## Abstract

For a closed connected set F in R^{n}, assume that there is a local regular Dirichlet form (a symmetric diffusion process) on F whose domain is included in a Lipschitz space or a Besov space on F. Under some condition for the order of the space and the Newtonian 1-capacity of F, we prove that there exists a symmetric diffusion process on R^{n} which moves like the process on F and like Brownian motion on R^{n} outside F. As an application, we will show that when F is a nested fractal or a Sierpinski carpet whose Hausdorff dimension is greater than n-2, we can construct Brownian motion penetrating the fractal. For the proof, we apply the technique developed in the theory of Besov spaces.

Original language | English |
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Pages (from-to) | 69-92 |

Number of pages | 24 |

Journal | Journal of Functional Analysis |

Volume | 170 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2000 Jan 10 |

Externally published | Yes |

## Keywords

- Lipschitz space; Besov space; capacity; trace theorem; Dirichlet form; diffusions on fractals

## ASJC Scopus subject areas

- Analysis