## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Stochastic Analysis and Related Topics in Kyoto: In honour of Kiyosi Itô, H. Kunita, S. Watanabe and Y. Takahashi, eds. (Tokyo: Mathematical Society of Japan, 2004), 189 - 207

### Homogenization on Finitely Ramified Fractals

#### Abstract

Let $Xt$ be a continuous time Markov chain on some finitely ramified fractal graph given by putting i.i.d. random resistors on each cell. We prove that under an assumption that a renormalization map of resistors has a non-degenerate fixed point, $\alpha^{-n} X_{\tau^n t}$ converges in law to a non-degenerate diffusion process on the fractal as $n \to \infty$, where $\alpha$ is a spatial scale and $\tau$ is a time scale of the fractal. Especially, when the fixed point of the renormalization map is unique, the diffusion is a constant time change of Brownian motion on the fractal. These results improve and extend our former results in [10].

#### Article information

**Dates**

Received: 17 February 2003

First available in Project Euclid:
3 January 2019

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1546542612

**Digital Object Identifier**

doi:10.2969/aspm/04110189

**Mathematical Reviews number (MathSciNet)**

MR2083710

**Zentralblatt MATH identifier**

1063.60104

#### Citation

Kumagai, Takashi. Homogenization on Finitely Ramified Fractals. Stochastic Analysis and Related Topics in Kyoto: In honour of Kiyosi Itô, 189--207, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/04110189. https://projecteuclid.org/euclid.aspm/1546542612