The homogenization problem for the Vicsek set

We discuss the homogenization problem for a particular class of nested fractals, called “tree-like” Vicsek sets, which are not covered by the class considered in Kumagai and Kusuoka (1996). Random irreducible conductivities are assigned to each cell in the infinite fractal lattice and then we consider the rescaled effective resistance. We only assume that the effective resistance across the unit cell has finite first moment and prove that the rescaled effective resistance converges. We also consider the limiting behaviour of the corresponding Markov chains under fractal “Brownian motion” scaling. By assuming finite first moment for the conductivity and finite fourth moments of the effective resistance across the unit cell, we show that for almost every environment, under the measure on the conductivities, the Markov chains converge weakly to a diffusion on the fractal. The limit process does not, in general, coincide with the “Brownian motion” on the original Vicsek set.