some exact results for the trapping of subdiffusive particles in one dimension we study a generalization of the standard trapping problem of random walk theory in which particles move subdiffusively on a one dimensional lattice we consider the cases in which the lattice is filled with a one sided and a two sided random distribution of static absorbing traps with concentration c the survival probability t that the random walker is not trapped by time t is obtained exactly in both versions of the problem through a fractional diffusion approach comparison with simulation results is made

on the asymptotic expansion of the empirical process of long memory moving averages let x n i 1 a i e n i where the e i are iid with mean 0 and finite fourth moment and the a i are regularly varying with index where 1 2 1 so that x n has long range dependence this covers an important class of the fractional arima process for r 0 let y n r n n 1 1 j1 jr r s 1 a js e n js y n 0 n 2 n r var y n r and f r the rth derivative of the distribution function of x n the y n r are uncorrelated and are stochastically decreasing in r for any positive integer p 2 1 1 it is shown under mild regularity conditions that with probability 1 n n 1 i x n x p r 0 1 r f r x y n r o n n p uniformly for all x r o 1 2 1 2 p 1 2 this generalizes a host of existing results and provides the vehicle for a number of statistical applications

molecular evolution modeled as a fractal statistical process abstract modeling the rate of nucleotide substitutions in dna as a dichotomous stochastic process with an inverse power law correlation function describes evolution by a fractal stochastic process fsp this fsp model agrees with recent findings on the relationship between the variance and mean number of synonymous and nonsynonymous substitutions in 49 different genes in mammals that being a power law increase in the ratio of the variance to the mean the index of dispersion with the number of substitutions in a protein the probability of a given number of substitutions occuring in a time t is determined by a fractional diffusion equation whose solution is a truncated levy distribution implying that evolution is a levy process in time and yields the same functional behavior for the variance in the number of substitutions as does the fsp model in addition to obtaining these relationships the fsp model implies lognormal statistics for the index of dispersion as a function of the mean number of substitutions in a protein which is confirmed in the regression of the fsp model to data lognormal statistics suggest that molecular evolution can be viewed as a multiplicative stochastic process rather than the linear additive processes of darwinian selection and drift

spectral properties of the cauchy process on half line and interval we study the spectral properties of the transition semigroup of the killed one dimensional cauchy process on the half line 0 a nd the interval 1 1 this process is related to the square root of one dimensional laplacian a d 2 dx 2 with a dirichlet exterior condition on a complement of a domain and to a mixed steklov problem in the half plane for the half line an explicit formula for generalized eigenfunctions of a is derived and then used to construct a spectral representation of a explicit formulas for the transition density of the killed cauchy process on the half line or the heat kernel of a in 0 and for the distribution of the first exit time from the half line follow the formula for is also used to construct approximations to eigenfunctions of a in the interval for the eigenvalues n of a in the interval the asymptotic formula n n 2 8 o 1 n is derived and all eigenvalues n are proved to be simple finally efficient numerical methods of estimation of eigenvalues n are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to the ninth decimal point

random walk in chemical space of cantor dust as a paradigm of superdiffusion we point out that the chemical space of a totally disconnected cantor dust k n ensuremath subset e n is a compact metric space c n with the spectral dimension d s d ensuremath ell ngd where d and d ensuremath ell n are the fractal and chemical dimensions of k n respectively hence we can define a random walk in the chemical space as a markovian gaussian process the mapping of a random walk in c n into k n ensuremath subset e n defines the quenched l evy flight on the cantor dust with a single step duration independent of the step length the equations describing the superdiffusion and diffusion reaction front propagation ruled by the local quenched l evy flight on k n ensuremath subset e n are derived the use of these equations to model superdiffusive phenomena observed in some physical systems in which propagators decay faster than algebraically is discussed

fractional levy stable motion can model subdiffusive dynamics we show in this paper that the sample time average mean squared displacement msd of the fractional levy stable motion behaves very differently from the corresponding ensemble average second moment while the ensemble average msd diverges for 2 the sample msd may exhibit either subdiffusion normal diffusion or superdiffusion thus h self similar levy stable processes can model either a subdiffusive diffusive or superdiffusive dynamics in the sense of sample msd we show that the character of the process is controlled by a sign of the memory parameter d h 1 we also introduce a sample p variation dynamics test which allows to distinguish between two models of subdiffusive dynamics finally we illustrate a subdiffusive behavior of the fractional levy stable motion on biological data describing the motion of individual fluorescently labeled mrna molecules inside live e coli cells but it may concern many other fields of contemporary experimental physics doi 10 1103 physreve 82 021130

