Analysis on fractal spaces and heat kernels

We discuss elements of Analysis on Ahlfors-regular metric spaces (in particular, on fractals) from the point of view on the notion of heat kernel. Such spaces are characterized by two parameters: the Hausdorff dimension and the walk dimension, where the latter determines the space/time scaling for a diffusion process. We present various approaches to the notion of the walk dimension, including those via Besov function spaces and via Markov jump processes. We also discuss heat kernel bounds for diffusion and jump processes on such spaces.