Description: This article is a reply to a comment by A. Helmstetter and D. Sornette about the article 'Power-Law Time Distribution of Large Earthquakes' from 2003.

Description: This article discusses memory effects in fractional Brownian motion with Hurst exponent H<1/3. Abstract: We study the regression to the origin of a walker driven by dynamically generated fractional Brownian motion (FBM) and we prove that when the FBM scaling, i.e., the Hurst exponent H<1/3, the emerging inverse power law is characterized by a power index that is a compelling signature of the infinitely extended memory of the system. Strong memory effects leads to the relation H=θ/2 between the Hurst exponent and the persistent exponent θ, which is different from the widely used relation H=1 - θ. The latter is valid for 1/3<H<1 and is known to be compatible with the renewal assumption.

Description: This article discusses noise-induced transition from anomalous to ordinary diffusion and the crossover time as a function of noise intensity. Abstract: We study the interplay between a deterministic process of weak chaos, responsible for the anomalous diffusion of a variable x, and a white noise of intensity ≡. The deterministic process of anomalous diffusion results from the correlated fluctuations of a statistical variable ξ between two distinct values +1 and -1, each of them characterized by the same waiting time distribution ψ(t), given by ψ(t)≃ t(-μ) with 2 < μ < 3, in the long-time limit. We prove that under the influence of a weak white noise of intensity ≡, the process of anomalous diffusion becomes normal at a time t(c) given by t(c) ~ 1/≡(β)(μ). Here β(μ) is a function of μ which depends on the dynamical generator of the waiting-time distribution ψ(t). We derive an explicit expression for β(μ) 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.

Description: This article discusses non-Markovian nonstationary completely positive open-quantum-system dynamics. Abstract: By modeling the interaction of a system with an environment through a renewal approach, we demonstrate that completely positive non-Markovian dynamics may develop some unexplored nonstandard statistical properties. The renewal approach is defined by a set of disruptive events, consisting in the action of a completely positive superoperator over the system density matrix. The random time intervals between events are described by an arbitrary waiting-time distribution. We show that, in contrast to the Markovian case, if one performs a system preparation (measurement) at an arbitrary time, the subsequent evolution of the density-matrix evolution is modified. The nonstationary character refers to the absence of an asymptotic master equation even when the preparation is performed at arbitrary long times. In spite of this property, we demonstrate that operator expectation values and operators correlations have the same dynamical structure, establishing the validity of a nonstationary quantum regression hypothesis. The nonstationary property of the dynamics is also analyzed through the response of the system to an external weak perturbation.

Description: This article discusses non-Poisson distribution of the time distances between two consecutive clusters of earthquakes. Abstract: With the help of the Diffusion Entropy technique the authors show the non-Poisson statistics of the distances between consecutive Omori's swarms of earthquakes. The authors give an analytical proof of the numerical results of an earlier paper [Mega et al., Phys. Rev. Lett. 90 (2003) 188501].