a comparison of estimators for 1 f noise abstract we use a monte carlo approach to investigate the performance of five different time series estimators of the exponent in 1 f noise we find that a maximum likelihood estimator is markedly superior to fourier regression methods and hurst exponent methods the results indicate that useful estimates of can be made from time series that are much shorter than generally presumed

quantifying long range correlations with a multiscale ordinal pattern approach abstract in this paper we use the ordinal patterns probabilities associated with fractional brownian motions for estimating the hurst exponent of artificially generated and experimentally measured data numerical analysis show a reliable estimation of this scaling parameter even when data with low resolution are analysed robustness to observational noise is also obtained several experimental applications allow us to confirm the practical utility of the proposed approach we contrast results obtained by implementing this multiscale symbolic tool with those obtained from the classical detrended fluctuation analysis

multidimensional solutions of space fractional diffusion equations green s functions and propagators for the multidimensional anisotropic spacetime fractional diffusion equation are obtained in integral form uniqueness of the solutions is studied in the framework

fractional diffusion probability distributions and random walk models abstract we present a variety of models of random walk discrete in space and time suitable for simulating random variables whose probability density obeys a space time fractional diffusion equation

dynamical model for dna sequences we address the problem of dna sequences developing a dynamical method based on the assumption that the statistical properties of dna paths are determined by the joint action of two processes one deterministic with long range correlations and the other random and ensuremath delta function correlated the generator of the deterministic evolution is a nonlinear map belonging to a class of maps recently tailored to mimic the processes of weak chaos that are responsible for the birth of anomalous diffusion it is assumed that the deterministic process corresponds to unknown biological rules that determine the dna path whereas the noise mimics the influence of an infinite dimensional environment on the biological process under study we prove that the resulting diffusion process if the effect of the random process is neglected is an ensuremath alpha stable l evy process with 12 we also show that if the diffusion process is determined by the joint action of the deterministic and the random process the correlation effects of the deterministic dynamics are cancelled on the short range scale but show up in the long range one we denote our prescription to generate statistical sequences as the copying mistake map cmm we carry out our analysis of several dna sequences and their cmm realizations with a variety of techniques and we especially focus on a method of regression to equilibrium which we call the onsager analysis with these techniques we establish the statistical equivalence of the real dna sequences with their cmm realizations we show that long range correlations are present in exons as well as in introns but are difficult to detect since the exon dynamics is shown to be determined by the entanglement of three distinct and independent cmm s

boundary problems for diffusion in a fluctuating potential abstract using a newly developed formalism adapted for diffusion in the presence of partially absorbing boundaries we treat the overdamped motion of a particle in a fluctuating potential and in a restricted space the potential is assumed to be linear with a slope switching between two possible values the switching process is taken as the standard markov alternating process we first construct the free dynamics propagator no boundaries which controls the evolution of the joint probability for both the particle s position and the state of the potential secondly at a given spatial point we affix a two channel boundary a given channel absorbs the trajectories arriving at the boundary with the potential in a given state we demonstrate the asymmetry of the filling process for the individual channels and we discuss their asymptotic occupation depending on the mean switching frequency finally we study the motion in a domain restricted by two boundaries of the above two channel type and the corresponding first passage time the simple problem setting allows for an explicit analysis of the correlation between the particle s motion and the potential switching process

fractional laplacian in bounded domains the fractional laplacian operator ensuremath ensuremath ensuremath delta ensuremath alpha 2 appears in a wide class of physical systems including l evy flights and stochastic interfaces in this paper we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval the implementation of boundary conditions is justified by appealing to two physical models namely hopping particles and elastic springs the eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions some analytical results concerning the structure of the eigenvalue spectrum are also obtained

fractional brownian motion and multifractional brownian motion of riemann liouville type the relationship between standard fractional brownian motion fbm and fbm based on the riemann liouville fractional integral or rl fbm is clarified the absence of stationary property in the increment process of rl fbm is compensated by a weaker property of local stationarity and the stationary property for the increments of the large time asymptotic rl fbm generalization of rl fbm to the rl multifractional brownian motion rl mbm can be carried out by replacing the constant holder exponent by a time dependent function rl mbm is shown to satisfy a weaker scaling property known as the local asymptotic self similarity this local scaling property can be translated into the small scale behaviour of the associated scalogram by using the wavelet transform

