Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Davies' method for anomalous diffusions


Authors: Mathav Murugan and Laurent Saloff-Coste
Journal: Proc. Amer. Math. Soc. 145 (2017), 1793-1804
MSC (2010): Primary 60J60, 60J35
DOI: https://doi.org/10.1090/proc/13324
Published electronically: October 18, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Davies' method of perturbed semigroups is a classical technique to obtain off-diagonal upper bounds on the heat kernel. However Davies' method does not apply to anomalous diffusions due to the singularity of energy measures. In this note, we overcome the difficulty by modifying the Davies' perturbation method to obtain sub-Gaussian upper bounds on the heat kernel. Our computations closely follow the seminal work of Carlen, Kusuoka and Stroock (1987). However, a cutoff Sobolev inequality due to Andres and Barlow (2015) is used to bound the energy measure.


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

  • [1] Sebastian Andres and Martin T. Barlow, Energy inequalities for cutoff functions and some applications, J. Reine Angew. Math. 699 (2015), 183-215. MR 3305925, https://doi.org/10.1515/crelle-2013-0009
  • [2] D. Bakry, T. Coulhon, M. Ledoux, and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995), no. 4, 1033-1074. MR 1386760, https://doi.org/10.1512/iumj.1995.44.2019
  • [3] 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, https://doi.org/10.1007/BFb0092537
  • [4] Martin T. Barlow, Heat kernels and sets with fractal structure, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 11-40. MR 2039950, https://doi.org/10.1090/conm/338/06069
  • [5] Martin T. Barlow and Richard F. Bass, Stability of parabolic Harnack inequalities, Trans. Amer. Math. Soc. 356 (2004), no. 4, 1501-1533 (electronic). MR 2034316, https://doi.org/10.1090/S0002-9947-03-03414-7
  • [6] Martin T. Barlow, Richard F. Bass, and Takashi Kumagai, Stability of parabolic Harnack inequalities on metric measure spaces, J. Math. Soc. Japan 58 (2006), no. 2, 485-519. MR 2228569
  • [7] 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, https://doi.org/10.1515/CRELLE.2009.005
  • [8] Oren Ben-Bassat, Robert S. Strichartz, and Alexander Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal. 166 (1999), no. 2, 197-217. MR 1707752, https://doi.org/10.1006/jfan.1999.3431
  • [9] 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
  • [10] Z.-Q. Chen, T. Kumagai, and J. Wang, Stability of heat kernel estimates for symmetric jump processes on metric measure spaces, arxiv:1604.04035
  • [11] E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math. 109 (1987), no. 2, 319-333. MR 882426, https://doi.org/10.2307/2374577
  • [12] E. B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995), no. 1, 141-169. MR 1346221, https://doi.org/10.1006/jfan.1995.1103
  • [13] Masatoshi Fukushima, Yōichi Ōshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR 1303354
  • [14] Alexander Grigor'yan, Jiaxin Hu, and Ka-Sing Lau, Estimates of heat kernels for non-local regular Dirichlet forms, Trans. Amer. Math. Soc. 366 (2014), no. 12, 6397-6441. MR 3267014, https://doi.org/10.1090/S0002-9947-2014-06034-0
  • [15] Alexander Grigor'yan, Jiaxin Hu, and Ka-Sing 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. MR 3417504, https://doi.org/10.2969/jmsj/06741485
  • [16] Masanori Hino, On singularity of energy measures on self-similar sets, Probab. Theory Related Fields 132 (2005), no. 2, 265-290. MR 2199293, https://doi.org/10.1007/s00440-004-0396-1
  • [17] Masanori Hino, On short time asymptotic behavior of some symmetric diffusions on general state spaces, Potential Anal. 16 (2002), no. 3, 249-264. MR 1885762, https://doi.org/10.1023/A:1014033208581
  • [18] J. Hu and X. Li, The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces, arxiv:1605.05548
  • [19] Takashi Kumagai, Recent developments of analysis on fractals [MR 2105988], Selected papers on analysis and related topics, Amer. Math. Soc. Transl. Ser. 2, vol. 223, Amer. Math. Soc., Providence, RI, 2008, pp. 81-95. MR 2441420
  • [20] Shigeo Kusuoka, Dirichlet forms on fractals and products of random matrices, Publ. Res. Inst. Math. Sci. 25 (1989), no. 4, 659-680. MR 1025071, https://doi.org/10.2977/prims/1195173187
  • [21] J. Lierl, Parabolic Harnack inequality on fractal-like metric measure Dirichlet spaces, arxiv:1509.04804v2
  • [22] Mathav Murugan and Laurent Saloff-Coste, Anomalous threshold behavior of long range random walks, Electron. J. Probab. 20 (2015), no. 74, 21. MR 3371433, https://doi.org/10.1214/EJP.v20-3989
  • [23] M. Murugan and L. Saloff-Coste, Heat kernel estimates for anomalous heavy-tailed random walks, arxiv:1512.02361

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 60J60, 60J35

Retrieve articles in all journals with MSC (2010): 60J60, 60J35


Additional Information

Mathav Murugan
Affiliation: Department of Mathematics, University of British Columbia and Pacific Institute for the Mathematical Sciences, Vancouver, British Columbia V6T 1Z2, Canada
Email: mathav@math.ubc.ca

Laurent Saloff-Coste
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
Email: lsc@math.cornell.edu

DOI: https://doi.org/10.1090/proc/13324
Received by editor(s): February 3, 2016
Received by editor(s) in revised form: May 26, 2016, and June 6, 2016
Published electronically: October 18, 2016
Additional Notes: Both authors were partially supported by NSF grant DMS 1404435
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society