On the Hausdorff Dimension of Regular Points of Inviscid Burgers Equation with Stable Initial Data

Consider an inviscid Burgers equation whose initial data is a Lévy α-stable process Z with α>1. We show that when Z has positive jumps, the Hausdorff dimension of the set of Lagrangian regular points associated with the equation is strictly smaller than 1/α, as soon as α is close to 1. This gives a partially negative answer to a Conjecture of Janicki and Woyczynski (J. Stat. Phys. 86(1–2):277–299, 1997). Along the way, we contradict a recent Conjecture of Z. Shi () about the lower tails of integrated stable processes.     

Universality in movie rating distributions

In this paper histograms of user ratings for movies (1 \bigstar\bigstar,...,10 \bigstar\bigstar) are analysed. The evolving stabilised shapes of histograms follow the rule that all are either double- or triple-peaked. Moreover, at most one peak can be on the central bins 2 \bigstar\bigstar,...,9 \bigstar\bigstar and the distribution in these bins looks smooth `Gaussian-like’ while changes at the extremes (1 \bigstar\bigstar and 10 \bigstar\bigstar) often look abrupt. It is shown that this is well approximated under the assumption that histograms are confined and discretised probability density functions of Lévy skew α-stable distributions. These distributions are the only stable distributions which could emerge due to a generalized central limit theorem from averaging of various independent random variables as which one can see the initial opinions of users. Averaging is also an appropriate assumption about the social process which underlies the process of continuous opinion formation. Surprisingly, not the normal distribution achieves the best fit over histograms observed on the web, but distributions with fat tails which decay as power-laws with exponent –(1+α) (a = \frac43)(\alpha=\frac{4}{3}). The scale and skewness parameters of the Lévy skew α-stable distributions seem to depend on the deviation from an average movie (with mean about 7.6 \bigstar\bigstar). The histogram of such an average movie has no skewness and is the most narrow one. If a movie deviates from average the distribution gets broader and skew. The skewness pronounces the deviation. This is used to construct a one parameter fit which gives some evidence of universality in processes of continuous opinion dynamics about taste.     

Random quantum magnets with broad disorder distribution

We study the critical behavior of Ising quantum magnets with broadly distributed random couplings (J), such that P(ln J) ∼ | ln J|^{-1 - α}, α > 1, for large | ln J| (Lévy flight statistics). For sufficiently broad distributions, α < , the critical behavior is controlled by a line of fixed points, where the critical exponents vary with the Lévy index, α. In one dimension, with = 2, we obtained several exact results through a mapping to surviving Riemann walks. In two dimensions the varying critical exponents have been calculated by a numerical implementation of the Ma-Dasgupta-Hu renormalization group method leading to ≈ 4.5. Thus in the region 2 < α < , where the central limit theorem holds for | ln J| the broadness of the distribution is relevant for the 2d quantum Ising model. Received 6 December 2000 and Received in final form 22 January 2001     

Structure of Shocks in Burgers Turbulence¶with Stable Noise Initial Data

Burgers equation can be used as a simplified model for hydrodynamic turbulence. The purpose of this paper is to study the structure of the shocks for the inviscid equation in dimension 1 when the initial velocity is given by a stable Lévy noise with index α∈ (1/2,2]. We prove that Lagrangian regular points exist (i.e. there are fluid particles that have not participated in shocks at any time between 0 and t) if and only if α≤ 1 and the noise is not completely asymmetric, and that otherwise the shock structure is discrete. Moreover, in the Cauchy case α= 1, we show that there are no rarefaction intervals, i.e. at time t >0$, there are fluid particles in any non-empty open interval. Received: 28 September 1998 / Accepted: 12 January 1999     

Verhulst model with Lévy white noise excitation

The transient dynamics of the Verhulst model perturbed by arbitrary non-Gaussian white noise is investigated. Based on the infinitely divisible distribution of the Lévy process we study the nonlinear relaxation of the population density for three cases of white non-Gaussian noise: (i) shot noise; (ii) noise with a probability density of increments expressed in terms of Gamma function; and (iii) Cauchy stable noise. We obtain exact results for the probability distribution of the population density in all cases, and for Cauchy stable noise the exact expression of the nonlinear relaxation time is derived. Moreover starting from an initial delta function distribution, we find a transition induced by the multiplicative Lévy noise, from a trimodal probability distribution to a bimodal probability distribution in asymptotics. Finally we find a nonmonotonic behavior of the nonlinear relaxation time as a function of the Cauchy stable noise intensity.     

