Multifractal features of random walks on random fractals

We find that the fluctuations of the probability density P(r,t) of random walks on random fractals, for fixed distance r and time t, have a broad logarithmic distribution. The average moments 〈Pq〉 scale is a multifractal way, 〈Pq〉{~}〈P〉^τ(q), where {τ}(q){~}q^γ, {γ}1. The exponent {γ} characterizes the structural disorder in the fractal. In chemical l space, the distribution of P(l,t) is narrow and 〈Pq〉{~}〈P〉q.