Marking (1,2) Points of the Brownian Web and Applications

The Brownian web (BW), which developed from the work of Arratia and then T\'{o}th and Werner, is a random collection of paths (with specified starting points) in one plus one dimensional space-time that arises as the scaling limit of the discrete web (DW) of coalescing simple random walks. Two recently introduced extensions of the BW, the Brownian net (BN) constructed by Sun and Swart, and the dynamical Brownian web (DyBW) proposed by Howitt and Warren, are (or should be) scaling limits of corresponding discrete extensions of the DW -- the discrete net (DN) and the dynamical discrete web (DyDW). These discrete extensions have a natural geometric structure in which the underlying Bernoulli left or right "arrow" structure of the DW is extended by means of branching (i.e., allowing left and right simultaneously) to construct the DN or by means of switching (i.e., from left to right and vice-versa) to construct the DyDW. In this paper we show that there is a similar structure in the continuum where arrow direction is replaced by the left or right parity of the (1,2) space-time points of the BW (points with one incoming path from the past and two outgoing paths to the future, only one of which is a continuation of the incoming path). We then provide a complete construction of the DyBW and an alternate construction of the BN to that of Sun and Swart by proving that the switching or branching can be implemented by a Poissonian marking of the (1,2) points.Comment: added 3 references to Sections 1, 2, 3; expanded explanations in Subsections 7.3, 7.4, 7.

MFGA-IDT2 workshop: Astrophysical and geophysical fluid mechanics: the impact of data on turbulence theories

