# Dirichlet forms on fractals and products of random matrices

@article{Kusuoka1989DirichletFO, title={Dirichlet forms on fractals and products of random matrices}, author={Shigeo Kusuoka}, journal={Publications of The Research Institute for Mathematical Sciences}, year={1989}, volume={25}, pages={659-680} }

The author studies Dirichlet forms on fractals. He constructs some local Dirichlet forms on abstract fractal sets by using products of random matrices. Also, he studies the martingale dimension of the associated diffusion processes and its self-similarity.

## 200 Citations

### Groups and analysis on fractals

- Mathematics
- 2005

We describe relation between analysis on fractals and the theory of self-similar groups. In particular, we focus on the construction of the Laplacian on limit sets of such groups in several concrete…

### Combinatorial and analytical problems for fractals and their graph approximations

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The recent field of analysis on fractals has been studied under a probabilistic and analytic point of view, but this work focuses on the analytic part developed by Kigami.

### Upper estimate of martingale dimension for self-similar fractals

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We study upper estimates of the martingale dimension dm of diffusion processes associated with strong local Dirichlet forms. By applying a general strategy to self-similar Dirichlet forms on…

### LAPLACIANS ON SELF-SIMILAR SETS AND THEIR SPECTRAL DISTRIBUTIONS

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This paper shows how to construct Laplacians on finitely ramified fractals, in particular, finitely ramified self-similar sets. The spectral distribution of such a iLaplacian is studied and an…

### Dirichlet forms on non-self-similar fractals: Hanoi attractors

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- 2014

We present the construction of a local and regular Dirichlet form on the so-called Hanoi attractor of parameter α for any α ∈ (0, 1 / 3). Since these fractals are non-self-similar sets, we need to…

### MARTINGALE DIMENSIONS FOR FRACTALS

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- 2008

We prove that the martingale dimensions for canonical diffusion processes on a class of self-similar sets including nested fractals are always one. This provides an affirmative answer to the…

### Distribution theory on P.C.F. fractals

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- 2009

We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. The results include basic…

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

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- 2009

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…

### Harmonic functions representation of Besov-Lipschitz functions on nested fractals

- Mathematics
- 2012

R. S. Strichartz proposes a discrete definition of Besov spaces on self-similar fractals having a regular harmonic structure. In this paper, we characterize some of these Holder-Zygmund and Besov-L…

### Harnack Inequality for p-Laplacians on Metric Fractals

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- 2005

By using the approach of the metric fractals, we prove a Harnack inequality for non-negative local supersolutions of p-Laplacians — associated to p-Lagrangians — on metric fractals whose homogeneous…

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