a framework for generating and analyzing movement paths on ecological landscapes the movement paths of individuals over landscapes are basically represented by sequences of points xi yi occurring at times ti theoretically these points can be viewed as being generated by stochastic processes that in the simplest cases are gaussian random walks on featureless landscapes generalizations have been made of walks that i take place on landscapes with features ii have correlated distributions of velocity and direction of movement in each time interval iii are levy processes in which distance or waiting time time between steps distributions have infinite moments or iv have paths bounded in space and time we begin by demonstrating that rather mild truncations of fat tailed step size distributions have a dramatic effect on dispersion of organisms where such truncations naturally arise in real walks of organisms bounded by space and more generally influenced by the interactions of physiological behavioral and ecological factors with landscape features these generalizations permit not only increased realism and hence greater accuracy in constructing movement pathways but also provide a biogeographically detailed epistemological framework for interpreting movement patterns in all organisms whether tossed in the wind or willfully driven we illustrate the utility of our framework by demonstrating how fission fusion herding behavior arises among individuals endeavoring to satisfy both nutritional and safety demands in heterogeneous environments we conclude with a brief discussion of potential methods that can be used to solve the inverse problem of identifying putative causal factors driving movement behavior on known landscapes leaving details to references in the literature

some properties of a random walk on a comb structure abstract we analyze transport properties of a random walk on a comb structure which serves as a model for a random walk on the backbone of a percolation cluster it is shown that the random walk along the x axis which is the analog of the backbone exhibits anomalous diffusion in that x 2 n n 1 2 and the expected number of x sites visited is proportional to n 1 4 for large n the distribution function is found to be a two dimensional gaussian if a field in the x direction so that diffusion is asymmetric the expected displacement is found to be asymptotically proportional to n 1 2

ap weights for nondoubling measures in rn and applications we study an analogue of the classical theory of ap weights in r n without assuming that the underlying measure is doubling then we obtain weighted norm inequalities for the centered hardy littlewood maximal function and corresponding weighted estimates for nonclassical calderon zygmund operators in the sense of ntv1 we also consider commutators of those calderon zygmund operators with bounded mean oscillation functions bmo extending the main result from crw finally we study self improving properties of poincare b m o type inequalities within this context more precisely we show that if f is a locally integrable function satisfying 1 q r q f fq d a q for all cubes q then it is possible to deduce higher l p integrability result of f assuming certain simple geometric condition on the functional a

instrumental variables regression with weak instruments this paper develops asymptotic distribution theory for instrumental variable regression when the partial correlation between the instruments and a single included endogenous variable is weak here modeled as local to zero asymptotic representations are provided for various instrumental variable statistics including the two stage least squares tsls and limited information maximum likelihood liml estimators and their t statistics the asymptotic distributions are found to provide good approximations to sampling distributions with just 20 observations per instrument even in large samples tsls can be badly biased but liml is in many cases approximately median unbiased the theory suggests concrete quantitative guidelines for applied work these guidelines help to interpret angrist and krueger s 1991 estimates of the returns to education whereas tsls estimates with many instruments approach the ols estimate of 6 the more reliable liml and tsls estimates with fewer instruments fall between 8 and 10 with a typical confidence interval of 6 14

assessing cross sectional correlation in panel data abstract consider drawing a sample of n experimental units where each unit is observed over t time periods are the draws independent a test statistic introduced by breusch and pagan 1980 is one measure used to quantify the amount of cross sectional correlation it is effective when t is large relative to n and as shown by breusch and pagan has desirable asymptotic in t properties in this paper we consider the case often encountered in the analysis of panel data where n is large relative to t here the breusch and pagan statistic does not enjoy the same desirable asymptotic in n properties in fact we show that the asymptotic distribution depends on the parent population even under the hypothesis of no cross sectional correlations thus we introduce a distribution free statistic that does not have this drawback asymptotic properties of the new statistic are established when the data are drawn from identical distributions and when the statistic is evaluated using residuals from a complex model both the breusch and pagan statistic and the new statistic are shown to be members of a more general family by considering this generalization finite sample properties of members of the family are easily established further it turns out that a statistic introduced by friedman 1937 is another special case of this family

