Recurrence time statistics in deterministic and stochastic dynamical systems in continuous time: A comparison

The dynamics of transitions between the cells of a finite phase-space partition is analyzed for deterministic and stochastic dynamical systems in continuous time. Special emphasis is placed on the dependence of mean recurrence time on the resolution τ between successive observations, in the limit $\overrightarrow{\tau }0.$ In deterministic systems the limit is found to exist, and to depend on only the intrinsic parameters of the underlying dynamics. In stochastic systems two different cases are identified, leading to a τ-independent behavior and a ${\tau }^{1/2}$ behavior, depending on whether a finite speed of propagation of the signals exists or not. An extension of the results to the second moment of the recurrence time is outlined.