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Homogeneous Dirichlet Forms on p.c.f. Fractals and their Spectral Asymptotics

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Abstract

We formulate a class of “homogeneous” Dirichlet forms (DF) that aims to explore those forms that do not satisfy the conventional energy self-similar identity (degenerate DFs). This class of DFs has been studied in Hambly Jones (J. Theoret. Probab., 15, 285–322 2002), Hambly and Kumagai (Potential Anal., 8, 359–397 1998), Hambly and Yang (J. Fractal Geom., 6, 1–51 2019) and Hattori et al. (Probab. Theory Related Fields, 100, 85–116 1994) in connection with the asymptotically one-dimensional diffusions on the Sierpinski gaskets (SG) and their generalizations. In this paper, we give a systematic study of such DFs and their spectral properties. We also emphasize the construction of some new homogeneous DFs. Moreover, a basic assumption on the resistance growth that was required in Hambly and Kumagai (Potential Anal., 8, 359–397 1998) to investigate the heat kernel and the existence of the “non-fixed point” limiting diffusion is verified analytically.

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References

  1. Barlow, M.: Diffusions on Fractals, vol. 1690 of Lect. Notes Math., pp. 1–121. Springer (1998)

  2. Barlow, M., Bass, R.: Transition densities for Brownian motion on the Sierpiński carpet. Probab. Theory Related Fields 91, 307–330 (1992)

    Article  MathSciNet  Google Scholar 

  3. Brouwer, L.E.J.: Zur Invarianz des n-dimensionalen Gebiets. Math. Ann. 72, 55–56 (1912)

    Article  MathSciNet  Google Scholar 

  4. Davies, E.B.: One-parameter Semigroups. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London Mathematical Society Monographs, 15 London-New York (1980)

  5. Davies, E.B: Heat Kernel and Spectral Theory. Cambridge University Press (1989)

  6. Deng, Q.-R., Lau, K.-S.: Open set condition and post-critically finite self-similar sets. Nonlinearity 21, 1227–1232 (2008)

    Article  MathSciNet  Google Scholar 

  7. Doyle, P., Snell, J.: Random Walks and Electric Networks, Carus Mathematical Monographs, vol. 22. Mathematical Association of America, Washington, DC (1984)

    Book  Google Scholar 

  8. Falconer, K.: Techniques in Fractal Geometry. Wiley (1997)

  9. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, vol. 19. Walter de Gruyter Studies in Mathematics, Berlin (2011). Second revised and extended edition ed.

    Google Scholar 

  10. Gu, Q., Lau, K.-S.: Dirichlet forms and convergence of Besov norms on self-similar sets. Ann. Acad. Sci. Fenn. Math. 45, 1–22 (2020)

    Article  MathSciNet  Google Scholar 

  11. Gu, Q., Lau, K.-S.: Dirichlet forms and critical exponents on fractals. Trans. Amer. Math. Soc. 373, 1619–1652 (2020)

    Article  MathSciNet  Google Scholar 

  12. Gu, Q., Lau, K.-S., Qiu, H.: On a recursive construction of Dirichlet form on the Sierpiński gasket. J. Math. Anal. Appl. 474, 674–692 (2019)

    Article  MathSciNet  Google Scholar 

  13. Hambly, B., Jones, O.: Asymptotically one-dimensional diffusion on the Sierpiński gasket and multi-type branching processes with varying environment. J. Theoret. Probab. 15, 285–322 (2002)

    Article  MathSciNet  Google Scholar 

  14. Hambly, B., Kumagai, T.: Heat kernel estimates and homogenization for asymptotically lower-dimensional processes on some nested fractals. Potential Anal. 8, 359–397 (1998)

    Article  MathSciNet  Google Scholar 

  15. Hambly, B., Kumagai, T.: Transition density estimates for diffusion processes on post critically finite self-similar fractals. Proc. London Math. Soc. (3) 78, 431–458 (1999)

    Article  MathSciNet  Google Scholar 

  16. Hambly, B., Metz, V., Teplyaev, A.: Self-similar energies on post-critically finite self-similar fractals. J. London Math. Soc. (2) 74, 93–112 (2006)

    Article  MathSciNet  Google Scholar 

  17. Hambly, B., Yang, W.: Degenerate limits for one-parameter families of non-fixed-point diffusions on fractals. J. Fractal Geom. 6, 1–51 (2019)

    Article  MathSciNet  Google Scholar 

  18. Hattori, K., Hattori, T., Watanabe, H.: Asymptotically one-dimensional diffusions on the Sierpiński gasket and the abc-gaskets. Probab. Theory Related Fields 100, 85–116 (1994)

    Article  MathSciNet  Google Scholar 

  19. Hu, J., Wang, X.-S.: Domains of Dirichlet forms and effective resistance estimates on p.c.f. fractals. Studia Math. 177, 153–172 (2006)

    Article  MathSciNet  Google Scholar 

  20. Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)

    Article  MathSciNet  Google Scholar 

  21. Jonsson, A.: Brownian motion on fractals and function spaces. Math. Zeit. 222, 495–504 (1996)

    Article  MathSciNet  Google Scholar 

  22. Kigami, J.: A harmonic calculus on the Sierpinski spaces. Japan J. Appl. Math. 6, 259–290 (1989)

    Article  MathSciNet  Google Scholar 

  23. Kigami, J.: Analysis on Fractals. Cambridge Univ Press (2001)

  24. Kigami, J.: Resistance forms, quasisymmetric maps and heat kernel estimates. Mem. Amer. Math. Soc., 216(1015) (2012)

  25. Kigami, J., Lapidus, L.: Weyl’s problem for the spectral distribution of Laplacians on p.c.f.self-similar fractals. Comm. Math. Phys. 158, 93–125 (1993)

    Article  MathSciNet  Google Scholar 

  26. Lindstrøm, T.: Brownian motion on nested fractals. Mem. Amer. Math. Soc., 83(420) (1990)

  27. Peirone, R.: Existence of self-similar energies on finitely ramified fractals. J. Anal. Math. 123, 35–94 (2014)

    Article  MathSciNet  Google Scholar 

  28. Strichartz, R.: Differential Equations on Fractals: A Tutorial. Princeton University Press (2006)

  29. Tetenov, A., Kamalutdinov, K., Vaulin, D.: Self-similar Jordan arcs which do not satisfy OSC. arXiv:1512.00290

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Acknowledgements

The authors thank Professor Sze-Man Ngai for going through the manuscript and making some valuable suggestions to improve the presentation. They are also grateful to the anonymous referee for providing many nice comments which leads to the improvement of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Qingsong Gu & Hua Qiu

  2. Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, China

    Ka-Sing Lau

  3. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15217, USA

    Ka-Sing Lau

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  1. Qingsong Gu
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  2. Ka-Sing Lau
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Correspondence to Hua Qiu.

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Qingsong Gu was supported by the National Natural Science Foundation of China (Grant No.12101303 and 12171354). Hua Qiu was supported by the National Natural Science Foundation of China, grant 12071213, and the Natural Science Foundation of Jiangsu Province in China, grant BK20211142.

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Gu, Q., Lau, KS. & Qiu, H. Homogeneous Dirichlet Forms on p.c.f. Fractals and their Spectral Asymptotics. Potential Anal 60, 219–252 (2024). https://doi.org/10.1007/s11118-022-10048-3

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  • DOI: https://doi.org/10.1007/s11118-022-10048-3

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