The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice ℤ3

We consider an Euclidean supersymmetric field theory in ℤ³ given by a supersymmetric Φ⁴ perturbation of an underlying massless Gaussian measure on scalar bosonic and Grassmann fields with covariance the Green’s function of a (stable) Lévy random walk in ℤ³. The Green’s function depends on the Lévy-Khintchine parameter with 0<α<2. For the Φ⁴ interaction is marginal. We prove for sufficiently small and initial parameters held in an appropriate domain the existence of a global renormalization group trajectory uniformly bounded on all renormalization group scales and therefore on lattices which become arbitrarily fine. At the same time we establish the existence of the critical (stable) manifold. The interactions are uniformly bounded away from zero on all scales and therefore we are constructing a non-Gaussian supersymmetric field theory on all scales. The interest of this theory comes from the easily established fact that the Green’s function of a (weakly) self-avoiding Lévy walk in ℤ³ is a second moment (two point correlation function) of the supersymmetric measure governing this model. The rigorous control of the critical renormalization group trajectory is a preparation for the study of the critical exponents of the (weakly) self-avoiding Lévy walk in ℤ³.     

The Effects of Time Delay on Stochastic Resonance in a Bistable System with Correlated Noises

The effects of time delay on stochastic resonance (SR) in a bistable system with time delay, correlated noises and periodic signal are studied by using the theory of signal-to-noise ratio (SNR). The expression of the SNR is derived under the adiabatic limit and the small delay time approximation. It is found that: (i) For the case of no correlations between multiplicative and additive noise, the delay time τ can enhance the SNR as a function of the multiplicative noise intensity α and it can restrain the SNR as a function of the additive noise intensity D; (ii) For the case of correlations between multiplicative and additive noise, τ can induce a minimum and maximum in curve of the SNR as a function of α, and can intensively restrain the SNR as a function of the D and there is a critical value of delay tim τ _c =0.1 in the height of the SNR peak with change of τ, i.e., when τ takes value blow τ _c , the τ boosts up the SNR as a function of the strength λ of correlations between multiplicative and additive noise, however, when τ takes value above τ _c , the τ restrains that.     

Stochastic resonance in multi-stable system driven by Lévy noise

In this paper, the stochastic resonance (SR) of a multi-stable system driven by Lévy noise is investigated by the mean signal-to-noise ratio gain (SNR-GM). The characteristics for resonant output of multi-stable system, governed by the system parameters (a and c), the noise amplification factor D of Lévy noise are investigated under different values of stability index α and asymmetry parameter β of Lévy noise. The results reveal that the parameter α is closer to 1, the amplitude of SNR-GM versus system parameter a (or c) is larger. The interval of SR presents a trend that the curve of SNR-GM shifts to the right with the increase of α especially when α > 1. In addition, the SNR-GM for different values of system parameter a (or c) exhibits a tendency to move to the left with the increase of system parameter c (or a). Finally, the simulation results prove that the proposed multi-stable model has better advantage than bistable system and monostable system in signal enhancement and SNR-GM performance.     

Stochastic resonance induced by a multiplicative periodic signal in a logistic growth model with correlated noises

The stochastic resonance (SR) phenomenon induced by a multiplicative periodic signal in a logistic growth model with correlated noises is studied by using the theory of signal-to-noise ratio (SNR) in the adiabatic limit. The expressions of the SNR are obtained. The effects of multiplicative noise intensity α and additive noise intensity D, and correlated intensity λ on the SNR are discussed respectively. It is found that the existence of a maximum in the SNR is the identifying characteristic of the SR phenomena. In comparison with the SR induced by additive periodic signal, some new features are found: (1) When SNR as a function of λ for fixed ratio of α and D, the varying of α can induce a stochastic multi-resonance, and can induce a re-entrant transition of the peaks in SNR vs λ; (2) There exhibits a doubly critical phenomenon for SNR vs D and λ, i.e., the increasing of D (or λ) can induce the critical phenomenon for SNR with respect to λ (or D); (3) The doubly stochastic resonance effect appears when α and D are simultaneously varying in SNR, i.e., the increment of one noise intensity can help the SR on another noise intensity come forth.      