efficient estimation and inference in linear pseudo panel data models we consider pseudo panel data models constructed from repeated cross sections in which the number of individuals per group is large relative to the number of groups and time periods first we show that when time invariant group fixed effects are neglected the ols estimator does not converge in probability to a constant but rather to a random variable second we show that while the fixed effects fe estimator is consistent the usual t statistic is not asymptotically normally distributed and we propose a new robust t statistic whose asymptotic distribution is standard normal third we propose efficient gmm estimators using the orthogonality conditions implied by grouping and we provide t tests that are valid even in the presence of time invariant group effects our monte carlo results show that the proposed gmm estimator is more precise than the fe estimator and that our new t test has good size and is powerful

an overview of semi continuity results on the spectral radius and positivity our aim is to provide a novelly comprehensive and unifying approach to showing the continuous dependence of the spectral radius of compact linear operators defined on banach spaces with specific attention to positive operators defined on normal banach spaces and emphasizing that the upper semi continuity generally holds unlike the lower semi continuity

anomalous diffusion and charge relaxation on comb model exact solutions abstract the random walks on the comb structure are considered it is shown that due to fingers a diffusion has an anomalous character that is an r m s displacement depends on time by a power way with exponent 1 2 the generalized diffusion equation for an anomalous case is deduced it essentially differs from a usual diffusion equation in the continuity equation form instead of the first time derivative the time derivative of fractal order 1 2 appears in the second part the charge relaxation on the comb structure is studied a non maxwell character is established the reason is that the electric field has three components but a charge may relax only along some conducting lines

1 2 order fractional fokker planck equation on comblike model from the generalized scheme of random walks on the comblike structure it is shown how a 1 2 order fractional fokker planck equation can be derived the operator method for the moments associated with the distribution function p x t is used to solve the resulting equation also the anomalous diffusion along the backbone of the structure has been considered

unequal twins probability distributions do not determine everything it is the common lore to assume that knowing the equation for the probability distribution function pdf of a stochastic model as a function of time tells the whole picture defining all other characteristics of the model we show that this is not the case by comparing two exactly solvable models of anomalous diffusion due to geometric constraints the comb model and the random walk on a random walk we show that though the two models have exactly the same pdfs they differ in other respects like their first passage time distributions their autocorrelation functions and their aging properties

diffusion on random comb structure effective medium approximation abstract the random walks on comb model of percolation clusters are studied the generalized diffusion equations in the form of fractal temporal derivatives are deduced the random distributions of fingers over lengths gaussian and power are studied in the effective medium approximation

numerical treatment of stochastic models used in statistical systems and financial markets in this paper by means of the variational iteration method numerical solutions are computed for some stochastic models without any linearization or weak assumptions two stochastic models the fokker planck equation for non equilibrium statistical systems and the black scholes model for pricing stock options are solved numerically in this approach the solution is found in the form of a convergent series with easily computed components the behavior of the approximate solutions is shown graphically

to hit or to pass it over remarkable transient behavior of first arrivals and passages for levy flights in finite domains the term levy flights was coined by benoit mandelbrot who thus poeticized stable levy random motion a markovian process with stationary independent increments distributed according to the stable levy probability law contrary to the brownian motion the trajectories of the stable levy motion are discontinous that is exhibit jumps this feature implies that the process of first passage through the boundary of a given space domain or the first escape is different from the process of first arrival hit at the boundary here we investigate the properties of first escapes and first arrivals for levy flights and explore how the asymptotic behavior of the corresponding passage and hit probabilities is sensitive to the size of the domain in particular we find that the survival probability to stay in a large enough finite domain has a universal sparre andersen temporal scaling which is transient and changes to an exponential non universal decay at longer times also the probability to arrive at a finite domain possesses a similar transient sparre andersen universality that turns into a non universal and slower power law decay in course of time finally we demonstrate that the probability density of the leapover length l over the boundary related to overshooting events has an intermediate asymptotics which is inherent for the escape from a semi infinite domain however for larger leapovers the probability density decays faster according to the law thus we find that the laws derived for the stable processes on the semi infinite domain manifest themselves as transients for levy flights on the finite domain

