A family of diffusion processes on Sierpinski carpets

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Abstract

We construct a family of diffusions Pα = {Px.} on the d-dimensional Sierpinski carpet F̂. The parameter α ranges over dH < α < ∞, where dH = log(3d - 1)/log 3 is the Hausdorff dimension of the d-dimensional Sierpinski carpet F̂. These diffusions Pα are reversible with invariant measures μ = μ[α]. Here, μ are Radon measures whose topological supports are equal to F̂ and satisfy self-similarity in the sense that μ(3A) = 3α · μ(A) for all A ∈ ℬ(F̂). In addition, the diffusion is self-similar and invariant under local weak translations (cell translations) of the Sierpinski carpet. The transition density p = p(t, x, y) is locally uniformly positive and satisfies a global Gaussian upper bound. In spite of these well-behaved properties, the diffusions are different from Barlow-Bass' Brownian motions on the Sierpinski carpet.

Original languageEnglish
Pages (from-to)275-310
Number of pages36
JournalProbability Theory and Related Fields
Volume119
Issue number2
DOIs
Publication statusPublished - Feb 2001
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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