Journal of the Mathematical Society of Japan

On the equivalence of parabolic Harnack inequalities and heat kernel estimates

Martin T. BARLOW, Alexander GRIGOR'YAN, and Takashi KUMAGAI

Full-text: Open access

Abstract

We prove the equivalence of parabolic Harnack inequalities and sub-Gaussian heat kernel estimates in a general metric measure space with a local regular Dirichlet form.

Article information

Source
J. Math. Soc. Japan, Volume 64, Number 4 (2012), 1091-1146.

Dates
First available in Project Euclid: 29 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1351516770

Digital Object Identifier
doi:10.2969/jmsj/06441091

Mathematical Reviews number (MathSciNet)
MR2998918

Zentralblatt MATH identifier
1281.58016

Subjects
Primary: 58J35: Heat and other parabolic equation methods
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 31C25: Dirichlet spaces

Keywords
Harnack inequality heat kernel estimate caloric function metric measure space volume doubling Dirichlet space

Citation

BARLOW, Martin T.; GRIGOR'YAN, Alexander; KUMAGAI, Takashi. On the equivalence of parabolic Harnack inequalities and heat kernel estimates. J. Math. Soc. Japan 64 (2012), no. 4, 1091--1146. doi:10.2969/jmsj/06441091. https://projecteuclid.org/euclid.jmsj/1351516770


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References

  • D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., 73 (1967), 890–896.
  • D. G. Aronson and J. Serrin, Local behaviour of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal., 25 (1967), 81–122.
  • M. T. Barlow, Diffusions on fractals, Lectures on probability theory and statistics, Lecture Notes in Math., 1690, Springer, Berlin, 1998, 1–121.
  • M. T. Barlow and R. F. Bass, The construction of the Brownian motion on the Sierpiński carpet, Ann. Inst. H. Poincaré Probab. Statist., 25 (1989), 225–257.
  • M. T. Barlow, R. F. Bass, Z.-Q. Chen and M. Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc., 361 (2009), 1963–1999.
  • M. T. Barlow, R. F. Bass and T. Kumagai, Stability of parabolic Harnack inequalities on metric measure spaces, J. Math. Soc. Japan, 58 (2006), 485–519.
  • M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields, 79 (1988), 543–623.
  • E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), 245–287.
  • T. Coulhon and A. Grigor'yan, On-diagonal lower bounds for heat kernels and Markov chains, Duke Math. J., 89 (1997), 133–199.
  • E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math., 92, Cambridge University Press, 1989.
  • E. B. Fabes and D. W. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal., 96 (1986), 327–338.
  • P. J. Fitzsimmons, B. M. Hambly and T. Kumagai, Transition density estimates for Brownian motion on affine nested fractals, Comm. Math. Phys., 165 (1994), 595–620.
  • M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter Stud. Math., 19, Walter de Gruyter & Co., Berlin, 1994.
  • A. Grigor'yan, The heat equation on non-compact Riemannian manifolds, Mat. Sb., 182 (1991), 55–87 (in Russian), Engl. transl. in Math. USSR-Sb., 72 (1992), 47–77.
  • A. Grigor'yan and J. Hu, Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces, Invent. Math., 174 (2008), 81–126.
  • A. Grigor'yan and J. Hu, Upper bounds of heat kernels on doubling spaces, preprint.
  • A. Grigor'yan, J. Hu and K.-S. Lau, Obtaining upper bounds of heat kernels from lower bounds, Comm. Pure Appl. Math., 61 (2008), 639–660.
  • A. Grigor'yan, J. Hu and K.-S. Lau, Heat kernels on metric spaces with doubling measure, In: Fractal Geometry and Stochastics IV, Progr. Probab, 61, Birkhäuser, 2009, pp.,3–44.
  • A. Grigor'yan and A. Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, Math. Ann., 324 (2002), 521–556.
  • A. K. Gushchin, Uniform stabilization of solutions of the second mixed problem for a parabolic equation, Mat. Sb. (N.S.), 119 (1982), 451–508 (in Russian), Engl. transl. in Math. USSR-Sb., 47 (1984), 439–498.
  • B. M. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc. (3), 78 (1999), 431–458.
  • W. Hebisch and L. Saloff-Coste, On the relation between elliptic and parabolic Harnack inequalities, Ann. Inst. Fourier (Grenoble), 51 (2001), 1437–1481.
  • J. Hu, On parabolic Harnack inequality, unpublished manuscript.
  • N. V. Krylov and M. V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161–175 (in Russian), Engl. transl. in Math. USSR-Izv., 16 (1981), 151–164.
  • E. M. Landis, Second order equations of elliptic and parabolic type, Nauka, Moscow, 1971 (in Russian), Engl. transl. in Transl. Math. Monogr., 171, Amer. Math. Soc., Providence RI, 1998.
  • P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153–201.
  • W. Littman, G. Stampacchia and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 43–77.
  • J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577–591.
  • J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., 17 (1964), 101–134.
  • Y. Oshima, On a construction of Markov processes associated with time dependent Dirichlet spaces, Forum Math., 4 (1992), 395–415.
  • Y. Oshima, Time-dependent Dirichlet forms and related stochastic calculus, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 7 (2004), 281–316.
  • Y. Oshima, Lecture notes on Dirichlet forms, Unpublished lecture notes.
  • L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices, 1992, 27–38.
  • L. Saloff-Coste, Aspects of Sobolev-Type Inequalities, London Math. Soc. Lecture Note Ser., 289, Cambridge University Press, 2002.
  • K.-T. Sturm, Analysis on local Dirichlet spaces, III, The parabolic Harnack inequality, J. Math. Pures Appl. (9), 75 (1996), 273–297.
  • J.-A. Yan, A formula for densities of transition functions, Séminaire de Probabilités, XXII, Lecture Notes in Math., 1321, Springer-Verlag, Berlin, 1988, pp.,92–100.