Transition density estimates for Brownian motion on affine nested fractals

Pat J. Fitzsimmons*, Ben M. Hambly, Takashi Kumagai

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

77 Citations (Scopus)


A class of affine nested fractals is introduced which have different scale factors for different similitudes but still have the symmetry assumptions of nested fractals. For these fractals estimates on the transition density for the Brownian motion are obtained using the associated Dirichlet form. An upper bound for the diagonal can be found using a Nash-type inequality, then probabilistic techniques are used to obtain the off-diagonal bound. The approach differs from previous treatments as it uses only the Dirichlet form and no estimates on the resolvent. The bounds obtained are expressed in terms of an intrinsic metric on the fractal.

Original languageEnglish
Pages (from-to)595-620
Number of pages26
JournalCommunications in Mathematical Physics
Issue number3
Publication statusPublished - 1994 Oct
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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