hierarchical spatio temporal coupling in fractional wanderings i continuous time weierstrass flights abstract the one dimensional continuous time weierstrass flights ctwf model is considered in the framework of the nonseparable continuous time random walks formalism ctrw a novel spatio temporal coupling is introduced by assuming that in each scale the probability density for the flight and for waiting are joined hence we treat the spatio temporal relations in terms of the self similar structure of the weierstrass process this stochastic structure is characterized by the spatial fractional dimension 1 representing the flights and the temporal one 1 representing the waiting time was assumed here as the only independent truncation range in this work we study the asymptotic properties of the ctwf model for example by applying the method of steepest descents we obtained the particle propagator in the approximate scaling form p x t t pt 2 f where the scalingfunction f pt exp const pt while the scaling variable x t pt 2 is large the principal result of our analysis is that the exponents and depend on more fundamental ones and what leads to a novel scaling as a result of competition between exponents and an enhanced dispersive or normal diffusion was recognized in distinction from the prediction of the separable ctrw model where the enhanced diffusion is lost and the dispersive one is strongly limited it should be noted that we compare here partially thermalized versions of both approaches where some initial fluctuations were also included in agreement with the spirit of the theory of the renewal processes having the propogator we calculated for example the particle mean square displacement and found its novel asymptotic scaling with time for enhanced diffusion given by t1 2 1 in distinction from its diverging for

eigenvalues of the fractional laplace operator in the interval abstract two term weyl type asymptotic law for the eigenvalues of the one dimensional fractional laplace operator 2 0 2 in the interval 1 1 is given the n th eigenvalue is equal to n 2 2 8 o 1 n simplicity of eigenvalues is proved for 1 2 l 2 and l properties of eigenfunctions are studied we also give precise numerical bounds for the first few eigenvalues

on the bahadur representation of sample quantiles for dependent sequences we establish the bahadur representation of sample quantiles for linear and some widely used nonlinear processes local fluctuations of empirical processes are discussed applications to the trimmed and winsorized means are given our results extend previous ones by establishing sharper bounds under milder conditions and thus provide new insight into the theory of empirical processes for dependent random variables

generalized space time fractional diffusion equation with composite fractional time derivative abstract we investigate the solution of space time fractional diffusion equations with a generalized riemann liouville time fractional derivative and riesz feller space fractional derivative the laplace and fourier transform methods are applied to solve the proposed fractional diffusion equation the results are represented by using the mittag leffler functions and the fox h function special cases of the initial and boundary conditions are considered numerical scheme and grunwald letnikov approximation are also used to solve the space time fractional diffusion equation the fractional moments of the fundamental solution of the considered space time fractional diffusion equation are obtained many known results are special cases of those obtained in this paper we investigate also the solution of a space time fractional diffusion equations with a singular term of the form x t 1 0

estimating the dimension of a fractal summary we suggest refinements of the box counting method which address the obvious problems caused by the incomplete information and inaccessibility of the limit a method for the statistical analysis of these corrected data is developed and tested on simulated and real data

cauchy noise and affiliated stochastic processes by departing from the previous attempt phys rev e 51 4114 1995 we give a detailed construction of conditional and perturbed markov processes under the assumption that the cauchy law of probability replaces the gaussian law appropriate for the wiener process as the model of primordial noise all considered processes are regarded as probabilistic solutions of the so called schrodinger interpolation problem whose validity is thus extended to the jump type processes and their step process approximants

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

hierarchical spatio temporal coupling in fractional wanderings ii diffusion phase diagram for weierstrass walks abstract the one dimensional weierstrass walks ww model is developed in the framework of the extended nonseparable continuous time random walk ctrw formalism 1 2 3 4 5 6 7 8 9 10 11 12 the ww model is a lacunary foundation of levy walks 6 7 8 9 10 11 12 generalized to a nonconstant velocity this nonconstant velocity is introduced by hierarchical coherent spatio temporal coupling adopted from the continuous time weierstrass flights ctwf model developed in the previous paper 13 hence for the probability density to pass by a walker in a single step a random displacement with finite velocity is constructed as a geometric series of the corresponding probability densities defined within a sequence of spatio temporal scales we calculated analytically and by monte carlo simulations the asymptotic in time mean square displacement msd of the walker obtaining very good agreement between both approaches also comparison with corresponding results of the ctwf model is made considering different types of the diffusion exponents we constructed a diffusion phase diagram on the plane defined by the spatial and temporal fractional dimensions which characterize our coupling we obtained a diffusion exponent as a function of these fractional dimensions covering all types of nonbiased diffusion known up to now from the dispersive one over the normal enhanced ballistic and hyperdiffusion up to the richardson law of diffusion which defines here a part of the borderline of the region where the msd diverges we observed that all kinds of anomalous diffusion are characterized by three types of diffusion exponents only for example we found an asymptotic scaling of msd to occur with time for enhanced diffusion which was discovered by us within the ctwf model but is valid for a much more extended range of spatio and temporal fractional dimensions