International audience1 Facts about the Workshop This workshop was convened on November 13-15 1995 by E. Falgarone and D. Schertzer within the framework of the Groupe de Recherche Mecanique des Fluides Geophysiques et Astrophysiques (GdR MFGA, Research Group of Geophysical and Astrophysical Fluid Mechanics) of Centre National de la Recherche Scientifique (CNRS, (French) National Center for Scientific Research). This Research Group is chaired by A. Babiano and the meeting was held at Ecole Normale Superieure, Paris, by courtesy of its Director E. Guyon. More than sixty attendees participated to this workshop, they came from a large number of institutions and countries from Europe, Canada and USA. There were twenty-five oral presentations as well as a dozen posters. A copy of the corresponding book of abstracts can be requested to the conveners. The theme of this meeting is somewhat related to the series of Nonlinear Variability in Geophysics conferences (NVAG1, Montreal, Aug. 1986; NVAG2, Paris, June 1988; NVAG3, Cargese (Corsica), September, 1993), as well as seven consecutive annual sessions at EGS general assemblies and two consecutive spring AGU meeting sessions devoted to similar topics. One may note that NVAG3 was a joint American Geophysical Union Chapman and European Geophysical Society Richardson Memorial conference, the first topical conference jointly sponsored by the two organizations. The corresponding proceedings were published in a special NPG issue (Nonlinear Processes in Geophysics 1, 2/3, 1994). In comparison with these previous meetings, MFGA-IDT2 is at the same time specialized to fluid turbulence and its intermittency, and an extension to the fields of astrophysics. Let us add that Nonlinear Processes in Geophysics was readily chosen as the appropriate journal for publication of these proceedings since this journal was founded in order to develop interdisciplinary fundamental research and corresponding innovative nonlinear methodologies in Geophysics. It had an appropriate editorial structure, in particular a large number of editors covering a wide range of methodologies, expertises and schools. At least two of its sections (Scaling and Multifractals, Turbulence and Diffusion) were directly related to the topics of the workshop, in any case contributors were invited to choose their editor freely. 2 Goals of the Workshop The objective of this meeting was to enhance the confrontation between turbulence theories and empirical data from geophysics and astrophysics fluids with very high Reynolds numbers. The importance of these data seems to have often been underestimated for the evaluation of theories of fully developed turbulence, presumably due to the fact that turbulence does not appear as pure as in laboratory experiments. However, they have the great advantage of giving access not only to very high Reynolds numbers (e.g. 1012 for atmospheric data), but also to very large data sets. It was intended to: (i) provide an overview of the diversity of potentially available data, as well as the necessary theoretical and statistical developments for a better use of these data (e.g. treatment of anisotropy, role of processes which induce other nonlinearities such as thermal instability, effect of magnetic field and compressibility ... ), (ii) evaluate the means of discriminating between different theories (e.g. multifractal intermittency models) or to better appreciate the relevance of different notions (e.g. Self-Organized Criticality) or phenomenology (e.g. filaments, structures), (iii) emphasise the different obstacles, such as the ubiquity of catastrophic events, which could be overcome in the various concerned disciplines, thanks to theoretical advances achieved. 3 Outlines of the Workshop During the two days of the workshop, the series of presentations covered many manifestations of turbulence in geophysics, including: oceans, troposphere, stratosphere, very high atmosphere, solar wind, giant planets, interstellar clouds... up to the very large scale of the Universe. The presentations and the round table at the end of the workshop pointed out the following: - the necessity of this type of confrontation which makes intervene numerical simulations, laboratory experiments, phenomenology as well as a very large diversity of geophysical and astrophysical data, - presumably a relative need for new geophysical data, whereas there have been recent astrophysical experiments which yield interesting data and exciting questions; - the need to develop a closer intercomparison between various intermittency models (in particular Log-Poisson /Log Levy models). Two main questions were underlined, in particular during the round table: - the behaviour of the extremes of intermittency, in particular the question of divergence or convergence of the highest statistical moments (equivalently, do the probability distributions have algebraic or more rapid falloffs?); - the extension of scaling ranges; in other words do we need to divide geophysics and astrophysics in many small (nearly) isotropic subranges or is it sufficient to use anisotropic scaling notions over wider ranges? 4 The contributions in this special issue Recalling that some of the most useful insights into the nature of turbulence in fluids have come from observations of geophysical flows, Van Atta gives a review of the impacts of geophysical turbulence data into theories. His paper starts from Taylor's inference of the nearly isotropy of atmospheric turbulence and the corresponding elegant theoretical developments by von Karman of the theory of isotropic turbulence, up to underline the fact that the observed extremely large intermittency in geophysical turbulence also raised new fundamental questions for turbulence theory. The paper discusses the potential contribution to theoretical development from the available or currently being made geophysical turbulence measurements, as well as from some recent laboratory measurements and direct numerical simulations of stably stratified turbulent shear flows. Seuront et al. consider scaling and multiscaling properties of scalar fields (temperature and phytoplankton concentration) advected by oceanic turbulence in both Eulerian and Lagrangian frameworks. Despite the apparent complexity linked to a multifractal background, temperature and fluorescence (i.e. phytoplankton biomass surrogate) fields are expressed over a wide range of scale by only three universal multifractal parameters, H, \alpha and C_l. On scales smaller than the characteristic scale of the ship, sampling is rather Eulerian. On larger scales, the drifting platform being advected by turbulent motions, sampling may be rather considered as Lagrangian. Observed Eulerian and Lagrangian universal multifractal properties of the physical and biological fields are discussed. Whereas theoretical models provide different scaling laws for fluid and MHD turbulent flows, no attempt has been done up to now to experimentally support evidence for these differences. Carbone et al. use measurements from the solar wind turbulence and from turbulence in ordinary fluid flows, in order to assess these differences. They show that the so-called Extended Self-Similarity (ESS) is evident in the solar wind turbulence up to a certain scale. Furthermore, up to a given order of the velocity structure functions, the scaling laws of MHD and fluids flows axe experimentally indistinguishable. However, differences can be observed for higher orders and the authors speculate on their origin. Dudok de Wit and Krasnosel'skikh present analysis of strong plasma turbulence in the vicinity of the Earth's bow shock with the help of magnetometer data from the AMPTE UKS satellite. They demonstrate that there is a departure from Gaussianity which could be a signature of multifractality. However, they point out that the complexity of plasma turbulence precludes a more quantitative understanding. Finally, the authors emphasise the fact that the duration of records prevents to obtain any reliable estimate of structure functions beyond the fourth order. Sylos Labini and Pietronero discuss the problem of galaxy correlations. They conclude from all the recently available three dimensional catalogues that the distribution of galaxies and clusters is fractal with dimension D ~ 2 up to the present observational limits without any tendency towards homogenization. This result is discussed in contrast to angular data analysis. Furthermore, they point out that the galaxy-cluster mismatch disappears when considering a multifractal distribution of matter. They emphasise that a new picture emerges which changes the standard ideas about the properties of the universe and requires a corresponding change in the related theoretical concepts. Chilla et al. investigate with the help of a laboratory experiment the possible influence of the presence of a large scale structure on the intermittency of small scale structures. They study a flow between coaxial co-rotating disks generating a strong axial vortex over a turbulent background. They show that the cascade process is preserved although strongly modified and they discuss the relevance of parameters developed for the description of intermittency in homogeneous turbulence to evaluate this modification

