Noise-induced transition from anomalous to ordinary diffusion: The crossover time as a function of noise intensity

We study the interplay between a deterministic process of weak chaos, responsible for the anomalous diffusion of a variable a, and a white noise of intensity Xi. The deterministic process of anomalous diffusion results from the correlated fluctuations of a statistical variable xi between two distinct values +1 and -1, each of them characterized by the same waiting time distribution phi(t), given by psi(t) similar or equal to t(-mu) with 2 < mu < 3, in the long-time limit. We prove that under the influence of a weak white noise of intensity Theta, the process of anomalous diffusion becomes normal at a time t(c) given by t(c) similar to 1/Xi(beta(mu)). Here beta(mu) is a function of mu which depends on the dynamical generator of the waiting-time distribution psi(t). We derive an explicit expression for beta(mu) in the case of two dynamical systems, a one-dimensional superdiffusive map and the standard map in the accelerating state. The theoretical prediction is supported by numerical calculations.