Local and non-local Dirichlet forms on the Sierpiński carpet
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- by Alexander Grigor’yan and Meng Yang PDF
- Trans. Amer. Math. Soc. 372 (2019), 3985-4030 Request permission
Abstract:
We give a purely analytic construction of a self-similar local regular Dirichlet form on the Sierpiński carpet using approximation of stable-like non-local closed forms which gives an answer to an open problem in analysis on fractals.References
- Sebastian Andres and Martin T. Barlow, Energy inequalities for cutoff functions and some applications, J. Reine Angew. Math. 699 (2015), 183–215. MR 3305925, DOI 10.1515/crelle-2013-0009
- 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, DOI 10.1007/BFb0092537
- M. T. Barlow, Analysis on the Sierpinski carpet, Analysis and geometry of metric measure spaces, CRM Proc. Lecture Notes, vol. 56, Amer. Math. Soc., Providence, RI, 2013, pp. 27–53. MR 3060498, DOI 10.1090/crmp/056/02
- Martin T. Barlow and Richard F. Bass, The construction of Brownian motion on the Sierpiński carpet, Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), no. 3, 225–257 (English, with French summary). MR 1023950
- M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. London Ser. A 431 (1990), no. 1882, 345–360. MR 1080496, DOI 10.1098/rspa.1990.0135
- Martin T. Barlow and Richard F. Bass, Transition densities for Brownian motion on the Sierpiński carpet, Probab. Theory Related Fields 91 (1992), no. 3-4, 307–330. MR 1151799, DOI 10.1007/BF01192060
- 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, DOI 10.4153/CJM-1999-031-4
- Martin T. Barlow, Richard F. Bass, Takashi Kumagai, and Alexander Teplyaev, Uniqueness of Brownian motion on Sierpiński carpets, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3, 655–701. MR 2639315, DOI 10.4171/jems/211
- M. T. Barlow, R. F. Bass, and J. D. Sherwood, Resistance and spectral dimension of Sierpiński carpets, J. Phys. A 23 (1990), no. 6, L253–L258. MR 1048762, DOI 10.1088/0305-4470/23/6/004
- 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, DOI 10.1002/cpa.20091
- Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields 79 (1988), no. 4, 543–623. MR 966175, DOI 10.1007/BF00318785
- Zhen-Qing Chen and Masatoshi Fukushima, Symmetric Markov processes, time change, and boundary theory, London Mathematical Society Monographs Series, vol. 35, Princeton University Press, Princeton, NJ, 2012. MR 2849840
- Gianni Dal Maso, An introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1201152, DOI 10.1007/978-1-4612-0327-8
- 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
- Alexander Grigor’yan and Jiaxin Hu, Heat kernels and Green functions on metric measure spaces, Canad. J. Math. 66 (2014), no. 3, 641–699. MR 3194164, DOI 10.4153/CJM-2012-061-5
- Alexander Grigor’yan and Jiaxin Hu, Upper bounds of heat kernels on doubling spaces, Mosc. Math. J. 14 (2014), no. 3, 505–563, 641–642 (English, with English and Russian summaries). MR 3241758, DOI 10.17323/1609-4514-2014-14-3-505-563
- 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. MR 1953538, DOI 10.1090/S0002-9947-03-03211-2
- 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, DOI 10.1016/j.jfa.2010.07.010
- 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, DOI 10.1090/S0002-9947-2014-06034-0
- 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, DOI 10.2969/jmsj/06741485
- Alexander Grigor′yan and Andras Telcs, Sub-Gaussian estimates of heat kernels on infinite graphs, Duke Math. J. 109 (2001), no. 3, 451–510. MR 1853353, DOI 10.1215/S0012-7094-01-10932-0
- Alexander Grigor’yan and András Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, Math. Ann. 324 (2002), no. 3, 521–556. MR 1938457, DOI 10.1007/s00208-002-0351-3
- Alexander Grigor’yan and Andras Telcs, Two-sided estimates of heat kernels on metric measure spaces, Ann. Probab. 40 (2012), no. 3, 1212–1284. MR 2962091, DOI 10.1214/11-AOP645
- Masanori Hino and Takashi Kumagai, A trace theorem for Dirichlet forms on fractals, J. Funct. Anal. 238 (2006), no. 2, 578–611. MR 2253734, DOI 10.1016/j.jfa.2006.05.012
- Ben M. Hambly, Takashi Kumagai, Shigeo Kusuoka, and Xian Yin Zhou, Transition density estimates for diffusion processes on homogeneous random Sierpinski carpets, J. Math. Soc. Japan 52 (2000), no. 2, 373–408. MR 1742797, DOI 10.2969/jmsj/05220373
- Jiaxin Hu, An introduction to the fractal analysis, Science Press, 2013.