A Renormalization Group Classification of Nonstationary and/or Infinite Second Moment Diffusive Processes

Anomalous diffusion processes are often classified by their mean square displacement. If the mean square displacement grows linearly in time, the process is considered classical. If it grows like t ^β with β<1 or β>1, the process is considered subdiffusive or superdiffusive, respectively. Processes with infinite mean square displacement are considered superdiffusive. We begin by examining the ways in which power-law mean square displacements can arise; namely via non-zero drift, nonstationary increments, and correlated increments. Subsequently, we describe examples which illustrate that the above classification scheme does not work well when nonstationary increments are present. Finally, we introduce an alternative classification scheme based on renormalization groups. This scheme classifies processes with stationary increments such as Brownian motion and fractional Brownian motion in the same groups as the mean square displacement scheme, but does a better job of classifying processes with nonstationary increments and/or processes with infinite second moments such as α-stable Lévy motion. A numerical approach to analyzing data based on the renormalization group classification is also presented.     

Stochastic dynamics and mean field approach in a system of three interacting species

The spatio-temporal dynamics of three interacting species, two preys and one predator, in the presence of two different kinds of noise sources is studied, by using Lotka-Volterra equations. A correlated dichotomous noise acts on the interaction parameter between the two preys, and a multiplicative white noise affects directly the dynamics of the three species. After analyzing the time behaviour of the three species in a single site, we consider a two-dimensional spatial domain, applying a mean field approach and obtaining the time behaviour of the first and second order moments for different multiplicative noise intensities. We find noise-induced oscillations of the three species with an anticorrelated behaviour of the two preys. Finally, we compare our results with those obtained by using a coupled map lattice (CML) model, finding a good qualitative agreement. However, some quantitative discrepancies appear, that can be explained as follows: i) different stationary values occur in the two approaches; ii) in the mean field formalism the interaction between sites is extended to the whole spatial domain, conversely in the CML model the species interaction is restricted to the nearest neighbors; iii) the dynamics of the CML model is faster since an unitary time step is considered.      

Non-Gaussian noise effects in the dynamics of a short overdamped Josephson junction

The role of thermal and non-Gaussian noise on the dynamics of driven short overdamped Josephson junctions is studied. The mean escape time of the junction is investigated considering Gaussian, Cauchy-Lorentz and Lévy-Smirnov probability distributions of the noise signals. In these conditions we find resonant activation and the first evidence of noise enhanced stability in a metastable system in the presence of Lévy noise. For Cauchy-Lorentz noise source, trapping phenomena and power law dependence on the noise intensity are observed.     

Complex networks from lévy noise

We construct complex networks from lévy noise (LN) using visibility algorithm proposed by Lucas lacasa el al. It is found that as the stability index α of the symmetric LN decreases, the corresponding complex network will transit from exponential network to long-tailed-degree-distribution one, and then to Gaussian one. The associated network for symmetric LN is the high clustering, hierarchy, and 18 community network. The properties of the associated networks for asymmetric LN except the skewness parameter β = −1 are similar with that for symmetric one. The associated network for the asymmetric LN with the skewness parameter β = −1 is always the exponential, high clustering, and hierarchy one with small k-clique communities.     

The analysis of stochastic resonance and bearing fault detection based on linear coupled bistable system under lévy noise

In this paper, the stochastic resonance (SR) phenomenon of the linear coupled bistable system induced by Lévy noise is analyzed. Meanwhile, the characteristics of Lévy noise is also analyzed according to its probability density functions (PDFs) of different stability index α, symmetry parameter β, scale parameter σ and location index μ. The mean of signal-noise ratio increase (MSNRI) is regarded as an index to measure the SR phenomenon. Then, the rules for MSNRI affected by noise intensity D are explored under different charastic indexes of Lévy noise, system parameters a, b, c and coupling coefficient r. The results are beneficial to the numerical simulation of single-frequency and multi-frequency weak signals detection based on single bistable system and linear coupled system respectively. It is found that the performance of the proposed system is better than single bistable system and results of bearing fault detection could also verify the conclusion.     

