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].