Probability densities of random walks in random systems

We review recent results for the probability distribution of random walkers in random systems, where diffusion is anomalous and the mean-square displacement scales with time as R²(t)∼t^{2 w}, d_w > 2. The random systems are characterized by structural disorder and by random transition rates. In general, the mean distribution function 〈P(r, t)〉 of the random walkers is a stretched Gaussian and scales as log[ P(r,t) P(r,0)]∼-[ r R(t)]^u, where u= d_w (d_w-1). On random fractals, the fluctuations of the density distribution P(r, t), for fixed distance r and time t, have a broad logarithmic distribution. The average moments 〈P^q〉 scale in a multifractal way as 〈P〉^τ(q), where τ(q)∼q^τ,γ<1. In contrast, in chemical l-space the fluctuations of P are narrow and 〈P^q〉∼〈P〉^q.