Homogenization on nested fractals

Abstract

We study the homogenization problem on nested fractals. Let Xt be the continuous time Markov chain on the pre-nested fractal given by putting i.i.d. random resistors on each cell. It is proved that under some conditions, α-nXtnEt converges in law to a constant time change of the Brownian motion on the fractal as n → ∞, where α is the contraction rate and tE is a time scale constant. As the Brownian motion on fractals is not a semi-martingale, we need a different approach from the well-developed martinga e method.