- 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, DOI 10.4064/sm177-2-5
- Alf Jonsson, Brownian motion on fractals and function spaces, Math. Z. 222 (1996), no. 3, 495–504. MR 1400205, DOI 10.1007/PL00004543
- Jun Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc. 335 (1993), no. 2, 721–755. MR 1076617, DOI 10.1090/S0002-9947-1993-1076617-1
- Jun Kigami, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR 1840042, DOI 10.1017/CBO9780511470943
- Jun Kigami, Harmonic analysis for resistance forms, J. Funct. Anal. 204 (2003), no. 2, 399–444. MR 2017320, DOI 10.1016/S0022-1236(02)00149-0
- Jun Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc. 216 (2012), no. 1015, vi+132. MR 2919892, DOI 10.1090/S0065-9266-2011-00632-5
- Takashi Kumagai and Karl-Theodor Sturm, Construction of diffusion processes on fractals, $d$-sets, and general metric measure spaces, J. Math. Kyoto Univ. 45 (2005), no. 2, 307–327. MR 2161694, DOI 10.1215/kjm/1250281992
- Shigeo Kusuoka and Zhou Xian Yin, Dirichlet forms on fractals: Poincaré constant and resistance, Probab. Theory Related Fields 93 (1992), no. 2, 169–196. MR 1176724, DOI 10.1007/BF01195228
- Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 83 (1990), no. 420, iv+128. MR 988082, DOI 10.1090/memo/0420
- I. McGillivray, Resistance in higher-dimensional Sierpiński carpets, Potential Anal. 16 (2002), no. 3, 289–303. MR 1885765, DOI 10.1023/A:1014035414658
- Katarzyna Pietruska-Pałuba, Some function spaces related to the Brownian motion on simple nested fractals, Stochastics Stochastics Rep. 67 (1999), no. 3-4, 267–285. MR 1729479, DOI 10.1080/17442509908834214
- Katarzyna Pietruska-Pałuba, On function spaces related to fractional diffusions on $d$-sets, Stochastics Stochastics Rep. 70 (2000), no. 3-4, 153–164. MR 1800954, DOI 10.1080/17442500008834250
- Katarzyna Pietruska-Pałuba, Limiting behaviour of Dirichlet forms for stable processes on metric spaces, Bull. Pol. Acad. Sci. Math. 56 (2008), no. 3-4, 257–266. MR 2481978, DOI 10.4064/ba56-3-8
- Meng Yang, Equivalent semi-norms of non-local Dirichlet forms on the Sierpiński gasket and applications, Potential Anal. 49 (2018), no. 2, 287–308. MR 3824963, DOI 10.1007/s11118-017-9657-3
Additional Information
- Alexander Grigor’yan
- Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
- MR Author ID: 203816
- Email: grigor@math.uni-bielefeld.de
- Meng Yang
- Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
- Address at time of publication: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
- MR Author ID: 1226000
- Email: ymeng@math.uni-bielefeld.de, meng-yang13@mails.tsinghua.edu.cn
- Received by editor(s): August 21, 2017
- Received by editor(s) in revised form: July 12, 2018
- Published electronically: May 9, 2019
- Additional Notes: The authors were supported by SFB701 and SFB1283 of the German Research Council (DFG)
The second author is the corresponding author - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 3985-4030
- MSC (2010): Primary 28A80
- DOI: https://doi.org/10.1090/tran/7753
- MathSciNet review: 4009425