Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


Estimates of heat kernels for non-local regular Dirichlet forms

Authors: Alexander Grigor’yan, Jiaxin Hu and Ka-Sing Lau
Journal: Trans. Amer. Math. Soc. 366 (2014), 6397-6441
MSC (2010): Primary 47D07; Secondary 28A80, 60J35
Published electronically: July 24, 2014
MathSciNet review: 3267014
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we present new heat kernel upper bounds for a certain class of non-local regular Dirichlet forms on metric measure spaces, including fractal spaces. We use a new purely analytic method where one of the main tools is the parabolic maximum principle. We deduce an off-diagonal upper bound of the heat kernel from the on-diagonal one under the volume regularity hypothesis, restriction of the jump kernel and the survival hypothesis. As an application, we obtain two-sided estimates of heat kernels for non-local regular Dirichlet forms with finite effective resistance, including settings with the walk dimension greater than $ 2$.

References [Enhancements On Off] (What's this?)

  • [1] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607-694. MR 0435594 (55 #8553)
  • [2] Martin T. Barlow, Diffusions on fractals, Lectures on probability theory and statistics (Saint-Flour, 1995) Lecture Notes in Math., vol. 1690, Springer, Berlin, 1998, pp. 1-121. MR 1668115 (2000a:60148),
  • [3] Martin T. Barlow and Richard F. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math. 51 (1999), no. 4, 673-744. MR 1701339 (2000i:60083),
  • [4] Martin T. Barlow, Richard F. Bass, Zhen-Qing Chen, and Moritz Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc. 361 (2009), no. 4, 1963-1999. MR 2465826 (2010e:60163),
  • [5] Martin T. Barlow, Thierry Coulhon, and Takashi Kumagai, Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs, Comm. Pure Appl. Math. 58 (2005), no. 12, 1642-1677. MR 2177164 (2006i:60106),
  • [6] Martin T. Barlow, Alexander Grigoryan, and Takashi Kumagai, Heat kernel upper bounds for jump processes and the first exit time, J. Reine Angew. Math. 626 (2009), 135-157. MR 2492992 (2009m:58077),
  • [7] Richard F. Bass and David A. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2933-2953 (electronic). MR 1895210 (2002m:60132),
  • [8] E. A. Carlen, S. Kusuoka, and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, suppl., 245-287 (English, with French summary). MR 898496 (88i:35066)
  • [9] Zhen-Qing Chen and Takashi Kumagai, Heat kernel estimates for stable-like processes on $ d$-sets, Stochastic Process. Appl. 108 (2003), no. 1, 27-62. MR 2008600 (2005d:60135),
  • [10] Zhen-Qing Chen and Takashi 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. MR 2357678 (2009e:60186),
  • [11] Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, Second revised and extended edition, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 2011. MR 2778606 (2011k:60249)
  • [12] Alexander Grigoryan, 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. MR 1659871 (99k:58195),
  • [13] Alexander Grigor'yan, Heat kernels and function theory on metric measure spaces (Paris, 2002), Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 143-172. MR 2039954 (2005c:60096),
  • [14] Alexander Grigoryan, Heat kernels on weighted manifolds and applications, The ubiquitous heat kernel, Contemp. Math., vol. 398, Amer. Math. Soc., Providence, RI, 2006, pp. 93-191. MR 2218016 (2007a:58028),
  • [15] Alexander Grigoryan and Jiaxin Hu, Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces, Invent. Math. 174 (2008), no. 1, 81-126. MR 2430977 (2009g:58029),
  • [16] A. Grigor'yan and J. Hu, Upper bounds of heat kernels on doubling spaces, Moscow Math. J. 14 (2014), no. 3, 505-563.
  • [17] Alexander Grigor'yan, Jiaxin Hu, and Ka-Sing Lau, Heat kernels on metric measure spaces and an application to semilinear elliptic equations, Trans. Amer. Math. Soc. 355 (2003), no. 5, 2065-2095 (electronic). MR 1953538 (2003j:60103),
  • [18] Alexander Grigor'yan, Jiaxin Hu, and Ka-Sing Lau, Comparison inequalities for heat semigroups and heat kernels on metric measure spaces, J. Funct. Anal. 259 (2010), no. 10, 2613-2641. MR 2679020 (2012c:58059),
