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

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

2009-06-23

arxiv.org/abs/0906.4251v1

Proceedings of the London Mathematical Society, vol. 100 (2010), 269-302.

Mathematics

Probability

33 pages, 3 figures

Scientific paper

10.1112/plms/pdp032

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.

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