Stieltjes Integrals of Hölder Continuous Functions with Applications to Fractional Brownian Motion

We give a new estimate on Stieltjes integrals of Hölder continuous functions and use it to prove an existence-uniqueness theorem for solutions of ordinary differential equations with Hölder continuous forcing. We construct stochastic integrals with respect to fractional Brownian motion, and establish sufficient conditions for its existence. We prove that stochastic differential equations with fractional Brownian motion have a unique solution with probability 1 in certain classes of Hölder-continuous functions. We give tail estimates of the maximum of stochastic integrals from tail estimates of the Hölder coefficient of fractional Brownian motion. In addition we apply the techniques used for ordinary Brownian motion to construct stochastic integrals of deterministic functions with respect to fractional Brownian motion and give tail estimates of its maximum.     

气液态分子运动的Hurst指数及相图研究

本文运用分子动力学模拟方法,模拟了不同温度条件下饱和气体和液体氩分子的运动。对气液状态下分子运动在速度空间的演化相图进行了比较。根据分形理论,对实际分子的分数布朗运动特性进行了讨论。应用本文提出的约化规模法,获得了不同温度条件下分子分数布朗运动的Hurst指数和轨迹分维数。饱和液相分子运动的Hurst指数并不随温度单调上升,存在一个极小值。     

具有列维飞行与布朗运动特征的循环竞争博弈及物种稳定共存条件

循环竞争博弈常被用来研究物种多样性.以前有关循环竞争博弈的研究工作所考虑的相互作用距离模式包括最近邻、取固定距离或一定距离以内的随机值,这与实际情况不相符.考虑到实际生物系统中物种个体做列维飞行与布朗运动的情况广泛存在,综合考虑了最近邻相互作用模式和列维飞行(布朗运动)长程相互作用模式,对循环竞争博弈及保持物种多样性的条件进行了研究.得到了最大飞行距离与选择概率的临界关系(包括Logistic式和指数式关系),进一步得到了幂指数与选择概率的临界关系,以及保持物种共存的条件.     

Degree distributions of the visibility graphs mapped from fractional Brownian motions and multifractal random walks

The dynamics of a complex system is usually recorded in the form of time series, which can be studied through its visibility graph from a complex network perspective. We investigate the visibility graphs extracted from fractional Brownian motions and multifractal random walks, and find that the degree distributions exhibit power-law behaviors, in which the power-law exponent α is a linear function of the Hurst index H of the time series. We also find that the degree distribution of the visibility graph is mainly determined by the temporal correlation of the original time series with minor influence from the possible multifractal nature. As an example, we study the visibility graphs constructed from three Chinese stock market indexes and unveil that the degree distributions have power-law tails, where the tail exponents of the visibility graphs and the Hurst indexes of the indexes are close to the α∼H linear relationship.     

长时相干效应与分形布朗运动

 我们研究了阻尼布朗粒子，在具有幂律长时相干C(t)~t^-β（0<β<1，1<β<2）的无规涨落力作用下的运动情况。我们发现它是作分形布朗运动，而不是作普通的布朗运动，而且，找出了分形布朗运动的有效Fokker-Planck方程，以及相应的精确解。于是第一次把长时相干效应和分形布朗运动建立了定量的联系。     

Burgers equation with self-similar gaussian initial data: Tail probabilities

The statistical properties of solutions of the one-dimensional Burgers equation in the limit of vanishing viscosity are considered when the initial velocity potential is fractional Brownian motion (FBM). We establish the asymptotic power-law order for log-probability of large values, both velocity and shock (amplitude of velocity discontinuity). This confirms the conjecture of U. Frisch and his collaborators. Rigorous results for this problem were previously derived for the case of Brownian motion using Markov techniques. Our approach is based on the intrinsic properties of FBM and the theory of extreme values for Gaussian processes.     

