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Heat Kernel Fluctuations for Stochastic Processes on Fractals and Random Media

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From Classical Analysis to Analysis on Fractals

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

It is well known that stochastic processes on fractal spaces or in certain random media exhibit anomalous heat kernel behaviour. One manifestation of such irregular behaviour is the presence of fluctuations in the short- or long-time asymptotics of the on-diagonal heat kernel. In this note, we review some examples for which such fluctuations are known to occur, including Brownian motion on certain deterministic or random fractals, and simple random walks on various examples of random graph trees, such as the incipient infinite cluster of critical percolation on a regular tree and low-dimensional uniform spanning trees. We also announce some new results that add the one-dimensional Bouchaud trap model to this class of examples.

In memory of Professor Robert Strichartz

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References

  1. S. Andres, D. A. Croydon, and T. Kumagai. Heat kernel fluctuations and quantitative homogenization for the one-dimensional Bouchaud trap model. In preparation, 2022.

    Google Scholar 

  2. O. Angel, D. A. Croydon, S. Hernandez-Torres, and D. Shiraishi. Scaling limits of the three-dimensional uniform spanning tree and associated random walk. Ann. Probab., 49(6):3032–3105, 2021.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. T. Barlow. Diffusions on fractals. In Lectures on probability theory and statistics (Saint-Flour, 1995), volume 1690 of Lecture Notes in Math., pages 1–121. Springer, Berlin, 1998.

    Book  Google Scholar 

  4. M. T. Barlow, D. A. Croydon, and T. Kumagai. Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree. Ann. Probab., 45(1):4–55, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. T. Barlow, D. A. Croydon, and T. Kumagai. Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree. Probab. Theory Related Fields, 181(1–3):57–111, 2021.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. T. Barlow and B. M. Hambly. Transition density estimates for Brownian motion on scale irregular Sierpinski gaskets. Ann. Inst. H. Poincaré Probab. Statist., 33(5):531–557, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. T. Barlow and J. Kigami. Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets. J. London Math. Soc. (2), 56(2):320–332, 1997.

    Google Scholar 

  8. M. T. Barlow and T. Kumagai. Random walk on the incipient infinite cluster on trees. Illinois J. Math., 50(1–4):33–65, 2006.

    MathSciNet  MATH  Google Scholar 

  9. M. T. Barlow and R. Masson. Spectral dimension and random walks on the two dimensional uniform spanning tree. Comm. Math. Phys., 305(1):23–57, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Ben Arous, M. Cabezas, J. Černý, and R. Royfman. Randomly trapped random walks. Ann. Probab., 43(5):2405–2457, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Ben Arous and T. Kumagai. Large deviations of Brownian motion on the Sierpinski gasket. Stochastic Process. Appl., 85(2):225–235, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  12. G. Ben Arous and J. Černý. Bouchaud’s model exhibits two different aging regimes in dimension one. Ann. Appl. Probab., 15(2):1161–1192, 2005.

    MathSciNet  MATH  Google Scholar 

  13. D. A. Croydon. Heat kernel fluctuations for a resistance form with non-uniform volume growth. Proc. Lond. Math. Soc. (3), 94(3):672–694, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. A. Croydon. Volume growth and heat kernel estimates for the continuum random tree. Probab. Theory Related Fields, 140(1–2):207–238, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. A. Croydon. Scaling limits of stochastic processes associated with resistance forms. Ann. Inst. Henri Poincaré Probab. Stat., 54(4):1939–1968, 2018.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. A. Croydon and B. Hambly. Self-similarity and spectral asymptotics for the continuum random tree. Stochastic Process. Appl., 118(5):730–754, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  17. D. A. Croydon, B. Hambly, and T. Kumagai. Time-changes of stochastic processes associated with resistance forms. Electron. J. Probab., 22:Paper No. 82, 41, 2017.

    Google Scholar 

  18. D. A. Croydon, B. M. Hambly, and T. Kumagai. Heat kernel estimates for FIN processes associated with resistance forms. Stochastic Process. Appl., 129(9):2991–3017, 2019.

    Article  MathSciNet  MATH  Google Scholar 

  19. D. A. Croydon and T. Kumagai. Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive. Electron. J. Probab., 13:no. 51, 1419–1441, 2008.

    Google Scholar 

  20. D. A. Croydon and D. Shiraishi. Scaling limit for random walk on the range of random walk in four dimensions. Ann. Inst. Henri Poincaré Probab. Stat., 59(1):166–184, 2023.

