Stochastic Ergodicity Breaking: a Random Walk Approach

The continuous time random walk (CTRW) model exhibits a non-ergodic phase when the average waiting time diverges. Using an analytical approach for the non-biased and the uniformly biased CTRWs, and numerical simulations for the CTRW in a potential field, we obtain the non-ergodic properties of the random walk which show strong deviations from Boltzmann--Gibbs theory. We derive the distribution function of occupation times in a bounded region of space which, in the ergodic phase recovers the Boltzmann--Gibbs theory, while in the non-ergodic phase yields a generalized non-ergodic statistical law.Comment: 5 pages, 3 figure

Dual random fragmentation and coagulation and an application to the genealogy of Yule processes

The purpose of this work is to describe a duality between a fragmentation associated to certain Dirichlet distributions and a natural random coagulation. The dual fragmentation and coalescent chains arising in this setting appear in the description of the genealogy of Yule processes.Comment: 14 page

Limit laws for distorted return time processes for infinite measure preserving transformations

We consider conservative ergodic measure preserving transformations on infinite measure spaces and investigate the asymptotic behaviour of distorted return time processes with respect to sets satisfying a type of Darling-Kac condition. We identify two critical cases for which we prove uniform distribution laws. For this we introduce the notion of uniformly returning sets and discuss some of their properties.Comment: 18 pages, 2 figure

The role of oxygen vacancies on the structure and the density of states of iron doped zirconia

In this paper we study, both with theoretical and experimental approach, the effect of iron doping in zirconia. Combining density functional theory (DFT) simulations with the experimental characterization of thin films, we show that iron is in the Fe3+ oxidation state and accordingly that the films are rich in oxygen vacancies (VO). VO favor the formation of the tetragonal phase in doped zirconia (ZrO2:Fe) and affect the density of state at the Fermi level as well as the local magnetization of Fe atoms. We also show that the Fe(2p) and Fe(3p) energy levels can be used as a marker for the presence of vacancies in the doped system. In particular the computed position of the Fe(3p) peak is strongly sensitive to the VO to Fe atoms ratio. A comparison of the theoretical and experimental Fe(3p) peak position suggests that in our films this ratio is close to 0.5. Besides the interest in the material by itself, ZrO2:Fe constitutes a test case for the application of DFT on transition metals embedded in oxides. In ZrO2:Fe the inclusion of the Hubbard U correction significantly changes the electronic properties of the system. However the inclusion of this correction, at least for the value U = 3.3 eV chosen in the present work, worsen the agreement with the measured photo-emission valence band spectra.Comment: 24 pages, 8 figure

Non-ergodic Intensity Correlation Functions for Blinking Nano Crystals

We investigate the non-ergodic properties of blinking nano-crystals using a stochastic approach. We calculate the distribution functions of the time averaged intensity correlation function and show that these distributions are not delta peaked on the ensemble average correlation function values; instead they are W or U shaped. Beyond blinking nano-crystals our results describe non-ergodicity in systems stochastically modeled using the Levy walk framework for anomalous diffusion, for example certain types of chaotic dynamics, currents in ion-channel, and single spin dynamics to name a few.Comment: 5 pages, 3 figure

Sign-time distribution for a random walker with a drifting boundary

We present a derivation of the exact sign-time distribution for a random walker in the presence of a boundary moving with constant velocity.Comment: 5 page

Breadth first search coding of multitype forests with application to Lamperti representation

We obtain a bijection between some set of multidimensional sequences and this of $d$-type plane forests which is based on the breadth first search algorithm. This coding sequence is related to the sequence of population sizes indexed by the generations, through a Lamperti type transformation. The same transformation in then obtained in continuous time for multitype branching processes with discrete values. We show that any such process can be obtained from a $d^2$ dimensional compound Poisson process time changed by some integral functional. Our proof bears on the discretisation of branching forests with edge lengths

Large deviations for clocks of self-similar processes

The Lamperti correspondence gives a prominent role to two random time changes: the exponential functional of a L\'evy process drifting to $\infty$ and its inverse, the clock of the corresponding positive self-similar process. We describe here asymptotical properties of these clocks in large time, extending the results of Yor and Zani

The influence of fractional diffusion in Fisher-KPP equations

We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the stan- dard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable L\'evy process, the front position is exponential in time. Our results provide a mathe- matically rigorous justification of numerous heuristics about this model