solving fractional schrodinger type spectral problems cauchy oscillator and cauchy well this paper is a direct offspring of the work of garbaczewski and stephanovich levy flights and nonlocal quantum dynamics j math phys 54 072103 2013 where basic tenets of the nonlocally induced random and quantum dynamics were analyzed a number of mentions were made with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so called fractional quantum mechanics spectral problems presently we give a decisive computer assisted proof for an exemplary finite and ultimately infinite cauchy well problem that spectral solutions proposed so far were plainly wrong as a constructive input we provide an explicit spectral solution of the finite cauchy well the infinite well emerges as a limiting case in a sequence of deepening finite wells the employed numerical methodology algorithm based on the strang splitting method has been tested for an exemplary cauchy oscillator problem whose analytic solution is available an impact of the inherent spatial nonlocality of motion generators upon computer assisted outcomes potentially defective in view of various cutoffs i e detailed eigenvalues and shapes of eigenfunctions has been analyzed

fractional schrodinger equation with riesz feller derivative for delta potentials the fractional schrodinger equation with the riesz feller derivative is discussed and solved when the potential involves delta functions some results in the literature are generalized

green function of the double fractional fokker planck equation path integral and stochastic differential equations the statistics of rare events the so called black swan events is governed by non gaussian distributions with heavy power like tails we calculate the green functions of the associated fokker planck equations and solve the related stochastic differential equations we also discuss the subject in the framework of path integration

dimensional influence on blow up in a superdiffusive medium thermal blow up in a superdiffusive medium with a localized energy source is examined for a spatial domain of infinite extent in one two and three dimensions an analysis of a nonlinear model of this problem reveals that the occurrence of a blow up depends upon the spatial dimension of the superdiffusive medium in one dimension a blow up always occurs in two and three dimensions a blow up can be avoided if the superdiffusive properties are sufficiently enhanced the asymptotic growth of the temperature near blow up is determined for energy sources whose output increases in either an algebraic or an exponential manner the results of the analysis have potential application to porous materials in which a pore filling agent is introduced to enhance the thermal transport properties

measuring complexity nonextensivity and chaos in the dna sequence of the major histocompatibility complex abstract we analyze 4 mb sequences of the major histocompatibility complex mhc which is a dna segment on chromosome 6 with high gene density controlling many immunological functions and associated with many diseases the analysis is based on modern theoretical and mathematical tools of complexity theory such as nonlinear time series analysis and tsallis non extensive statistics the results revealed that the dna complexity and self organization can be related to fractional dynamical nonlinear processes with low dimensional deterministic chaotic and non extensive statistical character which generate the dna sequences under the extremization of tsallis q entropy principle while it still remains an open question as to whether the dna walk is a fractional brownian motion fbm a static anomalous diffusion process or a non gaussian dynamical fractional anomalous diffusion process the results of this study testify for the latter providing also a possible explanation for the previously observed long range power law correlations of nucleotides as well as the long range correlation properties of coding and non coding sequences present in dna sequences

fractional diffusion exact representations of spectral functions for all the relevant transformed spaces i e fourier laplace and fourier laplace we present exact solutions of a fractional diffusion equation describing random transport on fractals the potential importance of such spectral representations lies in their applications to interpreting experimental measurements of anomalous diffusion processes in contrast to the well known asymptotic results the exact representations provide a much broader basis for comparison with data

phase transitions in a probabilistic cellular automaton growth kinetics and critical properties we investigate a discrete time kinetic model without detailed balance which simulates the phase segregation of a quenched binary alloy the model is a variation on the rothman keller cellular automaton in which particles of type a b move toward domains of greater concentration of a b modifications include a fully occupied lattice and the introduction of a temperature like parameter which endows the system with a stochastic evolution using computer simulations we examine domain growth kinetics in the two dimensional model for long times after a quench from disorder we find that the average domain sizer t t1 3 in agreement with the prediction of lifshitz slyozov wagner theory using a variety of methods we analyze the critical properties of the associated second order transition our analysis indicates that this model does not fall within either the ising or mean field classes

hurst exponent estimation of locally self similar gaussian processes using sample quantiles this paper is devoted to the introduction of a new class of consistent estimators of the fractal dimension of locally self similar gaussian processes these estimators are based on convex combinations of sample quantiles of discrete variations of a sample path over a discrete grid of the interval 0 1 we derive the almost sure convergence and the asymptotic normality for these estimators the key ingredient is a bahadur representation for sample quantiles of non linear functions of gaussians sequences with correlation function decreasing as k alpha l k for some alpha 0 and some slowly varying function l cdot

