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Heat Kernels and Green Functions on Metric Measure Spaces

Published online by Cambridge University Press:  20 November 2018

Alexander Grigor'yan  and
Jiaxin Hu
Alexander Grigor'yan
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany. e-mail: grigor@math.uni-bielefeld.de
Jiaxin Hu
Affiliation:
Department of Mathematical Sciences and Mathematical Sciences Center, Tsinghua University, Beijing 100084, China. e-mail: hujiaxin@mail.tsinghua.edu.cn
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Abstract

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We prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling property, the elliptic Harnack inequality, and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball that uses two-sided estimates of a Green function in a ball.

Keywords

35K0828A8031B0535J0846E3547D07Dirichlet formheat kernelGreen functioncapacity

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 3 , 01 June 2014 , pp. 641 - 699
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Armitage, D. H. and Gardiner, S. J., Classical potential theory. Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2001.Google Scholar
[2] Aronson, D. G., Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa. (3) 22(1968), 607–694; addendum 25(1971), 221–228.Google Scholar
[3] Barlow, M. T., Diffusions on fractals. In: Lectures on probability theory and statistics (Saint-Flour, 1995), Lecture Notes Math., 1690, Springer, Berlin, 1998, pp. 1–121.Google Scholar
[4] Barlow, M. T., Some remarks on the elliptic Harnack inequality. Bull. London Math. Soc. 37(2005), no. 2, 200–208. http://dx.doi.org/10.1112/S0024609304003893 Google Scholar
[5] Barlow, M. T. and Bass, R. F., Brownian motion and harmonic analysis on Sierpínski carpets. Canad. J. Math. 51(1999), no. 4, 673–744. http://dx.doi.org/10.4153/CJM-1999-031-4 Google Scholar
[6] Barlow, M. T., Bass, R. F., Chen, Z.-Q., and Kassmann, M., Non-local Dirichlet forms and symmetric jump processes. Trans. Amer. Math. Soc. 361(2009), no. 4, 1963–1999. http://dx.doi.org/10.1090/S0002-9947-08-04544-3 Google Scholar
[7] Barlow, M. T., Bass, R. F., Kumagai, T., and Teplyaev, A., Uniqueness of Brownian motion on Sierpinski carpets. J. Eur. Math. Soc. 12(2010), no. 3, 655–701.Google Scholar
[8] Barlow, M. T., Coulhon, T., and Kumagai, T., Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Comm. Pure Appl. Math. 58(2005), no. 12, 1642–1677. http://dx.doi.org/10.1002/cpa.20091 Google Scholar
[9] Barlow, M. T. and Perkins, E. A., Brownian motion on the Sierpinski gasket. Probab. Theory Related Fields 79(1988), no. 4, 543–623. http://dx.doi.org/10.1007/BF00318785 Google Scholar
[10] Bendikov, A. and Saloff-Coste, L., On-and-off diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces. Amer. J. Math. 122(2000), no. 6, 1205–1263. http://dx.doi.org/10.1353/ajm.2000.0043 Google Scholar
[11] Bliedtner, J. and Hansen, W., Potential theory. An analytic and probabilistic approach to balayage. Universitext, Springer-Verlag, Berlin, 1986.Google Scholar
[12] Carlen, E. A., Kusuoka, S., and Stroock, D.W., Upper bounds for symmetric Markov transition functions. Ann. Inst. Henri. Poincaré Probab. Statist. 23(1987), no. 2, 245–287.Google Scholar
[13] Chavel, I., Eigenvalues in Riemannian geometry. Pure and Applied Mathematics, 115, Academic Press, Orlando, FL, 1984.Google Scholar
[14] Coulhon, T., Heat kernel and isoperimetry on non-compact Riemannian manifolds. In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), Contemp. Math., 338, American Mathematical Society, Providence, RI, 2003, pp. 65–99.Google Scholar
[15] Davies, E. B., Heat kernels and spectral theory. Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1989.Google Scholar
[16] Fukushima, M., Oshima, Y., and Takeda, M., Dirichlet forms and symmetric Markov processes. Second revised and extended edition, de Gruyter Studies in Mathematics, 19, Walter de Gruyter & Co., Berlin, 2011.Google Scholar
[17] Grigor’yan, A., Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. (N.S.) 36(1999), no. 2, 135–249. http://dx.doi.org/10.1090/S0273-0979-99-00776-4 Google Scholar