estimating dynamic models from time series of independent cross sections the purpose of this paper is to analyze the estimation of dynamic models from time series of independent cross sections the population is divided into groups with fixed membership cohorts and the cohort sample means are used as a panel subject to measurement errors we propose measurement error corrected estimators and analyze their asymptotic properties we also calculate the asymptotic biases of the non corrected estimators for the ar 1 model to check up to what extent the measurement error correction is needed finally we carry out the monte carlo simulations to evaluate the performance of our estimators in finite samples

the fractional fokker planck equation on comb like model abstract the fractional fokker planck equation were used to describe the anomalous diffusion in external fields is derived using a comb like structure as a background model for the force free case the distribution function associated with space dependence diffusion coefficient along the backbone of the structure are obtained in a closed form of h function the operator method has been used to solve the fractional fokker planck equation taking the external field into account

mesoscopic description of random walks on combs combs are a simple caricature of various types of natural branched structures which belong to the category of loopless graphs and consist of a backbone and branches we study continuous time random walks on combs and present a generic method to obtain their transport properties the random walk along the branches may be biased and we account for the effect of the branches by renormalizing the waiting time probability distribution function for the motion along the backbone we analyze the overall diffusion properties along the backbone and find normal diffusion anomalous diffusion and stochastic localization diffusion failure respectively depending on the characteristics of the continuous time random walk along the branches and compare our analytical results with stochastic simulations

anomalous diffusion a dynamic perspective this paper investigates whether spontaneous stationary velocity fluctuations can lead to deviations from the regular fickian diffusion a kinematic analysis reveals that anomalous diffusion both fast and slow arises from long tailed velocity auto correlation functions vacf this infinite span of interdependence of the random velocity leads to the breakdown of the central limit theorem for particle displacements a generalized langevin equation which features a retarded friction has been used to describe the particle dynamics in the long time limit the analysis reveals that simple power law decay models for the friction kernel are adequate to yield the pathological vacfs which imply anomalous diffusion the fluctuation dissipation theorem is invoked to infer that a fractional noise gives rise to anomalous diffusion such a langevin equation represents a mean field description of disorder effects and the friction kernel then becomes a constitutive property of the medium

self improving properties of john nirenberg and poincare inequalities on spaces of homogeneous type abstract we give a condition which ensures that if one inequality of sobolev poincare type is valid then other stronger inequalities of a similar type also hold including weighted versions our main result includes many previously known results as special cases we carry out the analysis in the context of spaces of homogeneous type but the main result is new even in the usual euclidean setting

fluctuations in a levy flight gas we consider the density fluctuations of an ideal brownian gas of particles performing levy flights characterized by the index f we find that the fluctuations scale as n t t h where the hurst exponent h locks onto the universal value 1 4 for levy flights with a finite root mean square range f 2 for levy flights with a finite mean range but infinite root mean square range 1 f 2 the hurst exponent h 1 2f for infinite range levy flights f 1 the hurst exponent locks onto the value 1 2 the corresponding power spectrum scales with an exponent 1 2h independent of dimension

generalized time dependent schrodinger equation in two dimensions under constraints we investigate a generalized two dimensional time dependent schrodinger equation on a comb with a memory kernel a dirac delta term is introduced in the schrodinger equation so that the quantum motion along the x direction is constrained at y 0 the wave function is analyzed by using green s function approach for several forms of the memory kernel which are of particular interest closed form solutions for the cases of dirac delta and power law memory kernels in terms of fox h function as well as for a distributed order memory kernel are obtained further a nonlocal term is also introduced and investigated analytically it is shown that the solution for such a case can be represented in terms of infinite series in fox h functions green s functions for each of the considered cases are analyzed and plotted for the most representative ones anomalous diffusion signatures are evident from the presence of the power law tails the normalized green s functions obtained in this work are of broader interest

