Processos aleatórios não-markovianos: perfis de memória

One of the mechanisms responsible for the anomalous diffusion is the existence of long-range temporal correlations, for example, Fractional Brownian Motion and walk models according to Elephant memory and Alzheimer profiles, whereas in the latter two cases the walker can always "remember" of his first steps. The question to be elucidated, and the was the main motivation of our work, is if memory of the historic initial is condition for observation anomalous diffusion (in this case, superdiffusion). We give a conclusive answer, by studying a non-Markovian model in which the walkers memory of the past, at time t, is given by a Gaussian centered at time t=2 and standard deviation t which grows linearly as the walker ages. For large widths of we find that the model behaves similarly to the Elephant model; In the opposite limit (! 0), although the walker forget the early days, we observed similar results to the Alzheimer walk model, in particular the presence of amnestically induced persistence, characterized by certain log-periodic oscillations. We conclude that the memory of earlier times is not a necessary condition for the generating of superdiffusion nor the amnestically induced persistence and can appear even in profiles of memory that forgets the initial steps, like the Gausssian memory profile investigated here.