Random Walks and Diffusions on Fractals

Abstract

We investigate the asymptotic motion of a random walker, which at time n is at X(n), on certain “fractal lattices”. For the “Sierpinski lattice” in dimension d we show that as ℓ → ∞, the process Y_ℓ(t) ≡ X([(d+3)^ℓ t])/2^ℓ converges in distribution (so that, in particular, |X(n)| ~ n^γ, where γ = (ln 2)/ln(d + 3)) to a diffusion on the Sierpinski gasket, a Cantor set of Lebesgue measure zero. The analysis is based on a simple “renormalization group” type argument, involving self-similarity and “decimation invariance”.