Finite size correction for fixed word length Zipf analysis

Zipf’s original law deals with the statistics of ranked words in natural languages. It has recently been generalized to “words” defined as n-tuples of symbols derived by translation of real-valued univariate timeseries into a literal sequence. We verify that the rank-frequency plot of these words shows, for fractional Brownian motion, the previously found power laws, but with large finite length corrections. We verify a finite size scaling ansatz for these corrections and, as aresult, demonstrate greatly improved estimates of the (generalized) Zipf exponents. This allows us to find the correct relation between the Zipf exponent and the Hurst exponent characterizing the fractional Brownian motion.     

Parameter estimation for power-law distributions by maximum likelihood methods

Distributions following a power-law are an ubiquitous phenomenon. Methods for determining the exponent of a power-law tail by graphical means are often used in practice but are intrinsically unreliable. Maximum likelihood estimators for the exponent are a mathematically sound alternative to graphical methods.     

多标度分形上的输运方程

 利用质量守恒定律,得到在三维情况下多标度无序分形介质中的一般输运方程,并在讨论布朗运动和分数布朗运动以及在分形介质上的标准扩散方程的基础上,得到多标度无序分形介质中的分数阶输运方程。     

On the relation between the fractional Brownian motion and the fractional derivatives

The definition and simulation of fractional Brownian motion are considered from the point of view of a set of coherent fractional derivative definitions. To do it, two sets of fractional derivatives are considered: (a) the forward and backward and (b) the central derivatives, together with two representations: generalised difference and integral. It is shown that for these derivatives the corresponding autocorrelation functions have the same representations. The obtained results are used to define a fractional noise and, from it, the fractional Brownian motion. This is studied. The simulation problem is also considered.     

Power-law tails from multiplicative noise

We show that the well-known linear Langevin equation, modeling the Brownian motion and leading to a Gaussian stationary distribution of the corresponding Fokker-Planck equation, is changed by the smallest multiplicative noise. This leads to a power-law tail of the distribution for sufficiently large momenta. At finite ratio of the correlation strength for the multiplicative and the additive noises the stationary energy distribution becomes exactly the Tsallis distribution.     

Predicting the long tail of book sales: Unearthing the power-law exponent

The concept of the long tail has recently been used to explain the phenomenon in e-commerce where the total volume of sales of the items in the tail is comparable to that of the most popular items. In the case of online book sales, the proportion of tail sales has been estimated using regression techniques on the assumption that the data obeys a power-law distribution. Here we propose a different technique for estimation based on a generative model of book sales that results in an asymptotic power-law distribution of sales, but which does not suffer from the problems related to power-law regression techniques. We show that the proportion of tail sales predicted is very sensitive to the estimated power-law exponent. In particular, if we assume that the power-law exponent of the cumulative distribution is closer to 1.1 rather than to 1.2 (estimates published in 2003, calculated using regression by two groups of researchers), then our computations suggest that the tail sales of Amazon.com, rather than being 40% as estimated by Brynjolfsson, Hu and Smith in 2003, are actually closer to 20%, the proportion estimated by its CEO.     

How fast do stock prices adjust to market efficiency? Evidence from a detrended fluctuation analysis

In this paper we analyse price fluctuations with the aim of measuring how long the market takes to adjust prices to weak-form efficiency, i.e., how long it takes for prices to adjust to a fractional Brownian motion with a Hurst exponent of 0.5. The Hurst exponent is estimated for different time horizons using detrended fluctuation analysis–a method suitable for non-stationary series with trends–in order to identify at which time scale the Hurst exponent is consistent with the efficient market hypothesis. Using high-frequency share price, exchange rate and stock data, we show how price dynamics exhibited important deviations from efficiency for time periods of up to 15 min; thereafter, price dynamics was consistent with a geometric Brownian motion. The intraday behaviour of the series also indicated that price dynamics at trade opening and close was hardly consistent with efficiency, which would enable investors to exploit price deviations from fundamental values. This result is consistent with intraday volume, volatility and transaction time duration patterns.     