existence of strong and nontrivial solutions to strongly coupled elliptic systems abstract we establish the existence of strong solutions to a class of nonlinear strongly coupled and uniform elliptic systems consisting of more than two equations the existence of nontrivial and non constant solutions or pattern formations will also be studied

field induced drift and trapping in percolation networks the authors study the effects of a bias producing external field on a random walk on the infinite cluster in the percolation problem there are two competing physical effects drift and trapping which result in a drift velocity nu which rises and then falls as the field increases they study these effects on a one dimensional lattice with random length branches and on the diluted bethe lattice they calculate nu in the first model with a maximum allowed length for each branch in the limit that this cutoff length becomes infinite they find that the velocity vanishes identically above a finite critical value of the field for the bethe lattice they derive an upper bound on the critical field which varies as p pc 1 2 as the percolation concentration pc is approached in the one dimensional model they also investigate the anomalous regime in which the velocity vanishes they discuss the distribution of steady state times required to traverse n sites and find that it can be described in terms of a stable distribution of index x with superimposed oscillations the index x of the stable distribution is given by l xi where xi is a characteristic branch length and l is a bias induced length which describes the exponential buildup of the steady state density of particles towards the end of a branch

scaling properties of diffusion on comb like structures the authors study the scaling properties of the probability density of random walks on comb like structures they find that in contrast to the first passage time in which several exponents are needed to characterise their distribution the probability density can be described by a single exponent

implicit numerical approximation scheme for the fractional fokker planck equation in this paper we consider an anomalous subdiffusion process governed by fractional fokker planck equation an effective numerical method for approximating fokker planck equation in a bounded domain is presented the stability and convergence of the numerical method are analyzed some numerical examples are presented to show the application of the present technique the numerical results exhibit the good performance of our theoretical analysis

the diffusion drift equation on comb like structure from the generalized scheme of random walks on the comb like structure one can obtain the fractional fokker planck equation in the domain of fractal time evolution with critical exponent 0 1 the operator method for the moments associated with the density p x t is used to solve the obtained equation it is shown that due to fingers diffusion has an anomalous character the first two moments increase sub linearly in time

transport properties on a random comb we study the random walk of a particle in a random comb structure both in the presence of a biasing field and an the field free case we show that the mean field treatment of the quenched disorder can be exactly mapped on to a continuous time random walk ctrw on the backbone of the comb with a definite waiting time density we find an exact expression for this central quantity the green function for the ctrw is then obtained its first and second moments determine the drift and diffusion at all times we show that the drift velocity v vanishes asymptotically for power law and stretched exponential distributions of branch lengths on the comb whatever be the biasing field strength for an exponential branch length distribution v is a nonmonotonic function of the bias increasing initially to a maximum and then decreasing to zero at a critical value in the field free case anomalous diffusion occurs for a range of power law distributions of the branch length the corresponding exponent for the mean square displacement is obtained as is the asymptotic form of the positional probability distribution for the random walk we show that normal diffusion occurs whenever the mean branch length is finite and present a simple formula for the effective diffusion constant these results are extended to regular nonrandom combs as well the physical reason for anomalous drift or diffusion is traced to the properties of the distribution of a first passage time on a finite chain that controls the effective waiting time density of the ctrw

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

macroscopic current in fractional anomalous diffusion abstract fick s law which relates the macroscopic current with the density in ordinary diffusion is generalized to fractional diffusion where either the waiting time distribution or the displacement probability density exhibit power law decays these long tailed distributions give rise to respectively temporal or spatial nonlocality in connection between current and density in the case of power law displacement distributions the current is shown to also display such decays in the spatial coordinate an illustrative example is explicitly worked out in a one dimensional system

computer simulation study of the levy flight process abstract the random walk simulation of a levy flight shows a linear relation between the mean square displacement r 2 and time we have analyzed different aspects of this linearity it is shown that the restriction of jump length to a maximum value l m affects the diffusion coefficient even though it remains constant for l m greater than 1464 so this factor has no effect on the linearity in addition it is shown that the number of samples does not affect the results we have demonstrated that the relation between the mean square displacement and time remains linear in a continuous space while continuous variables just reduce the diffusion coefficient the results also imply that the movement of a levy flight particle is similar to the case in which the particle moves in each time step with an average jump length l finally it is shown that the non linear relation of the levy flight will be satisfied if we use a time average instead of an ensemble average the difference between the time average and ensemble average results shows that the levy distribution may be a non ergodic distribution

