A family of diffusion processes on Sierpinski carpets

We construct a family of diffusions P^α = {P_x.} on the d-dimensional Sierpinski carpet F̂. 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 F̂. These diffusions P^α 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) = 3^α · 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.