[18] Grigor’yan, A., Heat kernels on weighted manifolds and applications. In: The ubiquitous heat kernel, Contemporary Mathematics, 398, American Mathematical Society, Providence, RI, 2006, pp. 93–191.Google Scholar
[19] Grigor’yan, A. and Hu, J., Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces. Invent. Math. 174(2008), no. 1, 81–126. http://dx.doi.org/10.1007/s00222-008-0135-9 Google Scholar
[20] Grigor’yan, A., Upper bounds of heat kernels on doubling spaces. preprint, 2008. http://www.math.uni-bielefeld.de/_grigor/wf/uehtm/ue8.html Google Scholar
[21] Grigor’yan, A., Hu, J., and Lau, K.-S., Comparison inequalities for heat semigroups and heat kernels on metric measure spaces. J. Funct. Anal. 259(2010), no. 10, 2613–2641. http://dx.doi.org/10.1016/j.jfa.2010.07.010 Google Scholar
[22] Grigor’yan, A., Estimates of heat kernels for non-local regular Dirichlet forms. Trans. Amer. Math. Soc., to appear. http://www.math.uni-bielefeld.de/~grigor/jump.pdf Google Scholar
[23] Grigor’yan, A., Netrusov, Y., and Yau, S.-T., Eigenvalues of elliptic operators and geometric applications. In: Surveys in differential geometry, IX, Int. Press, Somerville, MA, 2004, pp. 147–217.Google Scholar
[24] Grigor’yan, A. and Telcs, A., sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109(2001), no. 3, 451–510. http://dx.doi.org/10.1215/S0012-7094-01-10932-0 Google Scholar
[25] Grigor’yan, A., Harnack inequalities and sub-Gaussian estimates for random walks. Math. Ann. 324(2002), no. 3, 521–556. http://dx.doi.org/10.1007/s00208-002-0351-3 Google Scholar
[26] Grigor’yan, A., Two-sided estimates of heat kernels on metric measure spaces. Ann. Probab. 40(2012), no. 3, 1212–1284. http://dx.doi.org/10.1214/11-AOP645 Google Scholar
[27] Hambly, B. M. and Kumagai, T., Transition density estimates for diffusion processes on post critically finite self-similar fractals. Proc. London Math. Soc. 78(1999), no. 2, 431–458. http://dx.doi.org/10.1112/S0024611599001744 Google Scholar
[28] Hebisch, W. and Saloff-Coste, L., On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble) 51(2001), no. 5, 1437–1481. http://dx.doi.org/10.5802/aif.1861 Google Scholar
[29] Heinonen, J., Lectures on analysis on metric spaces. Universitext, Springer-Verlag, New York, 2001.Google Scholar
[30] Kigami, J., Analysis on fractals. Cambridge Tracts in Mathematics, 143, Cambridge University Press, Cambridge, 2001.Google Scholar
[31] Kigami, J., Local Nash inequality and inhomogenous of heat kernels. Proc. London Math. Soc. 89(2004), no. 2, 525–544. http://dx.doi.org/10.1112/S0024611504014807 Google Scholar
[32] Li, P. and Yau, S.-T., On the parabolic kernel of the Schrüdinger operator Acta Math. 156(1986), no. 3–4, 153–201. http://dx.doi.org/10.1007/BF02399203 Google Scholar
[33] Nash, J., Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80(1958), 931–954. http://dx.doi.org/10.2307/2372841 Google Scholar
[34] Porper, F. O. and Eidel’man, S. D., Two-side estimates of the fundamental solutions of second-order parabolic equations and some applications of them. (Russian) Uspekhi Mat. Nauk 39(1984), no. 3(237), 107—156.Google Scholar
[35] Port, S. C. and Stone, C. J., Brownian motion and classical potential theory. Probability and Mathematical Statistics, Academic Press, New York-London, 1978.Google Scholar
[36] Saloff-Coste, L., Aspects of Sobolev-type inequalities. London Mathematical Society Lecture Note Series, 289, Cambridge University Press, Cambridge, 2002.Google Scholar
[37] Saloff-Coste, L., A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices 1992, no. 2, 27–38.Google Scholar
[38] Schoen, R. and -T Yau, S., Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994.Google Scholar
[39] Stroock, D.W., Estimates on the heat kernel for the second order divergence form operators. In: Probability theory. (Singapore, 1989), de Gruyter, Berlin, 1992, pp. 29–44.Google Scholar
[40] Schmuland, B., On the local property for positivity preserving coercive forms. In: Dirichlet forms and stochastic processes (Beijing, 1993), de Gruyter, Berlin, 1995, pp. 345–354.Google Scholar
[41] Varopoulos, N. Th., Hardy-Littlewood theory for semigroups. J. Funct. Anal. 63(1985), no. 2, 240–260. http://dx.doi.org/10.1016/0022-1236(85)90087-4 Google Scholar