, Springer, Heidelberg (2014), 147–207. Google Scholar

  • [25]

    A. Grigor’yan, J. Hu and K.-S. Lau, Generalized capacity, Harnack inequality and heat kernels of Dirichlet forms on metric measure spaces, J. Math. Soc. Japan 67 (2015), no. 4, 1485–1549. Web of ScienceCrossrefGoogle Scholar

  • [26]

    A. Grigor’yan and T. Kumagai, On the dichotomy in the heat kernel two sided estimates, Analysis on Graphs and its Applications, Proc. Sympos. Pure Math. 77, American Mathematical Society, Providence (2008), 199–210. Google Scholar

  • [27]

    A. Grigor’yan and A. Telcs, Two-sided estimates of heat kernels on metric measure spaces, Ann. Probab. 40 (2012), no. 3, 1212–1284. Web of ScienceCrossrefGoogle Scholar

  • [28]

    B. M. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. Lond. Math. Soc. (3) 78 (1999), no. 2, 431–458. CrossrefGoogle Scholar

  • [29]

    J. Hu and M. Zähle, Generalized Bessel and Riesz potentials on metric measure spaces, Potential Anal. 30 (2009), no. 4, 315–340. CrossrefWeb of ScienceGoogle Scholar

  • [30]

    J. Kigami, Volume doubling measures and heat kernel estimates on self-similar sets, Mem. Amer. Math. Soc. 932 (2009), 1–94. Google Scholar

  • [31]

    T. Kumagai, Some remarks for stable-like jump processes on fractals, Fractals in Graz 2001. Analysis, Dynamics, Geometry, Stochastics, Trends Math., Birkhäuser, Basel (2003), 185–196. Google Scholar

  • [32]

    U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123 (1994), no. 2, 368–421. CrossrefGoogle Scholar

  • [33]

    M. Murugan and L. Saloff-Coste, Heat kernel estimates for anomalous heavy-tailed random walks, preprint (2015), https://arxiv.org/abs/1512.02361.

  • [34]

    M. Murugan and L. Saloff-Coste, Davies’ method for anomalous diffusions, Proc. Amer. Math. Soc. 145 (2017), no. 4, 1793–1804. Web of ScienceGoogle Scholar

  • [35]

    A. Stós, Symmetric α-stable processes on d-sets, Bull. Pol. Acad. Sci. Math. 48 (2000), no. 3, 237–245. Google Scholar

About the article

Received: 2017-04-04

Revised: 2018-01-14

Published Online: 2018-02-08

Published in Print: 2018-09-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11371217

Funding Source: Ministry of Education of the People’s Republic of China

Award identifier / Grant number: 20130002110003

The first author was supported by NSFC no. 11371217, SRFDP no. 20130002110003.

Citation Information: Forum Mathematicum, Volume 30, Issue 5, Pages 1129–1155, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0072.

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