a bahadur type representation for empirical quantiles of a large class of stationary possibly infinite variance linear processes bahadur has obtained an asymptotic almost sure representation for empirical quantiles of independent and identically distributed random variables in this paper we present an analogous result for a large class of stationary linear processes

bayesian estimation of self similarity exponent in this study we propose a bayesian approach to the estimation of the hurst exponent in terms of linear mixed models even for unevenly sampled signals and signals with gaps our method is applicable we test our method by using artificial fractional brownian motion of different length and compare it with the detrended fluctuation analysis technique the estimation of the hurst exponent of a rosenblatt process is shown as an example of an h self similar process with non gaussian dimensional distribution additionally we perform an analysis with real data the dow jones industrial average closing values and analyze its temporal variation of the hurst exponent

levy flights in confining potentials we analyze confining mechanisms for levy flights when they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport two classes of stochastic jump type processes are considered those driven by langevin equation with levy noise and those named topological levy processes occurring in systems with topological complexity such as folded polymers or complex networks whose langevin representation is unknown and possibly nonexistent our major finding is that both above classes of processes stay in affinity and may share common stationary probability density even if their detailed dynamical behavior look different this near equilibrium observation seems to be generic to a broad class of levy noise driven processes such as e g superdiffusion on folded polymers geophysical flows and even climatic changes

long term memory in stock market prices a test for long term memory that is robust to short range dependence is developed it is a modification of the r s statistic and the relevant asymptotic sampling theory is derived via functional central limit theory contrary to previous findings when applied to daily and monthly stock returns indexes over several time periods this test yields no evidence of long range dependence once short range dependence is accounted for monte carlo experiments show that the modified r s test has power against at least two specific models of long term memory suggesting that models with short range dependence may adequately capture the behavior of historical stock returns copyright 1991 by the econometric society

diffusion equations for a markovian jumping process we consider a markovian jumping process which is defined in terms of the jump size distribution and the waiting time distribution with a position dependent frequency in the diffusion limit we assume the power law form for the frequency for small steps we derive the fokker planck equation and show the presence of the normal diffusion subdiffusion and superdiffusion for the l evy distribution of the step size we construct a fractional equation which possesses a variable coefficient and solve it in the diffusion limit then we calculate fractional moments and define the fractional diffusion coefficient as a natural extension to the cases with the divergent variance we also solve the master equation numerically and demonstrate that there are deviations from the l evy stable distribution for large wave numbers

time and space fractional partial differential equations the fundamental solution for time and space fractional partial differential operator dt a2 2 0 is given in terms of the fox s h function here the time fractional derivative in the sense of generalized functions distributions dt is defined by the convolution dt f t t f t where t t 1 and f t 0 as t 0 and the fractional n dimensional laplace operator 2 is defined by its fourier transform with respect to spatial variable f 2g x f g x the solutions for initial value problems for time and space fractional partial differential equation in the sense of caputo and riemann liouville time fractional derivatives respectively are obtained by the fundamental solution

the generalized cattaneo equation for the description of anomalous transport processes the cattaneo equation which describes a diffusion process with a finite velocity of propagation is generalized to describe anomalous transport three possible generalizations are proposed each one supported by a different scheme continuous time random walks non local transport theory and delayed flux force relation the properties of these generalizations are studied in both the long time and the short time regimes in the long time limit we recover the mean square displacement which is characteristic for these anomalous processes as expected the short time behaviour is modified in comparison to generalized diffusion equations

effect of trends on detrended fluctuation analysis scaling behavior we find that crossovers result from the competition between the scaling of the noise and the apparent scaling of the trend we study how the characteristics of these crossovers depend on i the slope of the linear trend ii the amplitude and period of the periodic trend iii the amplitude and power of the power law trend and iv the length as well as the correlation properties of the noise surprisingly we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws i e long range power law dependence of the position of the crossover on the parameters of the trends we show that the dfa result of noise with a trend can be exactly determined by the superposition of the separate results of the dfa on the noise and on the trend assuming that the noise and the trend are not correlated if this superposition rule is not followed this is an indication that the noise and the superposed trend are not independent so that removing the trend could lead to changes in the correlation properties of the noise in addition we show how to use dfa appropriately to minimize the effects of trends how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation or if the crossover is due to a trend without any transition in the dynamical properties of the noise

