This book is published open access.
In the ordinary theory of Sobolev spaces on domains of , the -energy is defined as the integral of . In this paper, we try to construct a -energy on compact metric spaces as a scaling limit of discrete -energies on a series of graphs approximating the original space. In conclusion, we propose a notion called conductive homogeneity under which one can construct a reasonable -energy if is greater than the Ahlfors regular conformal dimension of the space. In particular, if , then we construct a local regular Dirichlet form and show that the heat kernel associated with the Dirichlet form satisfies upper and lower sub-Gaussian type heat kernel estimates. As examples of conductively homogeneous spaces, we present new classes of square-based self-similar sets and rationally ramified Sierpiński crosses, where no diffusions were constructed before.