Exceptional Times for the Dynamical Discrete Web

The dynamical discrete web (DyDW),introduced in recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical time parameter \tau. The evolution is by independent updating of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed \tau. In this paper, we study the existence of exceptional (random) values of \tau where the paths of the web do not behave like usual random walks and the Hausdorff dimension of the set of exceptional such \tau. Our results are motivated by those about exceptional times for dynamical percolation in high dimension by H\"{a}ggstrom, Peres and Steif, and in dimension two by Schramm and Steif. The exceptional behavior of the walks in the DyDW is rather different from the situation for the dynamical random walks of Benjamini, H\"{a}ggstrom, Peres and Steif. For example, we prove that the walk from the origin S^\tau_0 violates the law of the iterated logarithm (LIL) on a set of \tau of Hausdorff dimension one. We also discuss how these and other results extend to the dynamical Brownian web, the natural scaling limit of the DyDW

Extreme values and fat tails of multifractal fluctuations

In this paper we discuss the problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, standard extreme value approach is not valid and classical tail exponent estimators should be interpreted cautiously. Extreme statistics associated with multifractal random processes turn out to be characterized by non self-averaging properties. Our considerations rely upon some analogy between random multiplicative cascades and the physics of disordered systems and also on recent mathematical results about the so-called multifractal formalism. Applied to financial time series, our findings allow us to propose an unified framemork that accounts for the observed multiscaling properties of return fluctuations, the volatility clustering phenomenon and the observed ``inverse cubic law'' of the return pdf tails

Universality of rain event size distributions

We compare rain event size distributions derived from measurements in climatically different regions, which we find to be well approximated by power laws of similar exponents over broad ranges. Differences can be seen in the large-scale cutoffs of the distributions. Event duration distributions suggest that the scale-free aspects are related to the absence of characteristic scales in the meteorological mesoscale.Comment: 16 pages, 10 figure

Components of multifractality in the Central England Temperature anomaly series

We study the multifractal nature of the Central England Temperature (CET) anomaly, a time series that spans more than 200 years. The series is analyzed as a complete data set and considering a sliding window of 11 years. In both cases, we quantify the broadness of the multifractal spectrum as well as its components defined by the deviations from the Gaussian distribution and the influence of the dependence between measurements. The results show that the chief contribution to the multifractal structure comes from the dynamical dependencies, mainly the weak ones, followed by a residual contribution of the deviations from Gaussianity. However, using the sliding window, we verify that the spikes in the non-Gaussian contribution occur at very close dates associated with climate changes determined in previous works by component analysis methods. Moreover, the strong non-Gaussian contribution found in the multifractal measures from the 1960s onwards is in agreement with global results very recently proposed in the literature.Comment: 21 pages, 10 figure