Fluctuations of the probability density of diffusing particles for different realizations of a random medium

We study the fluctuations of the probability density P(r,t) of diffusing particles to be at distance r at time t in the presence of random potentials, represented by random transition rates. We find an exact relation which expresses all the moments of P(r,t) in terms of its first moment, for both quenched and annealed disorder and for any dimension. From this relation follows that anomalous diffusion implies nontrivial behavior of the moments of P(r,t), such as an exponential divergence of the relative fluctuations for large r.