fractional schrodinger equation for a particle moving in a potential well in this paper the fractional schrodinger equation that contains the quantum riesz fractional derivative instead of the laplace operator is revisited for the case of a particle moving in the infinite potential well in the recent papers m jeng s l y xu e hawkins and j m schwarz on the nonlocality of the fractional schrodinger equation j math phys 51 062102 2010 10 1063 1 3430552 and s s bayin on the consistency of the solutions of the space fractional schrodinger equation j math phys 53 042105 2012 10 1063 1 4705268 published in this journal controversial opinions regarding solutions to the fractional schrodinger equation for a particle moving in the infinite potential well that were derived by laskin fractals and quantum mechanics chaos 10 780 790 2000 10 1063 1 1050284 have been given in this paper a thorough mathematical treatment of these matters is provided the problem under consideration is reformulated in terms of three integral equations with the power

relaxation and transport in fchc lattice gases fchc lattice gases are the basic models for studying flow problems in three dimensional systems this paper presents a self contained theoretical analysis and some computer simulations of such lattice gases extended to include an arbitrary number of rest particles with special emphasis on non semi detailed balance nsdb models the special fchc lattice symmetry guarantees isotropy of the navier stokes equations and enumerates the 12 spurious conservation laws staggered momenta the kinetic theory is based on the mean field approximation or the nonlinear boltzmann equation it is shown how calculation of the eigenvalues of the linearized boltzmann equation offers a simple alternative to the chapman enskog method or the multi time scale methods for calculating transport coefficients and relaxation rates the simulated values for the speed of sound in nsdb models slightly disagree with the boltzmann prediction

fractional diffusion and levy stable processes anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in time has been observed in all manner of physical and biological systems from macroscopic surface growth to dna sequences herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model this model is a simple two state process with long time correlations and is shown to produce a random walk described by an exact fractional diffusion equation fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of fox functions including l evy ensuremath alpha stable processes in the superdiffusive domain 1 21

levy flights in an infinite potential well as a hypersingular fredholm problem we study levy flights with arbitrary index 0 2 inside a potential well of infinite depth such a problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems the major technical tool is a transformation of the eigenvalue problem for initial fractional schrodinger equation into that for fredholm integral equation with hypersingular kernel the latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain d reducing the problem to the spectrum of a matrix of infinite dimensions the eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum

estimation of time dependent hurst exponents with variational smoothing and application to forecasting foreign exchange rates abstract hurst exponents depict the long memory of a time series for human dependent phenomena as in finance this feature may vary in the time it justifies modelling dynamics by multifractional brownian motions which are consistent with time dependent hurst exponents we improve the existing literature on estimating time dependent hurst exponents by proposing a smooth estimate obtained by variational calculus this method is very general and not restricted to the sole hurst framework it is globally more accurate and easier than other existing non parametric estimation techniques besides in the field of hurst exponents it makes it possible to make forecasts based on the estimated multifractional brownian motion the application to high frequency foreign exchange markets gbp chf sek usd cad aud jpy cny and sgd all against eur shows significantly good forecasts when the hurst exponent is higher than 0 5 what depicts a long memory feature the accuracy is higher

detecting long range correlations with detrended fluctuation analysis abstract we examine the detrended fluctuation analysis dfa which is a well established method for the detection of long range correlations in time series we show that deviations from scaling which appear at small time scales become stronger in higher orders of dfa and suggest a modified dfa method to remove them the improvement is necessary especially for short records that are affected by non stationarities furthermore we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of dfa

detrending moving average algorithm a closed form approximation of the scaling law abstract the hurst exponent h of long range correlated series can be estimated by means of the detrending moving average dma method the computational tool on which the algorithm is based is the generalized variance dma 2 1 n n i n n y i y n i 2 with y n i 1 n k 0 n y i k being the average over the moving window n and n the dimension of the stochastic series y i the ability to yield h relies on the property of dma 2 to vary as n 2 h over a wide range of scales e alessio a carbone g castelli v frappietro eur j phys b 27 2002 197 here we give a closed form proof that dma 2 is equivalent to c h n 2 h and provide an explicit expression for c h we furthermore compare the values of c h with those obtained by applying the dma algorithm to artificial self similar signals

