Convergence of Discrete Dirichlet Forms to Continuous Dirichlet Forms on Fractals

Abstract

It is well known that a Dirichlet form on a fractal structure can be defined as the limit of an increasing sequence of discrete Dirichlet forms, defined on finite subsets which fill the fractal. The initial form is defined on V(0), which is a sort of boundary of the fractal, and we have to require that it is an eigenform, i.e., an eigenvector of a particular nonlinear renormalization map for Dirichlet forms on V(0). In this paper, I prove that, provided an eigenform exists, even if the form on V(0) is not an eigenform, the corresponding sequence of discrete forms converges to a Dirichlet form on all of the fractal, both pointwise and in the sense of Γ-convergence (but these two limits can be different). The problem of Γ-convergence was first studied by S. Kozlov on the Gasket.