joint probability distributions for a class of non markovian processes we consider joint probability distributions for the class of coupled langevin equations introduced by fogedby h c fogedby phys rev e 50 1657 1994 we generalize well known results for the single time probability distributions to the case of n time joint probability distributions it is shown that these probability distribution functions can be obtained by an integral transform from distributions of a markovian process the integral kernel obeys a partial differential equation with fractional time derivatives reflecting the non markovian character of the process

levy flights in quenched random force fields l evy flights characterized by the microscopic step index f are for fl2 the case of rare events considered in short range and long range quenched random force fields with arbitrary vector character to first loop order in an expansion about the critical dimension 2f ensuremath 2 in the short range case and the critical fall off exponent 2f ensuremath 2 in the long range case by means of a dynamic renormalization group analysis based on the momentum shell integration method we determine flows fixed point and the associated scaling properties for the probability distribution and the frequency and wave number dependent diffusion coefficient unlike the case of ordinary brownian motion in a quenched force field characterized by a single critical dimension or fall off exponent d 2 two critical dimensions appear in the l evy case a critical dimension or fall off exponent d f below which the diffusion coefficient exhibits anomalous scaling behavior i e algebraic spatial behavior and long time tails and a critical dimension or fall off exponent d 2f ensuremath 2 below which the force correlations characterized by a nontrivial fixed point become relevant as a general result we find in all cases that the dynamic exponent z characterizing the mean square displacement locks onto the l evy index f independent of dimension and independent of the presence of weak quenched disorder

retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations we propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena whose diffusion exponent varies with time and which correspondingly cannot be viewed as self affine random processes possessing a unique hurst exponent we prove the positivity of the solutions of the proposed equations and establish their relation to the continuous time random walk theory we show that the distributed order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the wiener process and whose diffusion exponent decreases in time retarding subdiffusion this process may lead to superslow diffusion with the mean square displacement growing logarithmically in time we also demonstrate that the distributed order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time accelerating superdiffusion

anomalous transport in external fields continuous time random walks and fractional diffusion equations extended the continuous time random walk ctrw in a homogeneous velocity field and in arbitrary force fields is studied within the extended ctrw scheme anomalous transport properties due to long tailed waiting time or jump length distributions are consistently introduced the connections with generalised diffusion equations in a potential field are discussed these equations being of fractional order in particular the problems of a constant and a hookean linear force i e of a linear and a parabolic potential are worked out

continuous time random walk with correlated waiting times based on the langevin description of the continuous time random walk ctrw we consider a generalization of ctrw in which the waiting times between the subsequent jumps are correlated we discuss the cases of exponential and slowly decaying persistent power law correlations between the waiting times as two generic examples and obtain the corresponding mean squared displacements as functions of time in the case of exponential type correlations the sub diffusion at short times is slower than in the absence of correlations at long times the behavior of the mean squared displacement is the same as in uncorrelated ctrw for power law correlations we find subdiffusion characterized by the same exponent at all times which appears to be smaller than the one in uncorrelated ctrw interestingly in the limiting case of an extremely long power law correlations the sub diffusion exponent does not tend to zero but is bounded from below by the subdiffusion exponent corresponding to a short time behavior in the case of exponential correlations

a matricial boundary value problem which appears in the transport theory abstract we study the boundary value problem t a i p 0 p 0 where t t a a p 2 p p t tp t invertible the motivation to consider such problem comes from the transport theory the behavior of values of for which there is a nontrivial solution exceptional values is investigated using the indicator function this is an analytic hermitian valued function of the real parameter t which reflects in particular the characteristic properties of exceptional values

generalized viscoelastic models their fractional equations with solutions recently fractional calculus fc has encountered much success in the description of complex dynamics in particular fc has proved to be a valuable tool to handle viscoelastic aspects in this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models especially the maxwell the kelvin voigt the zener and the poynting thomson model to this end we introduce a fractional element in addition to the standard purely elastic and purely viscous elements as we proceed to show many of the fractional differential equations which we obtain by this construction method admit closed form analytical solutions in terms of fox h functions of the minag leffler type