fractional laplacians in bounded domains killed reflected censored and taboo levy flights the fractional laplacian 2 0 2 has many equivalent albeit formally different realizations as a nonlocal generator of a family of stable stochastic processes in r n on the other hand if the process is to be restricted to a bounded domain there are many inequivalent proposals for what a boundary data respecting fractional laplacian should actually be this ambiguity not only holds true for each specific choice of the process behavior at the boundary e g absorbtion reflection conditioning or boundary taboos but extends as well to its particular technical implementation dirichlet neumann etc problems the inferred jump type processes are inequivalent as well differing in their spectral and statistical characteristics which may strongly influence the ability of the formalism if uncritically adopted to provide an unambiguous description of real geometrically confined physical systems with disorder specifically that refers to their relaxation properties and the near equilibrium asymptotic behavior in the present paper we focus on levy flight induced jump type processes which are constrained to stay forever inside a finite domain this refers to a concept of taboo processes imported from brownian to levy stable contexts to so called censored processes and to reflected levy flights whose status still remains to be unequivocally settled as a by product of our fractional spectral analysis with reference to neumann boundary conditions we discuss disordered semiconducting heterojunctions as the bounded domain problem

stochastic models that separate fractal dimension and the hurst effect fractal behavior and long range dependence have been observed in an astonishing number of physical biological geological and socioeconomic systems time series profiles and surfaces have been characterized by their fractal dimension a measure of roughness and by the hurst coefficient a measure of long memory dependence both phenomena have been modeled and explained by self affine random functions such as fractional gaussian noise and fractional brownian motion the assumption of statistical self affinity implies a linear relationship between fractal dimension and hurst coefficient and thereby links the two phenomena this article introduces stochastic models that allow for any combination of fractal dimension and hurst coefficient associated software for the synthesis of images with arbitrary prespecified fractal properties and power law correlations is available the new models suggest a test for self affinity that assesses coupling and decoupling of local and global behavior

existence and regularity of a linear nonlocal fokker planck equation with growing drift abstract the nonlocal fokker planck equations for a class of stochastic differential equations with non gaussian stable levy motion in euclidean space are studied the existence and uniqueness of weak solution are obtained with rough drift the solution is shown to be smooth on spatial variable if all derivatives of the drift are bounded moreover the solution is jointly smooth on spatial and time variable if we assume further that the drift grows like a power of logarithm function at infinity

second order accuracy finite difference methods for space fractional partial differential equations abstract space fractional partial differential equations are used for simulations of for example diffusion of radioactive materials and financial and other models which are characterized by heavy tailed distributions a number of first order accuracy finite difference methods have been proposed in the present paper we introduce second order accuracy finite difference methods with dirichlet boundary conditions these methods have a parameter in these schemes and the parameter stabilizes the schemes this means that there exist various schemes with second order accuracy but the stability of each scheme is different in the present paper we introduce the most stable scheme for any fractional calculus order by choosing the optimal parameter in addition we describe a phenomenon whereby the expected accuracy cannot be obtained if the analytical solution can be expanded to a series having less than second order around boundaries this also happens in both existing methods and the proposed methods in the present paper we develop the stability conditions for the proposed schemes and numerical examples of second order accuracy and accuracy decay are shown

a particle model for spinodal decomposition we study a one dimensional lattice gas where particles jump stochastically obeying an exclusion rule and having a small drift toward regions of higher concentration we prove convergence in the continuum limit to a nonlinear parabolic equation whenever the initial density profile satisfies suitable conditions which depend on the strengtha of the drift there is a critical valuea c ofa fora a c the density values are unrestricted while fora a c they should all be to the right or to the left of a given interval a the diffusion coefficient of the limiting equation can be continued analytically to a and in the interior of a it has negative values which should correspond to particle aggregation phenomena we also show that the dynamics can be obtained as a limit of a kawasaki evolution associated to a kac potential the coefficienta plays the role of the inverse temperature the critical value ofa coincides with the critical inverse temperature in the van der waals limit and a with the spinodal region it is finally seen that in a scaling intermediate between the microscopic and the hydrodynamic the system evolves according to an integrodifferential equation the instanton solutions of this equation as studied by dal passo and de mottoni are then related to the phase transition region in the thermodynamic phase diagram analogies with the cahn hilliard equations are also discussed

characterizing surface smoothness via estimation of effective fractal dimension the fractal dimension d of stationary gaussian surfaces may be expressed very simply in terms of behaviour of the covariance function near the origin indeed only the covariance of line transect samples is required and that fact makes practical estimation of d relatively straightforward the case of non gaussian surfaces is more poorly understood but we might define effective fractal dimension in terms of the covariance function as though the surface were gaussian in the present paper we suggest simple practical methods for estimating effective fractal dimension based on the variogram like techniques proposed recently by taylor and taylor ours are founded on log linear regression

