Transition Density Asymptotics for Some Diffusion Processes with Multi-Fractal Structures

Martin T. Barlow (University of British Columbia)
Takashi Kumagai (Kyoto University)


We study the asymptotics as $t \to 0$ of the transition density of a class of $\mu$-symmetric diffusions in the case when the measure $\mu$ has a multi-fractal structure. These diffusions include singular time changes of Brownian motion on the unit cube.

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Pages: 1-23

Publication Date: March 16, 2001

DOI: 10.1214/EJP.v6-82


  1. Barlow, M. T. (1998), Diffusions on fractals, Lectures on Probability Theory and Statistics: Ecole d'Eté de Probabilités de Saint-Flour XXV, Springer, Berlin Heidelberg New York. Math. Review 2000a:60148
  2. Barlow, M. T. and Bass, R. F. (1989), The construction of Brownian motion on the Sierpinski carpet, Ann. Inst. H. Poincaré 25, 225-257. Math. Review 91d:60183
  3. Barlow, M. T. and Bass, R. F. (1990), Local times for Brownian motion on the Sierpinski carpet, Probab. Theory Relat. Fields 85, 91-104. Math. Review 91j:60129
  4. Barlow, M. T. and Bass, R. F. (1992), Transition densities for Brownian motion on the Sierpinski carpet, Probab. Theory Relat. Fields 91, 307-330. Math. Review 93k:60203
  5. Barlow, M. T. and Bass, R. F. (1999), Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math. 51, 673-744. Math. Review 2000i:60083
  6. Barlow, M. T., Bass, R. F. and Burdzy, K. (1997), Positivity of Brownian transition densities, Elect. Comm. in Probab. 2, 43-51. Math. Review 99e:60166
  7. Barlow, M. T. and Hambly, B. M. (1997), Transition density estimates for Brownian motion on scale irregular Sierpinski gasket, Ann. Inst. H. Poincaré 33, 531-557. Math. Review 98k:60137
  8. Barlow, M. T. and Perkins, E. A. (1988), Brownian motion on the Sierpinski gasket, Probab. Theory Relat. Fields 79, 543-624. Math. Review 89g:60241
  9. Fukushima, M., Oshima, Y. and Takeda, M. (1994), Dirichlet forms and symmetric Markov processes, de Gruyter, Berlin. Math. Review 96f:60126
  10. Fujita, T. (1990), Some asymptotic estimates of transition probability densities for generalized diffusion processes with self-similar speed measures, Publ. Res. Inst. Math. Sci. 26, 819-840. Math. Review 92f:60134
  11. Hambly, B. M. and Jones, O. D., Asymptotically one-dimensional diffusion on the Sierpinski gasket and multi-type branching processes with varying environment, 1998 (preprint) Math. Review number not available.
  12. Hambly, B. M., Kigami, J. and Kumagai, T. (2002), Multifractal formalisms for the local spectral and walk dimensions, Maths. Proc. Cam. Phil. Soc., to appear. Math. Review number not available.
  13. Hambly, B. M. and Kumagai, T. (1999), Transition density estimates for diffusions on post critically finite self-similar fractals, Proc. London Math. Soc. 78, 431-458. Math. Review 99m:60118
  14. Hambly, B. M., Kumagai, T., Kusuoka, S. and Zhou, X. Y. (2000), Transition density estimates for diffusion processes on homogeneous random Sierpinski carpets, J. Math. Soc. Japan 52, 373-408. Math. Review 2001e:60158
  15. Itô, K. and McKean, Jr. H. P. (1974), Diffusion processes and thier sample paths (Second printing), Springer, Berlin. Math. Review 33 #8031
  16. Kigami, J. (1993), Harmonic calculus on P.C.F. self-similar sets, Trans. Amer. Math. Soc. 335, 721-755. Math. Review 93d:39008
  17. Kigami, J. (2000), Markov property of Kusuoka-Zhou's Dirichlet forms on self-similar sets, J. Math. Sci. Univ. Tokyo 7, 27-33. Math. Review 1 749 979
  18. Kigami, J. (2001), Analysis on fractals, Cambridge Univ. Press, to appear. Math. Review number not available.
  19. Kigami, J. and Lapidus, M. L. (1993), Weyl's spectral problem for the spectral distribution of Laplacians on P.C.F. self-similar fractals, Comm. Math. Phys. 158, 93-125. Math. Review 94m:58225
  20. Kumagai, T. (1993), Regularity, closedness and spectral dimensions of the Dirichlet forms on P.C.F. self-similar sets, J. Math. Kyoto Univ. 33, 765-786. Math. Review 94i:28006
  21. Kusuoka, S. and Zhou, X. Y. (1992), Dirichlet forms on fractals: Poincaré constant and resistance, Probab. Theory Relat. Fields 93, 169-196. Math. Review 94e:60069
  22. Landkof, N. S. (1972), Foundations of Modern Potential Theory, Springer, New York-Heidelberg. Math. Review 50 #2520
  23. Metz, V. (1996), Renormalization contracts on nested frcatals, J. Reine Angew. Math. 480, 161-175. Math. Review 98b:31007
  24. Osada, H. (2001), A family of diffusion processes on Sierpinski carpets, Probab. Theory Relat. Fields 119, 275-310. Math. Review 1 818 249
  25. Sabot, C. (1997), Existence and uniqueness of diffusions on finitely ramified self-similar fractals, Ann. Sci. Ecole Norm. Sup. 30, 605-673. Math. Review 98h:60118

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