Diffusion on locally compact ultrametric spaces

We consider an ultrametric space with sufficiently many isometries and we construct a class of diffusion processes on the space as appropriate limits of discrete processes on the (open and closed) balls of the space. We show, using a version of the Lévy Khintchine formula adapted to this general context, that our construction includes all convolution semigroups associated to an unbounded Lévy measure. Finally we relate our construction to the construction of diffusion processes due to S. Albeverio and W. Karwowski on p-adic fields.