improved and more feasible numerical methods for riesz space fractional partial differential equations abstract in this article the numerical solutions of fractional order partial differential equations with riesz space fractional derivatives on a finite domain have been considered two kinds of equations have been considered the riesz fractional diffusion equation and the riesz fractional advection dispersion equation the riesz fractional diffusion equation is obtained from the standard diffusion equation by replacing the second order space derivative with the riesz fractional derivative of order where 1 2 the riesz fractional advection dispersion equation is obtained from the standard advection dispersion equation by replacing the first and second order space derivatives with the riesz fractional derivatives of order 1 1 and 1 2 respectively to obtain the solutions firstly the system of odes is obtained using the improved matrix transform method with respect to the space variable after this the 3 1 pade approximation can be used to construct the exponential matrix in the analytic solution of the ode and we obtain two difference schemes it has also been shown that the two difference schemes are unconditionally stable and feasible using the matrix analysis method finally some numerical results are given which demonstrate the effectiveness of the two difference schemes

the structure and analysis of spherical time dependent processes the class of all spherical processes is investigated irrespective of moments properties of these processes are studied including their structure representations and linear and nonlinear spaces a basic representation is x t a 1 2 y t t in t where ais nonnegative and y t is gaussian time series analysesnot requiring moments are set forth in both the time and frequency domains including spectral analysis solutions of forecasting problems are found to be linear when the process is spherical with the property that errors of the optimal forecasts are stochasticalay most concentrated about zero these developments apply to the alpha sub gaussian processes and are advanced for use with certain heavy tailed processes arising in economics engineering and the physical sciences

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

diffusion in heterogeneous media diffusion processes in heterogeneous media and biological systems in particular are riddled with the difficult theoretical issue of whether the true origin of anomalous behavior is renewal or memory or a special combination of the two accounting for the possible mixture of renewal and memory sources of subdiffusion is challenging from a computational point of view as well this problem is exacerbated by the limited number of techniques available for solving fractional diffusion equations with time dependent coefficients we propose an iterative scheme for solving fractional differential equations with time dependent coefficients that is based on a parametric expansion in the fractional index we demonstrate how this method can be used to predict the long time behavior of nonautonomous fractional differential equations by studying the anomalous diffusion process arising from a mixture of renewal and memory sources

fuzzy r d portfolio selection of interdependent projects global competition of markets has forced firms to invest in targeted r d projects so that resources can be focused on successful outcomes a number of options are encountered to select the most appropriate projects in an r d project portfolio selection problem the selection is complicated by many factors such as uncertainty interdependences between projects risk and long lead time that are difficult to measure our main concern is how to deal with the uncertainty and interdependences in project portfolio selection when evaluating or estimating future cash flows this paper presents a fuzzy multi objective programming approach to facilitate decision making in the selection of r d projects here we present a fuzzy tri objective r d portfolio selection problem which maximizes the outcome and minimizes the cost and risk involved in the problem under the constraints on resources budget interdependences outcome projects occurring only once and discuss how our methodology can be used to make decision support tools for optimal r d project selection in a corporate environment a case study is provided to illustrate the proposed method where the solution is done by genetic algorithm ga as well as by multiple objective genetic algorithm moga

theory of adiabatic fluctuations third order noise we consider the response of a dynamical system driven by external adiabatic fluctuations based on the adiabatic following approximation we have made a systematic separation of timescales to carry out an expansion in where is the strength of fluctuations and is the damping rate we show that the probability distribution functions obey the differential equations of motion which contain third order terms beyond the usual fokker planck terms leading to non gaussian noise the problem of adiabatic fluctuations in velocity space which is the counterpart of brownian motion for fast fluctuations has been solved exactly the characteristic function and the associated probability distribution function are shown to be of stable form the linear dissipation leads to a steady state which is stable and the variances and higher moments are shown to be finite

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

operations on orbits of unimodular vectors abstract taking the m power of an entry is a well defined operation on the unimodular vectors in an modulo addition operations if n is at least 3 for an arbitrary commutative ring a and any integer m

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

analogies between colored levy noise and random channel approach to disordered kinetics we point out some interesting analogies between colored levy noise and the random channel approach to disordered kinetics these analogies are due to the fact that the probability density of the levy noise source plays a similar role as the probability density of rate coefficients in disordered kinetics although the equations for the two approaches are not identical the analogies can be used for deriving new useful results for both problems the random channel approach makes it possible to generalize the fractional uhlenbeck ornstein processes fuo for space and time dependent colored noise we describe the properties of colored noise in terms of characteristic functionals which are evaluated by using a generalization of huber s approach to complex relaxation phys rev b 31 6070 1985 we start out by investigating the properties of symmetrical white noise and then define the levy colored noise in terms of a langevin equation with a levy white noise source we derive exact analytical expressions for the various characteristic functionals which characterize the noise and a functional fractional fokker planck equation for the probability density functional of the noise at a given moment in time second by making an analogy between the theory of colored noise and the random channel approach to disordered kinetics we derive fractional equations for the evolution of the probability densities of the random rate coefficients in disordered kinetics these equations serve as a basis for developing methods for the evaluation of the statistical properties of the random rate coefficients from experimental data special attention is paid to the analysis of systems for which the observed kinetic curves can be described by linear or nonlinear stretched exponential kinetics

a stochastic programming model using an endogenously determined worst case risk measure for dynamic asset allocation we present a new approach to asset allocation with transaction costs a multiperiod stochastic linear programming model is developed where the risk is based on the worst case payoff that is endogenously determined by the model that balances expected return and risk utilizing portfolio protection and dynamic hedging an investment portfolio similar to an option like payoff structure on the initial investment portfolio is characterized the relative changes in the expected terminal wealth worst case payoff and risk aversion are studied theoretically and illustrated using a numerical example this model dominates a static mean variance model when the optimal portfolios are evaluated by the sharpe ratio

spectral measure of the harmonic oscillator with α stable noise we study the coordinate velocity couple of the harmonic oscillator with stable noise as previously shown by sokolov et al the distribution of this couple is bivariate stable in this work we determine explicitly its associated spectral measure exhibiting directly both the non independence and non ellipticity of the coordinate velocity couple knowledge of the spectral measure allows to analyze and quantify the deviation from ellipticity

fractal landscapes in physics and biology the purpose of this talk is to describe some recent progress in three aspects of statistical physics that overlap biology the first of these represents a discovery about the real world for which we presently have no theoretical understanding 1 the second concerns the solution of a rich model of many body physics which should apply to a wide range of biological situations but presently applies primarily to mathematical ecology 2 the third concerns what on the one hand might be a mere numerical coincidence or on the other hand might represent evidence that diffusion limited aggregation describes the physical mechanism underlying the growth of nerve cells at least the quasi two dimensional nerve cells that grow in the retina 3

continuous and discrete stable processes the one sided levy stable probability densities and the discrete stable distributions form a doubly stochastic poisson transform pair this relationship facilitates the formulation of a class of continuous stable stochastic processes

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

a vague set based decision support approach for evaluating research funding programs abstract scientific research assessment sra is receiving increasing attention in both academic and industry more and more organizations are recognizing the importance of sra for the optimal use of scarce resources in this paper a vague set theory based decision support approach is proposed for sra specifically a family of parameterized s owa operator is developed for the aggregation of vague assessments the proposed approach is introduced to evaluate the research funding programs of the national natural science foundation of china nsfc it provides a soft and expansive way to help the decision maker in nsfc to make his decisions the proposed approach can also be used for some other agencies to make similar assessment

fractional correlation functions in simple viscoelastic liquids abstract we develop a hydrodynamic formulation of fractional fluctuations in a viscoelastic liquid whose longitudinal modulus is a scalar and only depends on time the method is based on the introduction of fractional time derivatives in the hydrodynamic equations due to the viscoelastic memory coupled generalized langevin equations for the fluctuations of the state variables are formulated and solved analytically for a power law memory kernel with long correlation noise the associated fractional fokker planck equation ffpe is also derived and it is shown that it exhibits a fluctuation dissipation theorem fdt the explicit analytic diffusion coefficient is calculated for power law viscoelasticity and is shown to be subdiffusive the fractional correlations for longitudinal velocity and density fluctuations are analytically calculated and used to obtain the light scattering spectrum and the intermediate scattering function of the viscoelastic fluid our model calculation predicts that the fractional effects on these properties are not small effects and might be measurable

properties of levy flights on an interval with absorbing boundaries we consider a l evy yer of orderthat starts from a point x0 on an interval 0 l with absorbing boundaries we 7nd a closed form expression for an arbitrary average quantity char acterizing the trajectory of the yer such as mean 7rst passage time average total path length probability to be absorbed by one of the boundaries using fractional di erential equations with a riesz kernel we 7nd exact analytical expressions for these quantities in the continuous limit we 7nd numerically the eigenfunctions and the eigenvalues of these equations we study how the results of monte carlo simulations of the l evy ights with di erent ight length distribu tions converge to the continuous approximations we show that if x0 is placed in the vicinity of absorbing boundaries the average total path length has a minimum near 1 corresponding to the cauchy distribution we discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere c 2001 elsevier science b v all rights reserved

fluctuations in multiplicative systems with jumps fluctuation properties of the langevin equation including a multiplicative power law noise and a quadratic potential are discussed the noise has the l evy stable distribution if this distribution is truncated the covariance can be derived in the limit of large time it falls exponentially covariance in the stable case studied for the cauchy distribution exhibits a weakly stretched exponential shape and can be approximated by the simple exponential the dependence of that function on system parameters is determined then we consider a dynamics which involves the above process and obeys the generalized langevin equation the same as for the gaussian case the resulting distributions possess power law tails which fall similarly to those for the driving noise whereas central parts can assume the gaussian shape moreover a process with the covariance 1 t at large time is constructed and the corresponding dynamical equation is solved diffusion properties of systems for both covariances are discussed

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 mixed r d projects and securities portfolio selection model the business environment is full of uncertainty allocating the wealth among various asset classes may lower the risk of overall portfolio and increase the potential for more benefit over the long term in this paper we propose a mixed single stage r d projects and multi stage securities portfolio selection model specifically we present a bi objective mixed integer stochastic programming model moreover we use semi absolute deviation risk functions to measure the risk of mixed asset portfolio based on the idea of moments approximation method via linear programming we propose a scenario generation approach for the mixed single stage r d projects and multi stage securities portfolio selection problem the bi objective mixed integer stochastic programming problem can be solved by transforming it into a single objective mixed integer stochastic programming problem a numerical example is given to illustrate the behavior of the proposed mixed single stage r d projects and multi stage securities portfolio selection model

soluble models for dynamics driven by a super diffusive noise keywords strategie de production first passage time randomly driven harmonic oscillator integrated brownian motion reference lpm article 2002 001view record in web of science record created on 2006 06 22 modified on 2016 08 08

anomalous diffusion in systems driven by the stable levy noise with a finite noise relaxation time and inertia dynamical systems driven by a general levy stable noise are considered the inertia is included and the noise represented by a generalized ornstein uhlenbeck process has a finite relaxation time a general linear problem the additive noise is solved the resulting distribution converges with time to the distribution for the white noise massless case moreover a multiplicative noise is discussed it can make the distribution steeper and the variance which is finite depends sublinearly on time subdiffusion for a small mass a white noise limit corresponds to the stratonovich interpretation on the other hand the distribution tails agree with the ito interpretation if the inertia is very large an escape time from the potential well is calculated

the fractional fick s law for non local transport processes fick s law is extensively adopted as a model for standard diffusion processes however requiring separation of scales it is not suitable for describing non local transport processes we discuss a generalized non local fick s law derived from the space fractional diffusion equation generating the levy feller statistics this means that the fundamental solutions can be interpreted as levy stable probability densities in the feller parameterization with index 1 2 and skewness 2 we explore the possibility of defining an equivalent local diffusivity by displaying a few numerical case studies concerning the relevant quantities flux and gradient it turns out that the presence of asymmetry 0 plays a fundamental role it produces shift of the maximum location of the probability density function and gives raise to phenomena of counter gradient transport

algebraically decaying noise in a system of particles with hard core interactions we record and analyze the noise experienced by a tracer particle in a one dimensional system of particles interacting with hard core interactions we find that the correlations of the noise are long range with an algebraic decay in time

functional characterization of generalized langevin equations we present an exact functional formalism to deal with linear langevin equations with arbitrary memory kernels and driven by an arbitrary noise structure characterized through its characteristic functional no other hypothesis is assumed over the noise neither do we use the fluctuation dissipation theorem we find that the characteristic functional of the linear process can be expressed in terms of noise functional and the green function of the deterministic memory like dissipative dynamics this yields a procedure for calculating the full kolmogorov hierarchy of the non markov process as examples we have characterized through the 1 time probability a noise induced interplay between the dissipative dynamics and the structure of different noises conditions that lead to non gaussian statistics and distributions with long tails are analysed the introduction of arbitrary fluctuations in fractional langevin equations has also been pointed out

stable equilibrium based on levy statistics a linear boltzmann equation approach to obtain further insight on possible power law generalizations of boltzmann equilibrium concepts we consider stochastic collision models the models are a generalization of the rayleigh collision model for a heavy one dimensional particle m interacting with ideal gas particles with a mass m m similar to previous approaches we assume elastic uncorrelated and impulsive collisions we let the bath particle velocity distribution function to be of general form namely we do not postulate a specific form of power law equilibrium we show under certain conditions that the velocity distribution function of the heavy particle is levy stable the maxwellian distribution being a special case we demonstrate our results with numerical examples the relation of the power law equilibrium obtained here to thermodynamics is discussed in particular we compare between two models a thermodynamic and an energy scaling approaches these models yield insight into questions like the meaning of temperature for power law equilibrium and into the issue of the universality of the equilibrium i e is the width of the generalized maxwellian distribution functions obtained here independent of coupling constant to the bath

an abelian group structure on orbits of unimodular squares in dimension 3 the subject of whether orbit sets of unimodular rows modulo the elementary subgroup um a re a have a group structure probably r r w x began first in 9 with the vaserstein symbol v um a re a a w a 3 3 e which was shown to be bijective when the stable dimension of a was less than or equal to 2 here the stable dimension of the ring a is in the sense of bass and equals the stable rank of a y 1 the reader could replace it by the more familiar krull dimension for simplicity the vaserstein symbol v and the grothendieck group w a of stably equivalent alternating e w x matrices of pfaffian 1 is described in 9 p 945 and briefly in the next section w x in 1 2 van der kallen has shown that if a is a commutative ring of stable dimension d then the coset space um a re a has the structure n n 1 4 of an abelian group provided n g max 3 d q 2 for topological situa2 tions like a being the ring of continuous functions c x r on a finite cw complex x of dimension d he has given some nice models namely the ny1 higher cohomotopy groups p x for this group w x in 5 it was shown that the vaserstein symbol was not injective for three dimensional rings so one could not expect to get a very nice group structure on the orbit set of unimodular 3 vectors over a three dimensional ring in this article we will prove that the orbit set of squares of

heuristic algorithms for the portfolio selection problem with minimum transaction lots the problem of selecting a portfolio has been largely faced in terms of minimizing the risk given the return while the complexity of the quadratic programming model due to markowitz has been overcome by the recent progress in algorithmic research the introduction of linear risk functions has given rise to the interest in solving portfolio selection problems with real constraints in this paper we deal with the portfolio problem with minimum transaction lots we show that in this case the problem of finding a feasible solution is independently of the risk function np complete moreover given the mixed integer linear model new heuristics are proposed which starting from the solution of the relaxed problem allow to find a solution close to the optimal one the algorithms are based on the construction of mixed integer subproblems using only a part of the securities available formulated using the information obtained from the solution of the relaxed problem the heuristics have been tested with respect to two disjoint time periods using real data from the milan stock exchange

mean absolute deviation portfolio optimization model and its applications to tokyo stock market the purpose of this paper is to demonstrate that a portfolio optimization model using the l1 risk mean absolute deviation risk function can remove most of the difficulties associated with the classical markowitz s model while maintaining its advantages over equilibrium models in particular the l1 risk model leads to a linear program instead of a quadratic program so that a large scale optimization problem consisting of more than 1 000 stocks may be solved on a real time basis numerical experiments using the historical data of nikkei 225 stocks show that the l1 risk model generates a portfolio quite similar to that of the markowitz s model within a fraction of time required to solve the latter

fractal rock slope dynamics anticipating a collapse time series of dilatometric measurements of relative displacements on rock cracks on stable and unstable sandstone slopes were analyzed the inherent dynamics of rock slopes lack any significant nonlinearity however the residuals obtained by removing meteorological influences are fat tailed non gaussian fluctuations with short range correlations in the case of stable slopes the fluctuations of unstable slopes exhibit self affine dynamics of fractional brownian motions with power law long range correlations and are characterized by asymptotic power law probability distributions with decay coefficients outside the range of stable l evy distributions

anomalous fluctuation properties we complement and extend our work on fluctuations arising in nonequilibrium systems in steady states driven by l evy noise h touchette and e g d cohen phys rev e 76 020101 r 2007 as a concrete example we consider a particle subjected to a drag force and a l evy white noise with tail index ensuremath mu 0 2 and calculate the probability distribution of the work done on the particle by the drag force as well as the probability distribution of the work dissipated by the dragged particle in a nonequilibrium steady state for 0l ensuremath mu l2 both distributions satisfy what we call an anomalous fluctuation property characterized by positive and negative fluctuations that asymptotically have the same probability for ensuremath mu 2 by contrast the work and dissipated work distributions satisfy the known conventional and extended fluctuation relations respectively which are both characterized by positive fluctuations that are exponentially more probable than negative fluctuations the difference between these different fluctuation behaviors is discussed in the context of large deviation theory experiments that could probe or reveal an anomalous fluctuation property are also discussed

on necklaces with hard non spherical beads abstract in spite of the fact that our world is three dimensional one dimensional statistical models are important for a number of reasons first in some cases they are exactly solvable while their higher dimensional counterparts are not consequently the one dimensional solutions may provide some insight into basic important mechanisms and useful approximation schemes then there are many instances where the description of a physical system as one dimensional or even better as quasi one dimensional is quite realistic in the present article two families of quasi one dimensional systems are presented although the two families are constructed to describe different physics their basic mathematical structure is similar the first family takes into the account the effect of hard core repulsion of non spherical particles it is prototyped by a necklace of hard non spherical beads that are free to move along the necklace and to rotate about it the second family takes into account the possibility of squeezing of hard objects because the system is not strictly one dimensional this is prototyped by a system of hard objects confined to a narrow pipe with a diameter which is somewhat larger than the size of the objects but not large enough to allow changing the order of the objects along the pipe the general treatment of such models is discussed and the exact configurational free energy is worked out for a specific example

linearly coupled quantum oscillators with levy stable noise abstract the influence of an external noise on two linearly coupled quantum oscillators is investigated the noise is described by the probability measure corresponding to a symmetric levy stable distribution in calculations the path integral technique is used this paper is a continuation of investigations started by bose datta and feinsilver

discrete random walk models for symmetric levy feller diffusion processes we propose a variety of models of random walk discrete in space and time suitable for simulating stable random variables of arbitrary index 0 2 in the symmetric case we show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous markovian stochastic processes which we refer to as levy feller diffusion processes

traveling length and minimal traveling time for flow through percolation networks with long range spatial correlations we study the distributions of traveling length l and minimal traveling time t mathrm min through two dimensional percolation porous media characterized by long range spatial correlations we model the dynamics of fluid displacement by the convective movement of tracer particles driven by a pressure difference between two fixed sites wells separated by euclidean distance r for strongly correlated pore networks at criticality we find that the probability distribution functions p l and p t mathrm min follow the same scaling ansatz originally proposed for the uncorrelated case but with quite different scaling exponents we relate these changes in dynamical behavior to the main morphological difference between correlated and uncorrelated clusters namely the compactness of their backbones our simulations reveal that the dynamical scaling exponents d l and d t for correlated geometries take values intermediate between the uncorrelated and homogeneous limiting cases where l ensuremath sim r d l and t mathrm min ensuremath sim r d t and l and t mathrm min are the most probable values of l and t mathrm min respectively

levy noise levy flights levy fluctuations this paper is concerned with a levy noise characterized for t 0 by the functional with 0 2 then levy flights can be defined through a stochastic differential equations rather than the usual chapmann kolmogorov equation we have used this functional approach to solve a plane rotor in the presence of levy noise the linear damped stochastic process driven by levy noise is revisited and its non autonomous and non markovian generalizations have been solved in the context of our functional analysis

a minimax portfolio selection rule with linear programming solution a new principle for choosing portfolios based on historical returns data is introduced the optimal portfolio based on this principle is the solution to a simple linear programming problem this principle uses minimum return rather than variance as a measure of risk in particular the portfolio is chosen that minimizes the maximum loss over all past observation periods for a given level of return this objective function avoids the logical problems of a quadratic nonmonotone utility function implied by mean variance portfolio selection rules the resulting minimax portfolios are diversified for normal returns data the portfolios are nearly equivalent to those chosen by a mean variance rule framing the portfolio selection process as a linear optimization problem also makes it feasible to constrain certain decision variables to be integer or 0 1 valued this feature facilitates the use of more complex decision making models including models with fixed transaction charges and models with boolean type constraints on allocations

generating scenario trees for multistage decision problems in models of decision making under uncertainty we often are faced with the problem of representing the uncertainties in a form suitable for quantitative models if the uncertainties are expressed in terms of multivariate continuous distributions or a discrete distribution with far too many outcomes we normally face two possibilities either creating a decision model with internal sampling or trying to find a simple discrete approximation of the given distribution that serves as input to the model this paper presents a method based on nonlinear programming that can be used to generate a limited number of discrete outcomes that satisfy specified statistical properties users are free to specify any statistical properties they find relevant and the method can handle inconsistencies in the specifications the basic idea is to minimize some measure of distance between the statistical properties of the generated outcomes and the specified properties we illustrate the method by single and multiple period problems the results are encouraging in that a limited number of generated outcomes indeed have statistical properties that are close to or equal to the specifications we discuss how to verify that the relevant statistical properties are captured in these specifications and argue that what are the relevant properties will be problem dependent

ornstein uhlenbeck cauchy process we combine earlier investigations of linear systems subject to levy fluctuations with recent attempts to give meaning to so called levy flights in external force fields we give a complete construction of the ornstein uhlenbeck cauchy process as a fully computable paradigm example of doob s stable noise supported ornstein uhlenbeck process despite the nonexistence of all moments we determine local characteristics forward drift of the process generators of forward and backward dynamics and relevant pseudodifferential evolution equations the induced nonstationary spatial process is proved to be markovian and quite apart from its inherent discontinuity defines an associated velocity process in a probabilistic sense

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