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

a continuous time generalization of the persistent random walk abstract we develop the formalism for a continuous time generalization of the persistent random walk by allowing the sojourn time to deviate from the exponential form found in standard discussions of this subject this generalization leads to evolution equations in the time domain that differ and are of higher order than the telegrapher s equation which is found in the case of the markovian persistent random walk

continuous time random walks and traveling fronts we present a geometric approach to the problem of propagating fronts into an unstable state valid for an arbitrary continuous time random walk with a fisher kolmogorov petrovski piskunov growth reaction rate we derive an integral hamilton jacobi type equation for the action functional determining the position of reaction front and its speed our method does not rely on the explicit derivation of a differential equation for the density of particles in particular we obtain an explicit formula for the propagation speed for the case of anomalous transport involving non markovian random processes

stationarity conservation laws for fractional differential equations with variable coefficients in this paper we study linear fractional differential equations with variable coefficients it is shown that by assuming some conditions for the coefficients the stationarity conservation laws can be derived the area where these are valid is restricted by the asymptotic properties of solutions of the respective equation applications of the proposed procedure include the fractional fokker planck equation in 1 1 and d 1 dimensional space and the fractional klein kramers equation

distributed order diffusion equations and multifractality models and solutions we study distributed order time fractional diffusion equations characterized by multifractal memory kernels in contrast to the simple power law kernel of common time fractional diffusion equations based on the physical approach to anomalous diffusion provided by the seminal scher montroll weiss continuous time random walk we analyze both natural and modified form distributed order time fractional diffusion equations and compare the two approaches the mean squared displacement is obtained and its limiting behavior analyzed we derive the connection between the wiener process described by the conventional langevin equation and the dynamics encoded by the distributed order time fractional diffusion equation in terms of a generalized subordination of time a detailed analysis of the multifractal properties of distributed order diffusion equations is provided

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

continuous time random walks simulation of continuous trajectories continuous time random walks have been developed as a straightforward generalization of classical random walk processes some ten years ago fogedby introduced a continuous representation of these processes by means of a set of langevin equations h c fogedby phys rev e 50 1657 1994 the present work is devoted to a detailed discussion of fogedby s model and presents its application for the robust numerical generation of sample paths of continuous time random walk processes

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

growing surfaces with anomalous diffusion results for the fractal kardar parisi zhang equation in this paper i study a model for a growing surface in the presence of anomalous diffusion also known as the fractal kardar parisi zhang equation fkpz this equation includes a fractional laplacian that accounts for the possibility that surface transport is caused by a hopping mechanism of a levy flight it is shown that for a specific choice of parameters of the fkpz equation the equation can be solved exactly in one dimension so that all the critical exponents which describe the surface that grows under fkpz can be derived for that case afterwards the self consistent expansion sce is used to predict the critical exponents for the fkpz model for any choice of the parameters and any spatial dimension it is then verified that the results obtained using sce recover the exact result in one dimension at the end a simple picture for the behavior of the fractal kpz equation is suggested and the upper critical dimension of this model is discussed

langevin equations for continuous time levy flights we consider the combined effects of a power law l evy step distribution characterized by the step index f and a power law waiting time distribution characterized by the time index g on the long time behavior of a random walker the main point of our analysis is a formulation in terms of coupled langevin equations which allows in a natural way for the inclusion of external force fields in the anomalous case for f2 and g1 the dynamic exponent z locks onto the ratio f g drawing on recent results on l evy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder for d below the critical dimension mathit d mathit c 2f 2 the disorder is relevant corresponding to a nontrivial fixed point for the force correlation function

fractional langevin model of memory in financial markets the separation of the microscopic and macroscopic time scales is necessary for the validity of ordinary statistical physics and the dynamical description embodied in the langevin equation when the microscopic time scale diverges the differential equations on the macroscopic level are no longer valid and must be replaced with fractional differential equations of motion in particular we obtain a fractional differential stochastic equation of motion after decades of statistical analysis of financial time series certain stylized facts have emerged including the statistics of stock price fluctuations having fat tails and their linear correlations in time being exceedingly short lived on the other hand the magnitude of these fluctuations and other such measures of market volatility possess temporal correlations that decay as an inverse power law one explanation of this long term memory is that it is a consequence of the time scale separation between microscopic and macroscopic economic variables we propose a fractional langevin equation as a dynamical model of the observed memory in financial time series

fractional diffusion waiting time distributions and cattaneo type equations we discuss a generalized diffusion equation resulting from an additive two state process in combination with an asymptotically fractal asymptotic power law waiting time distribution the obtained equation is an extension to previously discussed fractional diffusion equations our description leads to a mean squared displacement which describes enhanced subballistic transport for long times the short time behavior however is of a ballistic nature this separation into two domains results from the introduction of a time scale through the asymptotically fractal waiting time distribution this is also mirrored by the observation that for small times our generalized diffusion equation reduces to the standard cattaneo equation the asymptotic probability density is of compressed gaussian type and thus differs from the l evy tail generally found for these kinds of processes

