Poincaré recurrence times in stochastic mixing processes

Abstract

In the context of the discrete-time stochastic processes, this thesis presents new results on Poincaré recurrence theory. After a complete review of recent results, we present a new theorem on the exponential approximations for hitting and return times distributions. We show that the scaling parameter of the approximate distribution, called "potential well", brings fundamental informations about the structure of the target set. Moreover, we show that the asymptotic properties of the potential well influences several aspects of the recurrence times, such as limiting distributions and moments. Finally, we apply our results to obtain the waiting time spectrum as a function of the Rényi entropy for classes of processes not covered by previous works.