Construction of diffusion processes on fractals, $d$-sets, and general metric measure spaces

Abstract

We give a sufficient condition to construct non-trivial $\mu$-symmetric diffusion processes on a locally compact separable metric measure space $(M,\rho , \mu )$. These processes are associated with local regular Dirichlet forms which are obtained as continuous parts of $\Gamma$-limits for approximating non-local Dirichlet forms. For various fractals, we can use existing estimates to verify our assumptions. This shows that our general method of constructing diffusions can be applied to these fractals.