solutions of a class of non markovian fokker planck equations we show that a formal solution of a rather general non markovian fokker planck equation can be represented in a form of an integral decomposition and thus can be expressed through the solution of the markovian equation with the same fokker planck operator this allows us to classify memory kernels into safe ones for which the solution is always a probability density and dangerous ones when this is not guaranteed the first situation describes random processes subordinated to a wiener process while the second one typically corresponds to random processes showing a strong ballistic component in this case the non markovian fokker planck equation is only valid in a restricted range of parameters initial and boundary conditions

correlated continuous time random walk with time averaged waiting time abstract in this paper we study the dynamics of a correlated continuous time random walk with time averaged waiting time the mean square displacement msd shows this process is subdiffusive and generalized einstein relation holds we also get the asymptotic behavior of the probability density function pdf of this process is stretched gaussian at last by computing the time averaged msd we find ergodicity breaking occurs in this process

diffusion of epicenters of earthquake aftershocks omori s law and generalized continuous time random walk models the epidemic type aftershock sequence etas model is a simple stochastic process modeling seismicity based on the two best established empirical laws the omori law power law decay ensuremath sim 1 t 1 ensuremath theta of seismicity after an earthquake and gutenberg richter law power law distribution of earthquake energies in order to describe also the space distribution of seismicity we use in addition a power law distribution ensuremath sim 1 r 1 ensuremath mu of distances between triggered and triggering earthquakes the etas model has been studied for the last two decades to model real seismicity catalogs and to obtain short term probabilistic forecasts here we present a mapping between the etas model and a class of ctrw continuous time random walk models based on the identification of their corresponding master equations this mapping allows us to use the wealth of results previously obtained on anomalous diffusion of ctrw after translating into the relevant variable for the etas model we provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock specifically we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks the different regimes are fully characterized by the two exponents ensuremath theta and ensuremath mu our predictions are checked by careful numerical simulations we stress the distinction between the bare omori law describing the seismic rate activated directly by a mainshock and the renormalized omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock and so on in particular we predict that seismic diffusion or subdiffusion occurs and should be observable only when the observed omori exponent is less than 1 because this signals the operation of the renormalization of the bare omori law also at the origin of seismic diffusion in the etas model we present predictions and insights provided by the etas to ctrw mapping which suggest different ways for studying seismic catalogs finally we discuss the present evidence for our predicted subdiffusion of seismicity triggered by a main shock stressing the caveats and limitations of previous empirical works

anomalous diffusion in correlated continuous time random walks we demonstrate that continuous time random walks in which successive waiting times are correlated by gaussian statistics lead to anomalous diffusion with the mean squared displacement r2 t t2 3 long ranged correlations of the waiting times with a power law exponent 0 2 give rise to subdiffusion of the form r2 t t 1 in contrast correlations in the jump lengths are shown to produce superdiffusion we show that in both cases weak ergodicity breaking occurs our results are in excellent agreement with simulations

upper bound on the critical temperature in the 3d ising model the upper bound on the critical temperature in the ising model recently derived from callen s correlation equalities by sa barreto and o carroll 1983 and improved by monroe 1984 is further improved by a more economical use of messager and miracle sole inequalities starting from a different callen correlation equality

dissipative matrices and invariant maximal semidefinite subspaces abstract an explicit construction of invariant maximal semidefinite subspaces for a dissipative matrix in an indefinite scalar product is given necessary and sufficient conditions for uniqueness of such a subspace are given these subspaces play an important role in the explicit construction of pseudo canonical factorizations for certain rational matrix valued functions

inertial effects in anomalous dielectric relaxation of symmetrical top molecules the linear dielectric response of an assembly of noninteracting symmetrical top molecules each of which is free to rotate in space is evaluated in the context of fractional dynamics the infinite hierarchy of differentialrecurrence relations for the relaxation functions appropriate to the dielectric response is derived by using the underlying inertial fractional klein kramers equation on solving this hierarchy in terms of matrix continued fractions as in the normal rotational diffusion the complex dynamic susceptibility is obtained and is calculated for typical values of the model parameters for the limiting case of spherical top molecules the solution is obtained in terms of an ordinary continued fraction it is shown that the model can reproduce nonexponential anomalous dielectric relaxation behavior at low frequencies vt 1 where t is the debye relaxation time and the inclusion of inertial effects ensures that optical transparency is regained at very high frequencies in the far infrared region so that gordon s sum rule for integral dipolar absorption is satisfied