海杂波背景下的目标检测新方法

提出了一种基于分形布朗运动模型的S波段雷达海杂波分形维数提取方法.分析了基于记忆库混沌时间序列预测方法，引入一种改进核函数的支持向量机分类器.在此基础上，提出了一种新的海杂波背景下目标检测方法.应用S波段雷达实测海杂波数据，计算得到了该信号的分形维数与Lyapunov指数，验证了S波段雷达海杂波的混沌分形特性.仿真实验结果验证了该方法具有较强的检测能力和抗杂波性能. 关键词： 分形布朗运动 分形维数 记忆库预测方法 支持向量机分类器     

Generating synthetic time series from Bak-Sneppen co-evolution model mixtures

The Bak-Sneppen model of co-evolution is used to derive synthetic time series with a priori specified fractal dimension (or Hurst exponent) through a mixing of processes in various lattice dimensions. Both theoretical and numerical analyses concern the avalanches at the critical threshold and provide a model for time series reconstruction that can be tested as an alternative to the classical fractional Brownian motion (fBm) because of differences in properties. New results on critical threshold and avalanche structure are obtained up to Euclidean dimension d=6. The method involves a lattice-based structure and therefore is suitable for the application of parallel computing.     

Emergence of self-similarity in football dynamics

The multiplayer dynamics of a football game is analyzed to unveil self-similarities in the time evolution of player and ball positioning. Temporal fluctuations in both the team-turf boundary and the ball location are uncovered to follow the rules of fractional Brownian motion with a Hurst exponent of H ~ 0.7. The persistence time below which self-similarity holds is found to be several tens of seconds, implying a characteristic time scale that governs far-from-equilibrium motion on a playing field.     

Measurement of Hurst exponents for semiconductor laser phase dynamics

The phase dynamics of a semiconductor laser with optical feedback is studied by construction of the Hilbert phase from its experimentally measured intensity time series. The Hurst exponent is evaluated for the phase fluctuations and grows from 0.5 to approximately 0.7 (indicating fractional Brownian motion) as the feedback strength is increased. A comparison with numerical computations based on a delay-differential equation model shows excellent agreement and reveals the relative roles of spontaneous emission noise and deterministic dynamics for different feedback strengths.     

Structural and flow properties of binary media generated by fractional Brownian motion models

In the present paper, the structural and flow properties of binary media generated by two-dimensional lattices that follow fractional Brownian motion statistics are studied. A modification of the midpoint displacement and random addition method is employed in order to generate multicell binary media with sizes that are considerably larger than the correlation length of the medium. Several structural properties, such as the autocorrelation function, the surface area, and the percolation threshold, are studied for different values of porosity and degree of correlation. In addition, transport properties are investigated in the above media, by solving numerically the momentum and continuity equations, to determine the absolute permeability of the medium in directions parallel and normal to the fractional Brownian motion (fBm) plane. It is found that multicell fBm porous media possess very interesting structural properties that are functions of the Hurst exponent and porosity, and are independent of the lattice size, in contrast to the traditional single-cell fBm media. In addition, they exhibit stronger structural correlation, lower specific surface area, higher percolation threshold, and lower permeabilities than those of the corresponding single-cell porous media.     

分形理论在分子动力学模拟中的应用

本文基于分形理论提出了一个假说,认为实际流体分子的无规运动可以用分数布朗函数作为概率密度函数来描写,而其分数维数可根据分子运动的图像确定。本文以流体Ar作为对象进行了分子动力学模拟,根据分子动力学模拟的结果提取了运动的分数维数,构造了描述分子无规运动的分数布朗函数,并对所提出的假说进行了验证。     

Lévy-type complex diameter modulation in semiconductor nanowire growth

We have observed a non-periodic diameter modulation in SiC nanowire growth. The modulation was tentatively attributed to a stick-slip motion in a self-organized vapor-liquid-solid (VLS) growth, and was characterized by methods of time-series analysis. From the power-law decay behavior of a fat tail of the probability distribution of increments with the exponent 2.1±0.2, the modulation was interpreted as a Lévy flight. The degree of determinism is also discussed.