Abstract: Inverse (or projective) limit spaces appear in many areas of mathematical research, for instance in mathematical physics, dynamical systems and metric measure theory. These structures sometimes give rise to spaces that lack usual properties such as volume doubling. From a probabilistic point of view, natural questions to ask are the existence and the properties of diffusion processes occurring in them. In this talk we will consider a parametric family of diamond fractals as a toy model to illustrate the type of properties that the associated heat kernel and Dirichlet form present. Further, we will discuss which of these properties may extend to a larger class of inverse limit spaces.