A family of diffusion processes on Sierpinski carpets

Abstract

We construct a family of diffusions Pα = {Px} on the d-dimensional Sierpinski carpet F^. The parameter α ranges over d H < α < ∞, where d H = log(3 d − 1)/log 3 is the Hausdorff dimension of the d-dimensional Sierpinski carpet F^. These diffusions Pα are reversible with invariant measures μ = μ[α]. Here, μ are Radon measures whose topological supports are equal to F^ and satisfy self-similarity in the sense that μ(3A) = 3α·μ(A) for all A∈ℬ(F^). 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.