a new class of long tailed pausing time densities for the ctrw we present some asymptotic results for the family of pausing time densities having the asymptotic t property t t ln1 t t 1 in particular we show that for this class of pausing time densities the mean squared displacement r2 t is asymptotically proportional to ln t t and the asymptotic distribution of the displacement has a negative exponential form

stability of stationary transport equations with accretive collision operators abstract in this paper we consider transport equations with accretive collision operators we characterize when the equation has a unique solution and show that in this case the solution is stable under small perturbations of the collision operator and the initial value in one case in which there is more than one solution we show how to make a special selection of a solution which is then stable again under small perturbations of both the collision operator and the initial value the results obtained here parallel those obtained earlier for the case where the collision operator is positive semidefinite

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

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

bifurcation bimodality and finite variance in confined levy flights we investigate the statistical behavior of levy flights confined in a symmetric quartic potential well u x proportional variant x 4 at stationarity the probability density function features a distinct bimodal shape and decays with power law tails which are steep enough to give rise to a finite variance in contrast to free levy flights from a delta initial condition a bifurcation of the unimodal state is observed at t c 0 the nonlinear oscillator with potential u x ax 2 2 bx 4 4 a b 0 shows a crossover from unimodal to bimodal behavior at stationarity depending on the ratio a b

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

langevin picture of levy walks and their extensions in this paper we derive langevin picture of levy walks applying recent advances in the theory of coupled continuous time random walks we find a limiting process of the properly scaled levy walk next we introduce extensions of levy walks in which jump sizes are some functions of waiting times we prove that under proper scaling conditions such generalized levy walks converge in distribution to the appropriate limiting processes we also derive the corresponding fractional diffusion equations and investigate behavior of the mean square displacements of the limiting processes showing that different coupling functions lead to various types of anomalous diffusion

anomalous diffusion in the nonasymptotic regime we analyze some properties of the one dimensional levy flights assuming that the one step transition rates depend on the flight length x as p alpha x equivalent to x alpha 2 for flights on a finite 2m 1 site lattice we can define an effective size dependent diffusion coefficient d alpha m equivalent to m 1 alpha 1 1 alpha if alpha are well defined provided that alpha r 3 these moments exhibit a power law singularity if alpha 1 and r 2 3 the short and intermediate time properties of the generalized mean square displacement are then studied numerically this work suggests the conditions under which the asymptotic analytical formulas obtained in the literature by the use of the generalized central limit theorem could be applied to finite time experiments these formulas should work much better if alpha is close to zero than in the alpha 1 neighborhood

anomalous diffusion stable processes and generalized functions the evolution equations in real space and time corresponding to a class of anomalous diffusion processes arc examined as special cases evolution equations corresponding to stable processes are derived using the theory of generalized functions recovering some known results differently interpreted and an evolution law for stable processes of order unity

fractional calculus and continuous time finance ii the waiting time distribution we complement the theory of tick by tick dynamics of financial markets based on a continuous time random walk ctrw model recently proposed by scalas et al 4 and we point out its consistency with the behaviour observed in the waiting time distribution for bund future prices traded at liffe london

limit theorems for coupled continuous time random walks scaling limits of continuous time random walks are used in physics to model anomalous diffusion in which a cloud of particles spreads at a different rate than the classical brownian motion governing equations for these limit processes generalize the classical diffusion equation in this article we characterize scaling limits in the case where the particle jump sizes and the waiting time between jumps are dependent this leads to an efficient method of computing the limit and a surprising connection to fractional derivatives

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

scale invariant motion in intermittent chaotic systems we investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous time random walk ctrw approach two different ctrw models are considered i particles jump between sites turning points or ii particles move at a constant velocity between sites and choose a new direction at random for both models we study the mean squared displacement mathit r 2 t and the propagator p r t the probability to be at location r at time t having started at the origin at t 0 iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the ctrw framework and scaling arguments for the case of dispersive motion we discuss the problem of the stationary state and point out its relevance

fractional model equation for anomalous diffusion abstract in recent years the phenomenon of anomalous diffusion has attracted more and more attention one of the main impulses was initiated by de gennes idea of the ant in the labyrinth several authors presented asymptotic probability density functions for the location of a random walker on a fractal object as this density function and the time dependence of its second moment are now well established a modified diffusion equation providing the correct result is formulated the parameters of this fractional partial differential equation are uniquely determined by the fractal hausdorff dimension of the underlying object and the anomalous diffusion exponent the presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters a closed form solution is given in terms of fox s h function in the asymptotic case a halved diffusion equation can be established furthermore the differences to equations considered previously are discussed

