Energy measures and indices of Dirichlet forms, with applications to derivatives on some fractals

TL;DR

This paper introduces the concept of index for Dirichlet forms using energy measures, linking it to martingale dimension, and demonstrates the existence of derivatives on certain fractals with energy represented as squared integrals.

Contribution

It defines the index for regular Dirichlet forms via energy measures and establishes its equivalence with martingale dimension, also constructing derivatives on fractals.

Findings

Index of Dirichlet forms equals martingale dimension.

First-order derivatives exist on certain fractals.

Energy can be expressed as squared integrals of derivatives.

Abstract

We introduce the concept of index for regular Dirichlet forms by means of energy measures, and discuss its properties. In particular, it is proved that the index of strong local regular Dirichlet forms is identical with the martingale dimension of the associated diffusion processes. As an application, a class of self-similar fractals is taken up as an underlying space. We prove that first-order derivatives can be defined for functions in the domain of the Dirichlet forms and their total energies are represented as the square integrals of the derivatives.