A Stochastic Description for Extremal Dynamics

We show that extremal dynamics is very well modelled by the "Linear Fractional Stable Motion" (LFSM), a stochastic process entirely defined by two exponents that take into account spatio-temporal correlations in the distribution of active sites. We demonstrate this numerically and analytically using well-known properties of the LFSM. Further, we use this correspondence to write an exact expressions for an n-point correlation function as well as an equation of fractional order for interface growth in extremal dynamics.Comment: 4 pages LaTex, 3 figures .ep

Self-Similar Anisotropic Texture Analysis: the Hyperbolic Wavelet Transform Contribution

Textures in images can often be well modeled using self-similar processes while they may at the same time display anisotropy. The present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of Gaussian anisotropic selfsimilar processes. It will first be shown that accurate joint estimates of the anisotropy and selfsimilarity parameters are performed by replacing the standard 2D-discrete wavelet transform by the hyperbolic wavelet transform, which permits the use of different dilation factors along the horizontal and vertical axis. Defining anisotropy requires a reference direction that needs not a priori match the horizontal and vertical axes according to which the images are digitized, this discrepancy defines a rotation angle. Second, we show that this rotation angle can be jointly estimated. Third, a non parametric bootstrap based procedure is described, that provides confidence interval in addition to the estimates themselves and enables to construct an isotropy test procedure, that can be applied to a single texture image. Fourth, the robustness and versatility of the proposed analysis is illustrated by being applied to a large variety of different isotropic and anisotropic self-similar fields. As an illustration, we show that a true anisotropy built-in self-similarity can be disentangled from an isotropic self-similarity to which an anisotropic trend has been superimposed

Cascades infiniment divisibles voilées : au-delà des lois de puissance

Nous présentons les définitions et synthèses de processus stochastiques respectant des lois d'échelles voilées, qui s'écartent de façon contrôlée d'un comportement en loi de puissance. Nous définissons des bruit, mouvement et marche aléatoire issus de cascades infiniment divisibles (IDC) voilées. Nous étudions analytiquement le comportement des moments des accroissements de ces processus à travers les échelles. Ces résultats théoriques sont illustrés sur l'exemple d'une cascade log-Normale voilée. Les algorithmes de synthèse et les fonctions Matlab utilisés sont disponibles sur nos pages web.We address the definitions and synthesis of stochastic processes which possess warped scaling laws that depart from power law behaviors in a controlled manner. We define warped infinitely divisible cascading (IDC) noise, motion and random walk. We provide a theoretical derivation of the scaling behavior of the moments of their increments. We provide numerical simulations of a warped log-Normal cascade to illustrate these results. Algorithms for synthesis and Matlab functions are available from our web pages

Intermittent turbulence, noisy fluctuations and wavy structures in the Venusian magnetosheath and wake

Recent research has shown that distinct physical regions in the Venusian induced magnetosphere are recognizable from the variations of strength of the magnetic field and its wave/fluctuation activity. In this paper the statistical properties of magnetic fluctuations are investigated in the Venusian magnetosheath and wake regions. The main goal is to identify the characteristic scaling features of fluctuations along Venus Express (VEX) trajectory and to understand the specific circumstances of the occurrence of different types of scalings. For the latter task we also use the results of measurements from the previous missions to Venus. Our main result is that the changing character of physical interactions between the solar wind and the planetary obstacle is leading to different types of spectral scaling in the near-Venusian space. Noisy fluctuations are observed in the magnetosheath, wavy structures near the terminator and in the nightside near-planet wake. Multi-scale turbulence is observed at the magnetosheath boundary layer and near the quasi-parallel bow shock. Magnetosheath boundary layer turbulence is associated with an average magnetic field which is nearly aligned with the Sun-Venus line. Noisy magnetic fluctuations are well described with the Gaussian statistics. Both magnetosheath boundary layer and near shock turbulence statistics exhibit non-Gaussian features and intermittency over small spatio-temporal scales. The occurrence of turbulence near magnetosheath boundaries can be responsible for the local heating of plasma observed by previous missions

Numerical Schemes for Rough Parabolic Equations

This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0, 1) perturbed by a non-linear rough signal. It is the continuation of [8, 7], where the existence and uniqueness of a solution has been established. The approach combines rough paths methods with standard considerations on discretizing stochastic PDEs. The results apply to a geometric 2-rough path, which covers the case of the multidimensional fractional Brownian motion with Hurst index H \textgreater{} 1/3.Comment: Applied Mathematics and Optimization, 201

Statistical Tests of Distributional Scaling Properties for Financial Return Series

Existing empirical evidence of distributional scaling in financial returns has helped motivate the use of multifractal processes for modelling return processes. However, this evidence has relied on informal tests that may be unable to reliably distinguish multifractal processes from other related classes. The current paper develops a formal statistical testing procedure for determining which class of fractal process is most consistent with the distributional scaling properties in a given sample of data. Our testing methodology consists of a set of test statistics, together with a model-based bootstrap resampling scheme to obtain sample p-values. We demonstrate in Monte Carlo exercises that the proposed testing methodology performs well in a wide range of testing environments relevant for financial applications. Finally, the methodology is applied to study the scaling properties of a dataset of intraday equity index and exchange rate returns. The empirical results suggest that the scaling properties of these return series may be inconsistent with purely multifractal processes

Wavelets techniques for pointwise anti-Holderian irregularity

In this paper, we introduce a notion of weak pointwise Holder regularity, starting from the de nition of the pointwise anti-Holder irregularity. Using this concept, a weak spectrum of singularities can be de ned as for the usual pointwise Holder regularity. We build a class of wavelet series satisfying the multifractal formalism and thus show the optimality of the upper bound. We also show that the weak spectrum of singularities is disconnected from the casual one (denoted here strong spectrum of singularities) by exhibiting a multifractal function made of Davenport series whose weak spectrum di ers from the strong one

Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series

Notwithstanding the significant efforts to develop estimators of long-range correlations (LRC) and to compare their performance, no clear consensus exists on what is the best method and under which conditions. In addition, synthetic tests suggest that the performance of LRC estimators varies when using different generators of LRC time series. Here, we compare the performances of four estimators [Fluctuation Analysis (FA), Detrended Fluctuation Analysis (DFA), Backward Detrending Moving Average (BDMA), and centred Detrending Moving Average (CDMA)]. We use three different generators [Fractional Gaussian Noises, and two ways of generating Fractional Brownian Motions]. We find that CDMA has the best performance and DFA is only slightly worse in some situations, while FA performs the worst. In addition, CDMA and DFA are less sensitive to the scaling range than FA. Hence, CDMA and DFA remain "The Methods of Choice" in determining the Hurst index of time series.Comment: 6 pages (including 3 figures) + 3 supplementary figure

On the magnetic fields generated by experimental dynamos

We review the results obtained by three successful fluid dynamo experiments and discuss what has been learnt from them about the effect of turbulence on the dynamo threshold and saturation. We then discuss several questions that are still open and propose experiments that could be performed to answer some of them.Comment: 40 pages, 13 figure