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

time changed geometric fractional brownian motion and option pricing with transaction costs abstract this paper deals with the problem of discrete time option pricing by a fractional subdiffusive black scholes model the price of the underlying stock follows a time changed geometric fractional brownian motion by a mean self financing delta hedging argument the pricing formula for the european call option in discrete time setting is obtained

lagrangian statistical model for transport in highly heterogeneous velocity fields we define an effective lagrangian statistical model in phase space x t v for describing transport in highly heterogeneous velocity fields with complex spatial organizations the spatial markovian nature and temporal non markovian nature of lagrangian velocities leads to an effective transport description that turns out to be a correlated continuous time random walk this model correctly captures the lagrangian velocity correlation properties and is demonstrated to represent a forward model for predicting transport in highly heterogeneous porous media for different types of velocity organizations

stable equilibrium based on levy statistics stochastic collision models approach we investigate equilibrium properties of two very different stochastic collision models i the rayleigh particle and ii the driven maxwell gas for both models the equilibrium velocity distribution is a levy distribution the maxwell distribution being a special case we show how these models are related to fractional kinetic equations our work demonstrates that a stable power law equilibrium which is independent of details of the underlying models is a natural generalization of maxwell s velocity distribution

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

aging properties of an anomalously diffusing particule we report new results about the two time dynamics of an anomalously diffusing classical particle as described by the generalized langevin equation with a frequency dependent noise and the associated friction the noise is defined by its spectral density proportional to 1 at low frequencies with 0 1 subdiffusion or 1 2 superdiffusion using laplace analysis we derive analytic expressions in terms of mittag leffler functions for the correlation functions of the velocity and of the displacement while the velocity thermalizes at large times slowly in contrast to the standard brownian motion case 1 the displacement never attains equilibrium it ages we thus show that this feature of normal diffusion is shared by a subdiffusive or superdiffusive motion we provide a closed form analytic expression for the fluctuation dissipation ratio characterizing aging

diffusion on random site percolation clusters theory and nmr microscopy experiments with model objects quasi two dimensional random site percolation model objects were fabricated based on computer generated templates samples consisting of two compartments a reservoir of mathrm h 2 mathrm o gel attached to a percolation model object which was initially filled with mathrm d 2 mathrm o were examined with nuclear magnetic resonance microscopy for rendering proton spin density maps the propagating proton deuteron interdiffusion profiles were recorded and evaluated with respect to anomalous diffusion parameters the deviation of the concentration profiles from those expected for unobstructed diffusion directly reflects the anomaly of the propagator for diffusion on a percolation cluster the fractal dimension of the random walk d w evaluated from the diffusion measurements on the one hand and the fractal dimension d f deduced from the spin density map of the percolation object on the other permits one to experimentally compare dynamical and static exponents approximate calculations of the propagator are given on the basis of the fractional diffusion equation furthermore the ordinary diffusion equation was solved numerically for the corresponding initial and boundary conditions for comparison the anomalous diffusion constant was evaluated and is compared to the brownian case some ad hoc correction of the propagator is shown to pay tribute to the finiteness of the system in this way anomalous solutions of the fractional diffusion equation could experimentally be verified

correlated continuous time random walks combining scale invariance with long range memory for spatial and temporal dynamics standard continuous time random walk ctrw models are renewal processes in the sense that at each jump a new independent pair of jump length and waiting time are chosen globally anomalous diffusion emerges through scale free forms of the jump length and or waiting time distributions by virtue of the generalized central limit theorem here we present a modified version of recently proposed correlated ctrw processes where we incorporate a power law correlated noise on the level of both jump length and waiting time dynamics we obtain a very general stochastic model that encompasses key features of several paradigmatic models of anomalous diffusion discontinuous scale free displacements as in levy flights scale free waiting times as in subdiffusive ctrws and the long range temporal correlations of fractional brownian motion fbm we derive the exact solutions for the single time probability density functions and extract the scaling behaviours interestingly we find that different combinations of the model parameters lead to indistinguishable shapes of the emerging probability density functions and identical scaling laws our model will be useful for describing recent experimental single particle tracking data that feature a combination of ctrw and fbm properties

heterogeneous memorized continuous time random walks in an external force fields in this paper we study the anomalous diffusion of a particle in an external force field whose motion is governed by nonrenewal continuous time random walks with correlated memorized waiting times which involves reimann liouville fractional derivative or reimann liouville fractional integral we show that the mean squared displacement of the test particle x x which is dependent on its location x of the form el wakil and zahran chaos solitons fractals 12 1929 1935 2001 begin aligned langle mathbb x x 2 rangle t langle delta x x t 2 rangle 0 sim x theta t gamma quad 0 gamma 1 quad theta d w 2 end aligned 1 where d w 2 is the anomalous exponent the diffusion exponent gamma is dependent on the model parameters we obtain the fokker planck type dynamic equations and their stationary solutions are of the boltzmann gibbs form these processes obey a generalized einstein stokes smoluchowski relation and the second einstein relation we observe that the asymptotic behavior of waiting times and subordinations are of stretched gaussian distributions we also discuss the time averaged in the case of an harmonic potential and show that the process exhibits aging and ergodicity breaking

fractional dynamics at multiple times a continuous time random walk ctrw imposes a random waiting time between random particle jumps ctrw limit densities solve a fractional fokker planck equation but since the ctrw limit is not markovian this is not sufficient to characterize the process this paper applies continuum renewal theory to restore the markov property on an expanded state space and compute the joint ctrw limit density at multiple times

fractional diffusion model for force distribution in static granular media we present the results of a numerical and an experimental investigation of the probability distribution of normal contact forces in static packs of particles with two different hardnesses force distributions are computed and compared with existing models and experimental data it is found that the probability distribution function of normal contact forces p f is well described by a semiempirical model derived from a fractional diffusion equation this model reproduces most of the features common to force distributions observed in experimental and numerical studies including the finite value for p f as the forces tend to zero the results indicate that the fractional model fits well both the numerical and experimental data over a wide range of particle deformations in contrast to the existing models these results provide an insight into the physics of granular media and complement previous findings

upper bounds on tc for classical ising models using correlation inequalities following closely the work of sa barreto and o carroll 1983 the author improves the upper bounds they derive for the critical temperature of classical ising ferromagnetics the improvement is achieved by a more critical use of griffith s 1967 inequalities at one point along with a completely new use of messager and miracle sole 177 inequalities at another stage of the proof

self similar gaussian processes for modeling anomalous diffusion we study some gaussian models for anomalous diffusion which include the time rescaled brownian motion two types of fractional brownian motion and models associated with fractional brownian motion based on the generalized langevin equation gaussian processes associated with these models satisfy the anomalous diffusion relation which requires the mean square displacement to vary with t ensuremath alpha 0l ensuremath alpha l2 however these processes have different properties thus indicating that the anomalous diffusion relation with a single parameter is insufficient to characterize the underlying mechanism although the two versions of fractional brownian motion and time rescaled brownian motion all have the same probability distribution function the slepian theorem can be used to compare their first passage time distributions which are different finally in order to model anomalous diffusion with a variable exponent ensuremath alpha t it is necessary to consider the multifractional extensions of these gaussian processes

correlated continuous time random walk and option pricing in this paper we study a correlated continuous time random walk cctrw with averaged waiting time whose probability density function pdf is proved to follow stretched gaussian distribution then we apply this process into option pricing problem supposing the price of the underlying is driven by this cctrw we find this model captures the subdiffusive characteristic of financial markets by using the mean self financing hedging strategy we obtain the closed form pricing formulas for a european option with and without transaction costs respectively at last comparing the obtained model with the classical black scholes model we find the price obtained in this paper is higher than that obtained from the black scholes model a empirical analysis is also introduced to confirm the obtained results can fit the real data well

no go theorem for ergodicity and an einstein relation we provide a simple no go theorem for ergodicity and the generalized einstein relation for anomalous diffusion processes the theorem states that either ergodicity in the sense of equal time and ensemble averaged mean squared displacements msd is broken and or the generalized einstein relation for time averaged diffusivity and mobility is invalid which is in complete contrast to normal diffusion processes we also give a general relation for the time averages of drift and msd for ergodic in the msd sense anomalous diffusion processes showing that the ratio of these quantities depends on the measurement time the levy walk model is used to exemplify the no go theorem

linear systems with levy fluctuations abstract we have examined the response of multivariable linear systems to levy fluctuations the characteristic function of the response is obtained explicitly as a function of time using the characteristic function the fractional moments of the process are determined the nature of the product rule obeyed by the characteristic function of a general linear dynamical system is elucidated

numerical methods for the solution of partial differential equations of fractional order anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas to model this transport mechanism fractional order space derivative operators can be used here the numerical properties of partial differential equations of fractional order 1 2 are studied two numerical schemes an explicit and a semi implicit one are used in solving these equations two different discretization methods of the fractional derivative operator have also been used the accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order

systematic derivation of reaction diffusion equations with distributed delays and relations to fractional reaction diffusion equations and hyperbolic transport equations application to the theory of neolithic transition we introduce a general method for the systematic derivation of nonlinear reaction diffusion equations with distributed delays we study the interactions among different types of moving individuals atoms molecules quasiparticles biological organisms etc the motion of each species is described by the continuous time random walk theory analyzed in the literature for transport problems whereas the interactions among the species are described by a set of transformation rates which are nonlinear functions of the local concentrations of the different types of individuals we use the time interval between two jumps the transition time as an additional state variable and obtain a set of evolution equations which are local in time in order to make a connection with the transport models used in the literature we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so called generalized master equations the method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations hyperbolic reaction diffusion equations and delay differential reaction diffusion equations thus in the analysis of a given problem we can fit to the data the type of reaction diffusion equation and the corresponding physical and kinetic parameters the method is illustrated as a test case by the study of the neolithic transition we introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction diffusion equation for the population density we derive a delay evolution equation for the rate of advance of agriculture which illustrates an application of our analysis

subordinated continuous time ar processes and their application to modeling behavior of mechanical system abstract many real data exhibit behavior adequate to subdiffusion processes very often it is manifested by so called trapping events the visible evidence of subdiffusion we observe not only in financial time series but also in technical data in this paper we propose a model which can be used for description of such kind of data the model is based on the continuous time autoregressive time series with stable noise delayed by the infinitely divisible inverse subordinator the proposed system can be applied to real datasets with short time dependence visible jumps and mentioned periods of stagnation in this paper we extend the theoretical considerations in analysis of subordinated processes and propose a new model that exhibits mentioned properties we concentrate on the main characteristics of the examined subordinated process expressed mainly in the language of the measures of dependence which are main tools used in statistical investigation of real data we present also the simulation procedure of the considered system and indicate how to estimate its parameters the theoretical results we illustrate by the analysis of real technical data

stochastic processes with stable distributions in random environments the asymptotic behavior in random environments of random flights with stable distribution laws is analyzed by the field theoretic renormalization group random force fields with isotropic divergenceless curl free and unconstrained pair correlation functions with both finite and infinite correlation length are considered stability of the effective distribution laws in the scaling limit is determined and the scaling dimension of time is calculated in the ensuremath varepsilon mathit d mathit c d expansion where mathit d mathit c is the critical dimension of the model textcopyright 1996 the american physical society

indefinite sturm liouville problems and half range completeness where the function h changes sign on i such problems occur in certain physical models particularly in transport theory and statistical physics see 2 9 11 131 and references therein as for the standard eigenvalue problem it is true under quite general conditions and not difficult to prove that there is a set of eigenfunctions un which is complete in an appropriate hilbert space of functions this is known as full range completeness for the physical problems the question of interest is generally that of half range completeness to formulate this we set

stochastic foundations of fractional dynamics it is shown that fractional diffusion equations arise very naturally as the limiting dynamic equations of all continuous time random walks with decoupled temporal and spatial memories and with either temporal or spatial scale invariance fractal walks thus enlarging their stochastic foundations hitherto restricted to a particular case of fractal walk r hilfer and l anton phys rev e 51 r848 1995

fractional fokker planck equation and black scholes formula in composite diffusive regime in statistical physics anomalous diffusion plays an important role whose applications have been found in many areas in this paper we introduce a composite diffusive fractional brownian motion x h t xh s t 0 h 1 driven by anomalous diffusions as a model of asset prices and discuss the corresponding fractional fokker planck equation and black scholes formula we obtain the fractional fokker planck equation governing the dynamics of the probability density function of the composite diffusive fractional brownian motion and find the black scholes differential equation driven by the stock asset x h t and the corresponding black scholes formula for the fair prices of european option

answer to an open problem proposed by r metzler and j klafter in a positive answer to the open problem proposed by r metzler and j klafter it is proved that the asymptotic shape of the solution for a wide class of fractional fokker planck equations is a stretched gaussian for the initial condition being a pulse function in the homogeneous and heterogeneous fractal structures whose mean square displacement behaves like x 2 t t and x 2 t x t 0 1 respectively

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

mean first passage time for anomalous diffusion when the random force acting on a particle diffusing in an interval 0 l and subjected to a constant external force is a gaussian white noise the brownian mean squared displacement is described by the seminal relation x 2 2 mathrm dt ensuremath gamma with ensuremath gamma 1 however for more complicated random forces the diffusion may be slower ensuremath gamma l1 subdiffusion or faster ensuremath gamma g1 superdiffusion than the normal diffusion for both these cases we calculated the mean free passage time mfpt the time needed to reach one of the traps at boundaries the simple formulas for the different diffusive regimes are compared quantitatively for the simplest case of the absence of an external field and for an initial position in the middle of the interval it turns out that the mfpt s for anomalous diffusion can be both larger or smaller than that for normal diffusion depending on the values of the length of the interval and the diffusion coefficient moreover the mfpt can show nonmonotonic changes with the degree of departure from normal diffusion

fractional diffusion equation on fractals one dimensional case and asymptotic behaviour a fractional differential equation is studied and its application for describing diffusion on random fractal structures is considered it represents the simplest generalization of the fractional diffusion equation valid in euclidean systems the solution of the fractional equation in one dimension is discussed and compared with exact results for the fractional brownian motion and the one dimensional version of the standard diffusion equation on fractals in higher dimensions it correctly describes the asymptotic scaling behaviour of the probability density function on random fractals as obtained recently by using scaling arguments and exact enumeration calculations for the infinite percolation cluster at criticality

fractional diffusion equation on fractals three dimensional case and scattering function a fractional equation for diffusion in isotropic and homogeneous fractal structures is discussed it generalizes the fractional diffusion equation valid for d dimensional euclidean systems the asymptotic behaviour of the probability density function is obtained exactly analytical expressions are derived for the scattering and relaxation functions which can be studied by x ray and neutron scattering experiments on fractals

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

fractional dynamics from the ordinary langevin equation we consider the usual langevin equation depending on an internal time this parameter is substituted by a first passage time of a self similar markov process then the gaussian process is parent and the hitting time process is directing the probability to find the resulting process at the real time is defined by the integral relationship between the probability densities of the parent and directing processes the corresponding master equation becomes the fractional fokker planck equation we show that the resulting process has non markovian properties all its moments are finite the fluctuation dissipation relation and the h theorem hold

anomalous diffusion in one dimension abstract in view of the interest in the occurrence of anomalous diffusion r 2 t t 2h 0 1 2 in several physical circumstances we study anomalous diffusion per se in terms of exactly solvable one dimensional models the basic idea is to exploit the fact that temporal correlations lead directly to anomalous diffusion and provide solvable analogues of more realistic physical situations we first derive a general equation for a deterministic trajectory xe t that comprehensively characterizes the diffusive motion by finding the e quantiles of the time dependent probability distribution the class of all diffusion processes or equivalently symmetric random walks for which x e t t 1 2 and subsequently x e t t h is identified explicit solutions are presented for families of such processes considering random walks whose step sequences in time are governed by renewal processes and proceeding to the continuum limit a true generalization of brownian motion the latter corresponds to the limiting value h 1 2 is obtained explicitly x2 t t2h the diffusive spread of the initial condition is given by xe t th and the first passage time from the origin to the point x has a stable levy distribution with an exponent equal to h

stochastic foundation of normal and anomalous cattaneo type transport abstract we investigate the connection of the cattaneo equation and the stochastic continuous time random walk ctrw theory we show that the velocity model in a ctrw scheme is suited to derive the standard cattaneo equation and allows in principle for a generalisation to anomalous transport as a result for a broad waiting time distribution with diverging mean we find a strong memory to the initial condition of the system the ballistic behaviour subsists also for long times only if a characteristic waiting time exists a non ballistic enhanced motion is found in the limit of long times no transition to subdiffusion can be found

wiener hopf factorization inverse fourier transforms and exponentially dichotomous operators abstract for a certain class of matrix functions that are analytic on the real line but not at infinity an explicit method of factorization and of constructing inverse fourier transforms is developed this method is applied to invert wiener hopf integral equations on the half line and the full line the results obtained extend analogous results of the authors for rational matrix functions and for functions that are analytic on the real line and at infinity the analysis is based on an infinite dimensional realization theorem which involves operators that are a direct sum of two infinitesimal generators of c 0 semigroups of negative exponential type one of which has support on the negative half line and the other on the positive half line the latter operators are called exponentially dichotomous and the study of their properties forms an essential part of the paper

continuous time random walk with jump length correlated with waiting time abstract a coupled continuous time random walk ctrw model is proposed in which the jump length of a walker is correlated with waiting time the power law distribution is chosen as the probability density function of waiting time and the gaussian like distribution as the probability density function of jump length normal diffusion subdiffusion and superdiffusion can be realized within the present model it is shown that the competition between long tailed distribution and correlation of jump length and waiting time will lead to different diffusive behavior

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

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