The cone of diffusions on finitely ramified fractals Original Research Article

Like Brownian motion on View the MathML source one would like to identify a “natural” (unique) diffusion on a fractal. Equivalently, we look for abstract Dirichlet integrals (local, regular, irreducible, symmetric Dirichlet forms) on strongly connected, finitely ramified, self-similar fractals. These forms have to be “self-similar” in the sense that they scale by a fixed constant when the fractal is scaled by one of its defining contractions. Two necessary and sufficient uniqueness criteria are derived. In the case of ambiguity we describe the shape and the location of the cone of self-similar Dirichlet forms. Technically, we analyze a superlinear renormalization map Λ which is non-expansive with respect to Hilbertʹs projective metric h. It contracts h-distances by irreducibility or super-additivity.