## Abstract

We construct a family of diffusions P^{α} = {P_{x}.} on the d-dimensional Sierpinski carpet F̂. The parameter α ranges over d_{H} < α < ∞, where d_{H} = log(3^{d} - 1)/log 3 is the Hausdorff dimension of the d-dimensional Sierpinski carpet F̂. These diffusions P^{α} are reversible with invariant measures μ = μ^{[α]}. Here, μ are Radon measures whose topological supports are equal to F̂ and satisfy self-similarity in the sense that μ(3A) = 3^{α} · μ(A) for all A ∈ ℬ(F̂). In addition, the diffusion is self-similar and invariant under local weak translations (cell translations) of the Sierpinski carpet. The transition density p = p(t, x, y) is locally uniformly positive and satisfies a global Gaussian upper bound. In spite of these well-behaved properties, the diffusions are different from Barlow-Bass' Brownian motions on the Sierpinski carpet.

Original language | English |
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Pages (from-to) | 275-310 |

Number of pages | 36 |

Journal | Probability Theory and Related Fields |

Volume | 119 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2001 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty