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

on the parametric description of log growth rates of cities sizes of four european countries and the usa abstract we have studied the parametric description of the distribution of the log growth rates of the sizes of cities of france germany italy spain and the usa we have considered several parametric distributions well known in the literature as well as some others recently introduced there are some models that provide similar excellent performance for all studied samples the normal distribution is not the one observed empirically

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

dispersive estimates for the dirac equation in an aharonov bohm field abstract we prove local smoothing and weighted strichartz estimates for the dirac equation with a aharonov bohm potential the proof inspired by 12 relies on an explicit representation of the solution built in terms of spectral projections

the global cauchy problem for the nls and nlkg with small rough data abstract by using the unit cube decomposition to the frequency spaces we study the cauchy problem for the nonlinear schrodinger equation and the nonlinear klein gordon equation some global well posedness results are obtained for the small cauchy data in some modulation spaces m p 1 s

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

global strong solution to the thirring model in critical space abstract we obtain a strong solution in charge critical space l 2 r of the thirring system and federbusch equations in one space dimension by using solution representation of the models the uniqueness is obtained for the solution l 0 t l 2 r l 4 r a decay of local charge and asymptotic behavior of the field can be shown directly

estimate for generalized unimodular multipliers on modulation spaces abstract in this paper we consider the operator e i t h d where h defined on r n is a c r n 0 positive homogeneous function with degree 0 and r r is a smooth function satisfying the following a1 there exists a constant m 1 0 such that for all n 0 n 0 r r m 1 r 1 a2 there exists a constant m 2 0 such that for all n 0 r r m 2 0 r 1 we prove the boundedness of the operator e i t h d on the modulation spaces and obtain its asymptotic estimate as t goes to infinity as applications we give the grow up rate of the solutions for the cauchy problems for the generalized wave klein gordon and schrodinger equations

levy statistics in a hamiltonian system enhanced diffusion in a hamiltonian system is studied in terms of the continuous time random walk formulation for l evy walks the previous l evy walk scheme is extended i to include interruptions by periods of temporal localization and ii to describe motion in two dimensions we analyze a case of conservative motion in a two dimensional periodic potential numerical calculations of the mean squared displacements and the propagators for intermediate energies are consistent with the l evy walk description

estimates on modulation spaces for schrodinger evolution operators with quadratic and sub quadratic potentials abstract in this paper we give new estimates for the solution to the schrodinger equation with quadratic and sub quadratic potentials in the framework of modulation spaces

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

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

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

path integral approach to fractional levy motion we give an explicit and simple derivation of the propagator for fractional levy motion using path integral methods some known forms of propagators like for brownian motion are recovered in limiting cases we also discuss the associated kinetic equations as well as some extensions

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

levy walk approach to anomalous diffusion abstract the transport properties of levy walks are discussed in the framework of continuous time random walks ctrw with coupled memories this type of walks may lead to anomalous diffusion where the mean squared displacement r2 t t with 1 we focus on the enhanced diffusion limit 1 in one dimension and present our results on r2 t the mean number of distinct sites visited s t and p r t the probability of being at position r at time t

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

path integral representation for fractional brownian motion fractional brownian motion fbm is a generalization of the usual brownian motion a path integral representation that has recently been suggested for it is shown to be not for the fbm but for a different generalization of the brownian motion a new path integral representation is given and its measure has fractional derivatives of the path in it the measure shows that the process is gaussian but is in general non markovian even though brownian motion itself is markovian it is shown how the propagator for the motion of free fbm may be evaluated this is somewhat more complex than for the usual path integrals due to the occurrence of fractional derivatives we also find the propagator in the presence of a linear absorption potential and for fbm on a ring

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

solving fractional integral equations by the haar wavelet method haar wavelets for the solution of fractional integral equations are applied fractional volterra and fredholm integral equations are considered the proposed method also is used for analysing fractional harmonic vibrations the efficiency of the method is demonstrated by three numerical examples

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

response characteristics of a fractional oscillator abstract the integral equation of motion of a driven fractional oscillator is obtained by generalizing the corresponding equation of motion of a driven harmonic oscillator to include integrals of arbitrary order according to the methods of fractional calculus the green s function solution for the fractional oscillator is obtained in terms of mittag leffler functions using laplace transforms the response and resonance characteristics of the fractional oscillator are studied for several cases of forcing function

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

vector valued multiparameter singular integrals and pseudodifferential operators abstract we consider multiparameter singular integrals and pseudodifferential operators acting on mixed norm bochner spaces l p 1 p n r n 1 r n n x where x is a umd banach space satisfying pisier s property these geometric conditions are shown to be necessary we obtain a vector valued version of a result by r fefferman and stein also providing a new inductive proof of the original scalar valued theorem then we extend a result of bourgain on singular integrals in umd spaces with an unconditional basis to a multiparameter situation finally we carry over a result of yamazaki on pseudodifferential operators to the bochner space setting improving the known vector valued results even in the one parameter case

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

the growth of business firms theoretical framework and empirical evidence we introduce a model of proportional growth to explain the distribution pg g of business firm growth rates the model predicts that pg g is exponential in the central part and depicts an asymptotic power law behavior in the tails with an exponent 3 because of data limitations previous studies in this field have been focusing exclusively on the laplace shape of the body of the distribution in this article we test the model at different levels of aggregation in the economy from products to firms to countries and we find that the predictions of the model agree with empirical growth distributions and size variance relationships

operators of fractional integration and their applications the main purpose of this paper is to present a systematic and historical account of the investigations carried out by various authors in the field of fractional calculus and its applications several interesting results relevant to the present investigation are also considered

construction of relativistic quantum fields in the framework of white noise analysis we construct a class of euclidean invariant distributions h indexed by a function h holomorphic at zero these generalized functions can be considered as generalized densities w r t the white noise measure and their moments fulfill all osterwalder schrader axioms except for reflection positivity the case where f s h is 12s2 s r is a levy characteristic is considered in rev math phys 8 763 1996 under this assumption the moments of the euclidean invariant distributions h can be represented as moments of a generalized white noise measure ph here we enlarge this class by convolution with kernels g coming from euclidean invariant operators g the moments of the resulting euclidean invariant distributions hg also fulfill all osterwalder schrader axioms except for reflection positivity for no nontrivial case we succeeded in proving reflection positivity nevertheless an analytic extension to wightman functions can be performed these functions fulfill all wightman axioms except for the po

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

a note on pareto tails and lognormal body of us cities size distribution abstract luckstead and devadoss 2017 fitted a distribution made up of lower tail pareto lognormal body and upper tail pareto to city sizes in the us here we propose a class of distributions providing better fits the fits are assessed in terms of akaike information criterion bayesian information criterion and probability plots

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

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

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

convolution theorems with weights analogues of young s inequality and the convolution theorem are shown to hold when the lp and l p q spaces have underlying measure defined in terms of power weights

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

time frequency analysis of the dirac equation abstract the purpose of this paper is to investigate several issues concerning the dirac equation from a time frequency analysis perspective more precisely we provide estimates in weighted modulation and wiener amalgam spaces for the solutions of the dirac equation with rough potentials we focus in particular on bounded perturbations arising as the weyl quantization of suitable time dependent symbols as well as on quadratic and sub quadratic non smooth functions hence generalizing the results in 40 we then prove local well posedness on the same function spaces for the nonlinear dirac equation with a general nonlinearity including power type terms and the thirring model for this study we adopt the unifying framework of vector valued time frequency analysis 56 most of the preliminary results are stated under general assumptions and hence they may be of independent interest

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

some new strichartz estimates for the schrodinger equation abstract we deal with fixed time and strichartz estimates for the schrodinger propagator as an operator on wiener amalgam spaces we discuss the sharpness of the known estimates and we provide some new estimates which generalize the classical ones as an application we present a result on the wellposedness of the linear schrodinger equation with a rough time dependent potential

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

fractional derivative quantum fields at positive temperature abstract this paper considers fractional generalization of finite temperature klein gordon kg field and vector potential in covariant gauge and static temporal gauge fractional derivative quantum field at positive temperature can be regarded as a collection of infinite number of fractional thermal oscillators generalized riemann zeta function regularization and heat kernel techniques are used to obtain the high temperature expansion of free energy associated with the fractional kg field we also show that quantization of the fractional derivative fields can be carried out by using the parisi wu stochastic quantization

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

stochastic processes associated with kms states abstract a general mathematical framework is presented for the connection between quantum statistical mechanics and periodic stochastic processes stochastically positive kms systems periodic os positive stochastic processes periodic positive semigroup structures are defined and shown to be equivalent the objects of the tomita takesaki theory are explicitly exhibited in terms of the associated stochastic process and the statements of the theory are proven a two sided markov property for the stochastic process is related to a cyclicity property of the kms state perturbations are constructed by the feynman kac nelson formula the general framework is applied to kms states given by density matrices

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

unimodular fourier multipliers for modulation spaces abstract we investigate the boundedness of unimodular fourier multipliers on modulation spaces surprisingly the multipliers with general symbol e i where 0 2 are bounded on all modulation spaces but in general fail to be bounded on the usual l p spaces as a consequence the phase space concentration of the solutions to the free schrodinger and wave equations are preserved as a byproduct we also obtain boundedness results on modulation spaces for singular multipliers sin for 0

the pareto law of incomes an explanation and an extension abstract a stochastic model for the generation of observed income distributions is used to provide an explanation for the pareto law of incomes analysis of the model also yields a prediction of paretian power law behaviour in the lower tail of the distribution and this is shown to occur in a number of empirical distributions a tractable four parameter distribution is derived and shown to fit extremely well to a number of different empirical income distributions

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

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

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

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

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

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

remarks on fourier multipliers and applications to the wave equation exploiting continuity properties of fourier multipliers on modulation spaces and wiener amalgam spaces we study the cauchy problem for the nlw equation local wellposedness for rough data in modulation spaces and wiener amalgam spaces is shown the results formulated in the framework of modulation spaces refine those in a benyi k a okoudjou local well posedness of nonlinear dispersive equations on modulation spaces preprint april 2007 available at arxiv 0704 0833v1 the same arguments may apply to obtain local wellposedness for the nlkg equation

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 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

on the pointwise convergence of the integral kernels in the feynman trotter formula we study path integrals in the trotter type form for the schrodinger equation where the hamiltonian is the weyl quantization of a real valued quadratic form perturbed by a potential v in a class encompassing that considered by albeverio and ito in celebrated papers of fourier transforms of complex measures essentially v is bounded and has the regularity of a function whose fourier transform is in l 1 whereas the strong convergence in l 2 in the trotter formula as well as several related issues at the operator norm level are well understood the original feynman s idea concerned the subtler and widely open problem of the pointwise convergence of the corresponding probability amplitudes that are the integral kernels of the approximation operators we prove that for the above class of potentials such a convergence at the level of the integral kernels in fact occurs uniformly on compact subsets and for every fixed time except for certain exceptional time values for which the kernels are in general just distributions actually theorems are stated for potentials in several function spaces arising in harmonic analysis with corresponding convergence results proofs rely on banach algebras techniques for pseudo differential operators acting on such function spaces

continuity properties for modulation spaces with applications to pseudo differential calculus i we discuss continuity for weighted modulation spaces andprove that many such spaces can be obtained in a canonicalway from the corresponding standard modulation spaces we also discussthe trace operator a a 0 acting on modulationspaces the results are used to get inclusions betweenmodulation spaces and besov spaces and proving continuityfor pseudo differential operators and toeplitz operators

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

path integral representation of fractional harmonic oscillator abstract fractional oscillator process can be obtained as the solution to the fractional langevin equation there exist two types of fractional oscillator processes based on the choice of fractional integro differential operators namely weyl and riemann liouville an operator identity for the fractional differential operators associated with the fractional oscillators is derived it allows the solution of fractional langevin equations to be obtained by simple inversion the relationship between these two fractional oscillator processes is studied the operator identity also plays an important role in the derivation of the path integral representation of the fractional oscillator processes relevant quantities such as two point and n point functions can be calculated from the generating functions

fokker planck equation and path integral representation of the fractional ornstein uhlenbeck process with two indices this paper considers the fokker planck equation and path integral formulation of the fractional ornstein uhlenbeck process parametrized by two indices the effective fokker planck equation of this process is derived from the associated fractional langevin equation the path integral representation of the process is constructed and the basic quantities are evaluated

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

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

endpoint estimates and global existence for the nonlinear dirac equation with potential abstract we prove endpoint estimates with angular regularity for the wave and dirac equations perturbed with a small potential the estimates are applied to prove global existence for the cubic dirac equation perturbed with a small potential for small initial h 1 data with additional angular regularity this implies in particular global existence in the critical energy space h 1 for small radial data

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