A family of diffusion processes on Sierpinski carpets

We construct a family of diffusions P-alpha = {P-x } on the d-dimensional Sierpinski carpet (F) over cap. The parameter alpha ranges over d(H) < <alpha> < <infinity>, where d(H) = log(3(d) - 1)/log3 is the Hausdorff dimension of the d-dimensional Sierpinski carpet (F) over cap. These diffusions P-alpha are reversible with invariant measures mu = mu (\ alpha \). Here, mu are Radon measures whose topological supports are equal to (F) over cap and satisfy self-similarity in the sense that mu (3A) = 3(alpha).mu (A) for all A is an element of B((F) over cap). 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.