levy walks with applications to turbulence and chaos abstract diffusion on fractal structures has been a popular topic of research in the last few years with much emphasis on the sublinear behavior in time of the mean square displacement of a random walker another type of diffusion is encountered in turbulent flows with the mean square displacement being superlinear in time we introduce a novel stochastic process called a levy walk which generalizes fractal brownian motion to provide a statistical theory for motion in the fractal media which exists in a turbulent flow the levy walk describes random but still correlated motion in space and time in a scaling fashion and is able to account for the motion of particles in a hierarchy of coherent structures we apply our model to the description of fluctuating fluid flow when kolmogorov s 5 3 law for homogeneous turbulence is used to determine the memory of the levy walk then richardson s 4 3 law of turbulent diffusion follows in the mandelbrot absolute curdling limit if as suggested by mandelbrot that turbulence is isotropic but fractal then intermittency corrections to the 5 3 law follow in a natural fashion the same process with a different space time scaling provides a description of chaos in a josephson junction

estimating long run relationships from dynamic heterogeneous panels in panel data four procedures are widely used pooling aggregating averaging group estimates and cross section regression in the static case if the coefficients differ randomly all four procedures give unbiased estimates of coefficient means in the dynamic case when the coefficients differ across groups pooling and aggregating give inconsistent and potentially highly misleading estimates of the coefficients though the cross section can provide consistent estimates of the long run effects the theoretical results on the properties of the four procedures are illustrated by uk labour demand functions for 38 industries over 30 years

a quasi differencing approach to dynamic modelling from a time series of independent cross sections we motivate and describe a gmm method of estimating linear dynamic models from a time series of independent cross sections this involves subjecting the model to a quasi differencing transformation across pairs of individuals that belong to the same group desirable features of the model include the fact that i no aggregation is involved ii dynamic response parameters can vary across groups iii the presence of unobserved individual specific heterogeneity is explicitly allowed for and iv the gmm estimators are derived by realistically assuming group size asymptotics the monte carlo experiments conducted to assess the finite sample performances of the proposed estimators provide us with encouragement

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

on a fractional model for earthquakes fractional calculus is suitable for systems with memory and for fractal systems earthquakes have both properties it is fair to claim that fractional model is very efficient for earthquake molding a fractional model for earthquake is presented and solved numerically

validity of the mean field approximation for diffusion on a random comb we consider unbiased diffusion on a random comb structure an infinitely long backbone with loopless branches of arbitrary length emanating from it if t ifmmode pm else textpm fi j 0 mathrm mathit w mathit t 0 is the mean time averaged over all random walks for first passage from an arbitrary origin 0 on the backbone to either of the sites j or j on it in a given realization of the structure the exact diffusion constant for the problem is defined as k mathrm lim mathit j ensuremath rightarrow mathrm ensuremath infty mathit j 2 1 mathit t 0 mathrm mathit c where mathrm mathit c stands for the configuration average over the realizations of the random comb the diffusion constant in the mean field approximation is given by mathit k mathrm mf mathrm lim mathit j ensuremath rightarrow mathrm ensuremath infty mathit j 2 mathit t 0 mathrm mathit c we compute mathit t 0 exactly for an arbitrary realization of the comb and then show rigorously that owing to the suppression of the relative fluctuations in mathit t 0 in the thermodynamic limit j ensuremath rightarrow ensuremath infty we have mathit k mathrm mf k whenever the moments of certain random variables ensuremath gamma l ensuremath alpha ensuremath beta are finite here the site dependent random variables l ensuremath alpha and ensuremath beta are respectively the branch length stay probability at the tip of a branch and the backbone to branch jump probability finally we discuss different situations in which k will not be equal to mathit k mathrm mf although the transport remains diffusive as opposed to those in which anomalous diffusion occurs textcopyright 1996 the american physical society