Moment equations for a spatially extended system of two competing species

The dynamics of a spatially extended system of two competing species in the presence of two noise sources is studied. A correlated dichotomous noise acts on the interaction parameter and a multiplicative white noise affects directly the dynamics of the two species. To describe the spatial distribution of the species we use a model based on Lotka-Volterra (LV) equations. By writing them in a mean field form, the corresponding moment equations for the species concentrations are obtained in Gaussian approximation. In this formalism the system dynamics is analyzed for different values of the multiplicative noise intensity. Finally by comparing these results with those obtained by direct simulations of the time discrete version of LV equations, that is coupled map lattice (CML) model, we conclude that the anticorrelated oscillations of the species densities are strictly related to non-overlapping spatial patterns.     

Correlated Lvy flight in external force fields

The correlated Lvy flight is studied analytically in terms of the fractional Fokker-Planck equation and simulated numerically by using the Langevin equation,where the usual white Lvy noise is generalized to an Ornstein-Uhlenbeck Lvy process(OULP)with a correlation timeτc.We analyze firstly the stable behavior of OULP.The probability density function of Lvy flight particle driven by the OULP in a harmonic potential is exactly obtained,which is also a Lvy-type one withτc-dependence width;when the particle is bounded by a quartic potential,its stationary distribution has a bimodality shape and becomes noticeable with the increase of τc.     

Power, Lévy, exponential and Gaussian-like regimes in autocatalytic financial systems

We study by theoretical analysis and by direct numerical simulation the dynamics of a wide class of asynchronous stochastic systems composed of many autocatalytic degrees of freedom. We describe the generic emergence of truncated power laws in the size distribution of their individual elements. The exponents α of these power laws are time independent and depend only on the way the elements with very small values are treated. These truncated power laws determine the collective time evolution of the system. In particular the global stochastic fluctuations of the system differ from the normal Gaussian noise according to the time and size scales at which these fluctuations are considered. We describe the ranges in which these fluctuations are parameterized respectively by: the Lévy regime α < 2, the power law decay with large exponent ( α > 2), and the exponential decay. Finally we relate these results to the large exponent power laws found in the actual behavior of the stock markets and to the exponential cut-off detected in certain recent measurement. Received 29 July 2000 and Received in final form 25 September 2000     

Anomalous Pulsation

Subordinating regular diffusion – namely, Brownian motion – to random time flows generated by Lévy noises may result in anomalous diffusion. Motivated by this phenomena, and by the recent interest in the phenomena of blinking in various physical systems, we explore the subordination of regular stochastic pulsation – namely, Poisson process – to random time flows generated by Lévy noises. We show that such subordination may yield, analogous to the case of diffusion, anomalous pulsation. Anomalous pulsation displays the following anomalous behaviors, which are impossible in the case of regular pulsation: (i) simultaneous emission of multiple pulses; (ii) non-linear local pulsation rates; (iii) clustering of pulsation epochs.     

Alternative interpretations of power-law distributions found in nature

We investigate two inherently different classes of probability density functions (pdfs) that share the common property of power law tails: the α-stable Le?vy process and the linear Markov diffusion process with additive and multiplicative Gaussian noise. Dynamical processes described by these distributions cannot be uniquely identified as belonging to one or the other class either by diverging variance due to power-law tails in the pdf or by the possible existence of skew. However, there are distinguishing features that may be found in sufficiently well sampled time series. We examine these features and discuss how they may guide the development of proper approximations to equations of motion underlying dynamical systems. An additional result of this research was the identification of a variable describing the relative importance of the multiplicative and independent additive noise forcing in our linear Markov process. The distribution of this variable is generally skewed, depending on the level of correlation between the additive and multiplicative noise.