solutions for a fractional nonlinear diffusion equation spatial time dependent diffusion coefficient and external forces we analyze a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation we start our investigation by considering the linear case and the nonlinear case afterward the linear case is discussed taking fractional time and spatial derivatives into account in a unified approach we also discuss the modifications that emerge by employing simple drifts and the diffusion coefficient given by d x t d t x for the nonlinear case we study scaling behavior of the time in connection with the asymptotic behavior for the solution of the nonlinear fractional diffusion equation

fractals with point impact in functional linear regression this paper develops a point impact linear regression model in which the trajectory of a continuous stochastic process when evaluated at a sensitive time point is associated with a scalar response the proposed model complements and is more interpretable than the functional linear regression approach that has become popular in recent years the trajectories are assumed to have fractal self similar properties in common with a fractional brownian motion with an unknown hurst exponent bootstrap confidence intervals based on the least squares estimator of the sensitive time point are developed misspecification of the point impact model by a functional linear model is also investigated non gaussian limit distributions and rates of convergence determined by the hurst exponent play an important role

average time spent by levy flights and walks on an interval with absorbing boundaries we consider a levy flyer of order alpha that starts from a point x 0 on an interval o l with absorbing boundaries we find a closed form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed these two quantities are equivalent to the mean first passage times for levy flights and levy walks respectively using fractional differential equations with a riesz kernel we find exact analytical expressions for both quantities in the continuous limit we show that numerical solutions for the discrete levy processes converge to the continuous approximations in all cases except the case of alpha 2 and the cases of x 0 0 and x 0 l for alpha 2 when the second moment of the flight length distribution exists our result is replaced by known results of classical diffusion we show that if x 0 is placed in the vicinity of absorbing boundaries the average total length has a minimum at alpha 1 corresponding to the cauchy distribution we discuss the relevance of this result to the problem of foraging which has received recent attention in the statistical physics literature

persistent random walk of cells involving anomalous effects and random death the purpose of this paper is to implement a random death process into a persistent random walk model which produces sub ballistic superdiffusion levy walk we develop a stochastic two velocity jump model of cell motility for which the switching rate depends upon the time which the cell has spent moving in one direction it is assumed that the switching rate is a decreasing function of residence running time this assumption leads to the power law for the velocity switching time distribution this describes the anomalous persistence of cell motility the longer the cell moves in one direction the smaller the switching probability to another direction becomes we derive master equations for the cell densities with the generalized switching terms involving the tempered fractional material derivatives we show that the random death of cells has an important implication for the transport process through tempering of the superdiffusive process in the long time limit we write stationary master equations in terms of exponentially truncated fractional derivatives in which the rate of death plays the role of tempering of a levy jump distribution we find the upper and lower bounds for the stationary profiles corresponding to the ballistic transport and diffusion with the death rate dependent diffusion coefficient monte carlo simulations confirm these bounds

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

levy targeting and the principle of detailed balance we investigate confining mechanisms for levy flights under premises of the principle of detailed balance in this case the master equation of the jump type process admits a transformation to the levy schrodinger semigroup dynamics akin to a mapping of the fokker planck equation into the generalized diffusion equation this sets a correspondence between above two stochastic dynamical systems within which we address a stochastic targeting problem for an arbitrary stability index e 0 2 of symmetric levy drivers namely given a probability density function specify the semigroup potential and thence the jump type dynamics for which this pdf is actually a long time asymptotic target solution of the master equation here an asymptotic behavior of different motion scenarios ceases to depend on that is exemplified by considering gaussian and cauchy family target pdfs a complementary problem of the reverse engineering is analyzed given a priori a semigroup potential quantify how sensitive upon the choice of the driver is an asymptotic behavior of solutions of the associated master equation and thus an invariant pdf itself this task is accomplished for so called family of levy oscillators

levy scaling the diffusion entropy analysis applied to dna sequences we address the problem of the statistical analysis of a time series generated by complex dynamics with the diffusion entropy analysis dea n scafetta p hamilton and p grigolini fractals 9 193 2001 this method is based on the evaluation of the shannon entropy of the diffusion process generated by the time series imagined as a physical source of fluctuations rather than on the measurement of the variance of this diffusion process as done with the traditional methods we compare the dea to the traditional methods of scaling detection and prove that the dea is the only method that always yields the correct scaling value if the scaling condition applies furthermore dea detects the real scaling of a time series without requiring any form of detrending we show that the joint use of dea and variance method allows to assess whether a time series is characterized by levy or gauss statistics we apply the dea to the study of dna sequences and prove that their large time scales are characterized by levy statistics regardless of whether they are coding or noncoding sequences we show that the dea is a reliable technique and at the same time we use it to confirm the validity of the dynamic approach to the dna sequences proposed in earlier work