    Article  MathSciNet  MATH  Google Scholar 

  21. L. R. G. Fontes, M. Isopi, and C. M. Newman. Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension. Ann. Probab., 30(2):579–604, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  22. I. Fujii and T. Kumagai. Heat kernel estimates on the incipient infinite cluster for critical branching processes. In Proceedings of RIMS Workshop on Stochastic Analysis and Applications, RIMS Kôkyûroku Bessatsu, B6, pages 85–95. Res. Inst. Math. Sci. (RIMS), Kyoto, 2008.

    Google Scholar 

  23. M. Fukushima and T. Shima. On a spectral analysis for the Sierpiński gasket. Potential Anal., 1(1):1–35, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  24. P. J. Grabner and W. Woess. Functional iterations and periodic oscillations for simple random walk on the Sierpiński graph. Stochastic Process. Appl., 69(1):127–138, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  25. B. M. Hambly. Brownian motion on a homogeneous random fractal. Probab. Theory Related Fields, 94(1):1–38, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  26. B. M. Hambly. Brownian motion on a random recursive Sierpinski gasket. Ann. Probab., 25(3):1059–1102, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  27. B. M. Hambly. On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets. Probab. Theory Related Fields, 117(2):221–247, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  28. B. M. Hambly and T. Kumagai. Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gaskets. Stochastic Process. Appl., 92(1):61–85, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  29. N. Kajino. Non-regularly varying and non-periodic oscillation of the on-diagonal heat kernels on self-similar fractals. In Fractal geometry and dynamical systems in pure and applied mathematics. II. Fractals in applied mathematics, volume 601 of Contemp. Math., pages 165–194. Amer. Math. Soc., Providence, RI, 2013.

    Google Scholar 

  30. N. Kajino. On-diagonal oscillation of the heat kernels on post-critically finite self-similar fractals. Probab. Theory Related Fields, 156(1–2):51–74, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Kigami. Analysis on fractals, volume 143 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2001.

    Book  MATH  Google Scholar 

  32. J. Kigami and M. L. Lapidus. Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals. Comm. Math. Phys., 158(1):93–125, 1993.

    Google Scholar 

  33. T. Kumagai. Short time asymptotic behaviour and large deviation for Brownian motion on some affine nested fractals. Publ. Res. Inst. Math. Sci., 33(2):223–240, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  34. T. Kumagai. Random walks on disordered media and their scaling limits, volume 2101 of Lecture Notes in Mathematics. Springer, Cham, 2014. Lecture notes from the 40th Probability Summer School held in Saint-Flour, 2010, École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School].

    Google Scholar 

  35. R. Pemantle. Choosing a spanning tree for the integer lattice uniformly. Ann. Probab., 19(4):1559–1574, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  36. O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118:221–288, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  37. D. Shiraishi and S. Watanabe. Volume and heat kernel fluctuations for the three-dimensional uniform spanning tree. Preprint available at arXiv:2211.15031, 2022.

    Google Scholar 

  38. R. S. Strichartz. Differential equations on fractals. Princeton University Press, Princeton, NJ, 2006. A tutorial.

    Google Scholar 

  39. W. Woess. Random walks on infinite graphs and groups, volume 138 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2000.

    Book  MATH  Google Scholar 

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Acknowledgements

This research was supported by JSPS Grant-in-Aid for Scientific Research (A) 22H00099, JSPS Grant-in-Aid for Scientific Research (C) 19K03540, and the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.

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Authors and Affiliations

  1. Department of Mathematics, The University of Manchester, Manchester, UK

    Sebastian Andres

  2. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

    David Croydon

  3. Department of Mathematics, Waseda University, Tokyo, Japan

    Takashi Kumagai

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  1. Sebastian Andres
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  2. David Croydon
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Correspondence to Takashi Kumagai .

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Editors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Patricia Alonso Ruiz

  2. Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany

    Michael Hinz

  3. Department of Mathematics, Tufts University, Medford, MA, USA

    Kasso A. Okoudjou

  4. Department of Mathematics, University of Connecticut, Storrs, CT, USA

    Luke G. Rogers

  5. Department of Mathematics, University of Connecticut, Storrs, CT, USA

    Alexander Teplyaev

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Andres, S., Croydon, D., Kumagai, T. (2023). Heat Kernel Fluctuations for Stochastic Processes on Fractals and Random Media. In: Alonso Ruiz, P., Hinz, M., Okoudjou, K.A., Rogers, L.G., Teplyaev, A. (eds) From Classical Analysis to Analysis on Fractals. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-37800-3_12

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