## Advanced Studies in Pure Mathematics

### Homogenization on Finitely Ramified Fractals

Takashi Kumagai

#### 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
First available in Project Euclid: 3 January 2019

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