  • [19] Alexander Grigor'yan, Xueping Huang, and Jun Masamune, On stochastic completeness of jump processes, Math. Z. 271 (2012), no. 3-4, 1211-1239. MR 2945605,
  • [20] Alexander Grigoryan and Takashi Kumagai, On the dichotomy in the heat kernel two sided estimates, Analysis on graphs and its applications, Proc. Sympos. Pure Math., vol. 77, Amer. Math. Soc., Providence, RI, 2008, pp. 199-210. MR 2459870 (2010j:31012)
  • [21] 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), no. 2, 431-458. MR 1665249 (99m:60118),
  • [22] Ben M. Hambly and Takashi Kumagai, Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries, Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, Proc. Sympos. Pure Math., vol. 72, Amer. Math. Soc., Providence, RI, 2004, pp. 233-259. MR 2112125 (2005k:60141)
  • [23] Jiaxin Hu, An analytical approach to heat kernel estimates on strongly recurrent metric spaces, Proc. Edinb. Math. Soc. (2) 51 (2008), no. 1, 171-199. MR 2391637 (2009a:58026),
  • [24] Jiaxin Hu and Takashi Kumagai, Nash-type inequalities and heat kernels for non-local Dirichlet forms, Kyushu J. Math. 60 (2006), no. 2, 245-265. MR 2268236 (2008d:60102),
  • [25] Jiaxin Hu and Xingsheng Wang, Domains of Dirichlet forms and effective resistance estimates on p.c.f. fractals, Studia Math. 177 (2006), no. 2, 153-172. MR 2285238 (2009j:28021),
  • [26] J. Hu and M. Zähle, Generalized Bessel and Riesz potentials on metric measure spaces, Potential Anal. 30 (2009), no. 4, 315-340. MR 2491456 (2010j:31014),
  • [27] Jun Kigami, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR 1840042 (2002c:28015)
  • [28] Jun Kigami, Harmonic analysis for resistance forms, J. Funct. Anal. 204 (2003), no. 2, 399-444. MR 2017320 (2004m:31010),
  • [29] Jun Kigami, Volume doubling measures and heat kernel estimates on self-similar sets, Mem. Amer. Math. Soc. 199 (2009), no. 932, viii+94. MR 2512802 (2010e:28007)
  • [30] Takashi Kumagai, Estimates of transition densities for Brownian motion on nested fractals, Probab. Theory Related Fields 96 (1993), no. 2, 205-224. MR 1227032 (94e:60068),
  • [31] Takashi Kumagai, Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms, Publ. Res. Inst. Math. Sci. 40 (2004), no. 3, 793-818. MR 2074701 (2005g:60122)
  • [32] Shigeo Kusuoka, A diffusion process on a fractal, Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), Academic Press, Boston, MA, 1987, pp. 251-274. MR 933827 (89e:60149)
  • [33] Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153-201. MR 834612 (87f:58156),
  • [34] Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 83 (1990), no. 420, iv+128. MR 988082 (90k:60157)
  • [35] Jun Masamune and Toshihiro Uemura, Conservation property of symmetric jump processes, Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 3, 650-662 (English, with English and French summaries). MR 2841069 (2012f:60291),
  • [36] Andrzej Stós, Symmetric $ \alpha $-stable processes on $ d$-sets, Bull. Polish Acad. Sci. Math. 48 (2000), no. 3, 237-245. MR 1779007 (2002f:60152)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 47D07, 28A80, 60J35

Retrieve articles in all journals with MSC (2010): 47D07, 28A80, 60J35

Additional Information

Alexander Grigor’yan
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

Jiaxin Hu
Affiliation: Department of Mathematical Sciences, and Mathematical Sciences Center, Tsinghua University, Beijing 100084, People’s Republic of China

Ka-Sing Lau
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Keywords: Heat kernel, non-local Dirichlet form, effective resistance
Received by editor(s): September 23, 2011
Received by editor(s) in revised form: November 13, 2012
Published electronically: July 24, 2014
Additional Notes: The first author was supported by SFB 701 of the German Research Council (DFG) and the Grants from the Department of Mathematics and IMS of CUHK
The second author was supported by NSFC (Grant No. 11071138, 11271122), SFB 701 and the HKRGC Grant of CUHK. The second author is the corresponding author
The third author was supported by the HKRGC Grant, the Focus Investment Scheme of CUHK, and also by the NSFC (no. 11171100, 11371382)
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society