Abstract: The present talk aims to illustrate different ways to construct (in some sense natural) diffusion processes on fractals using as an example a parametric family of generalized diamond fractals. These spaces arise as scaling limits of diamond hierarchical lattices. The latter are studied in the physics literature in relation to random polymers, Ising and Potts models among others.

In the case of constant parameters, one can exploit the self-similarity of the space to obtain a canonical Dirichlet form and a diffusion process. This approach is common to many fractal settings and was taken in earlier investigations due to Hambly and Kumagai. We will outline this construction and the properties of the diffusion process and the heat kernel that were obtained there.

Alternatively, a diamond fractal can also be regarded as an inverse limit of metric measure graphs and a canonical diffusion process can be constructed through a procedure proposed by Barlow and Evans. Following this approach it turns out that it is possible to give a rather explicit expression of the associated heat kernel, that is in particular uniformly continuous and admits an analytic continuation.