design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

a second order difference scheme for the time fractional substantial diffusion equation abstract in this work a second order approximation of the fractional substantial derivative is presented by considering a modified shifted substantial grunwald formula and its asymptotic expansion moreover the proposed approximation is applied to a fractional diffusion equation with fractional substantial derivative in time with the use of the fourth order compact scheme in space we give a fully discrete grunwald letnikov formula based compact difference scheme and prove its stability and convergence by the energy method under smooth assumptions in addition the problem with nonsmooth solution is also discussed and an improved algorithm is proposed to deal with the singularity of the fractional substantial derivative numerical examples show the reliability and efficiency of the scheme

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the langevin equation when the microscopic time scale diverges the differential equations on the macroscopic level are no longer valid and must be replaced with fractional differential equations of motion in particular we obtain a fractional differential stochastic equation of motion after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

tempered stable levy motion and transient super diffusion the space fractional diffusion equation models anomalous super diffusion its solutions are transition densities of a stable levy motion representing the accumulation of power law jumps the tempered stable levy motion uses exponential tempering to cool these jumps a tempered fractional diffusion equation governs the transition densities which progress from super diffusive early time to diffusive late time behavior this article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift a temporal and spatial second order crank nicolson method is developed based on a finite difference formula for tempered fractional derivatives a new exponential rejection method for simulating tempered levy stables is presented to facilitate particle tracking codes

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

spectral methods for tempered fractional differential equations in this paper we first introduce fractional integral spaces which possess some features i when 0 1 functions in these spaces are not required to be zero on the boundary ii the tempered fractional operators are equivalent to the riemann liouville operator in the sense of the norm spec tral galerkin and petrov galerkin methods for tempered fractional advection problems and tempered fractional diffusion problems can be developed as the classical spectral galerkin and petrov galerkin methods error analysis is provided and numerically confirmed for the tempered fractional advection and diffusion problems

transport in the spatially tempered fractional fokker planck equation a study of truncated levy flights in super diffusive transport in the presence of an external potential is presented the study is based on the spatially tempered fractional fokker planck tffp equation in which the fractional diffusion operator is replaced by a tempered fractional diffusion tfd operator we focus on harmonic quadratic potentials and periodic potentials with broken spatial symmetry the main objective is to study the dependence of the steady state probability density function pdf and the current in the case of periodic potentials on the level of tempering and on the order of the fractional derivative in space an expansion of the tfd operator for large is presented and the corresponding equation for the coarse grained pdf is obtained the steady state pdf solution of the tffp equation for a harmonic potential is computed numerically in the limit the pdf approaches the expected boltzmann distribution however nontrivial departures from this distribution are observed for finite 0 truncations and 2 in the study of periodic potentials we use two complementary numerical methods a finite difference scheme based on the grunwald letnikov discretization of the truncated fractional derivatives and a fourier based spectral method in the limit the pdfs converges to the boltzmann distribution and the current vanishes however for 2 the pdf deviates from the boltzmann distribution and a finite non equilibrium ratchet current appears for any 0 the current is observed to converge exponentially in time to the steady state value the steady state current exhibits algebraical decay with as j for 1 75 however for 1 5 the steady state current decays exponentially with as j e in the presence of an asymmetry in the tfd operator the tempering can lead to a current reversal a detailed numerical study is presented on the dependence of the current on and the physical parameters of the system

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

ldpc codes for fading gaussian broadcast channels in this work we study coding over a class of two user broadcast channels bcs with additive white gaussian noise and multiplicative fading known at the receivers only joint decoding of low density parity check ldpc codes is analyzed the message update rule at the mapping node linking the users codes is derived and is found to exhibit an interesting soft interference cancellation property high performance codes are found using the differential evolution optimization technique and extrinsic information transfer analysis adapted to our multiuser setting the optimized codes have rates very close to the boundary of the achievable region for binary constrained input for both faded and unfaded channels simulation results for moderate block lengths show that our codes operate within less than 1 db of their respective threshold

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

applications of inverse tempered stable subordinators abstract the inverse tempered stable subordinator is a stochastic process that models power law waiting times between particle movements with an exponential tempering that allows all moments to exist this paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time fractional diffusion equation and its folded density solves a tempered time fractional telegraph equation two explicit formulae for the density function are developed and applied to compute explicit solutions to tempered fractional cauchy problems where a tempered fractional derivative replaces the first derivative in time several examples are given including tempered fractional diffusion equations on bounded or unbounded domains and the probability distribution of a tempered fractional poisson process it is shown that solutions to the tempered fractional diffusion equation have a cusp at the origin

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

linear maps preserving the spectral radius abstract let x be a complex banach space if b x b x is a surjective linear map such that a and a have the same spectral radius for every a b x then c where is either an algebra automorphism or an antiautomorphism of b x and c is a complex constant such that c 1

extrinsic information transfer functions model and erasure channel properties extrinsic information transfer exit charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems a model is introduced that applies to decoding problems including the iterative decoding of parallel concatenated turbo codes serially concatenated codes low density parity check ldpc codes and repeat accumulate ra codes exit functions are defined using the model and several properties of such functions are proved for erasure channels one property expresses the area under an exit function in terms of a conditional entropy a useful consequence of this result is that the design of capacity approaching codes reduces to a curve fitting problem for all the aforementioned codes a second property relates the exit function of a code to its helleseth klove levenshtein information functions and thereby to the support weights of its subcodes the relation is via a refinement of information functions called split information functions and via a refinement of support weights called split support weights split information functions are used to prove a third property that relates the exit function of a linear code to the exit function of its dual

time changed ornstein uhlenbeck process the ornstein uhlenbeck process is one of the most popular systems used for financial data description however this process has also been examined in the context of many other phenomena in this paper we consider the so called time changed ornstein uhlenbeck process in which time is replaced by an inverse subordinator of general infinite divisible distribution time changed processes nowadays play an important role in various fields of mathematical physics chemistry and biology as well as in finance in this paper we examine the main characteristics of the time changed ornstein uhlenbeck process such as the covariance function moreover we also prove the formula for a generalized fractional fokker planck equation that describes the one dimensional probability density function of the analyzed system for three cases of subordinators we show the special forms of obtained general formulas furthermore we mention how to simulate the trajectory of the ornstein uhlenbeck process delayed by a general inverse subordinator

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

governing equations and solutions of anomalous random walk limits fachbereich mathematik university of dortmund 44221 dortmund germany received 5 june 2002 published 30 december 2002 continuous time random walks model anomalous diffusion coupling allows the magnitude of particlejumps to depend on the waiting time between jumps governing equations for the long time scaling limits ofthese models are found to have fractional powers of coupled space and time differential operators explicitsolutions and scaling properties are presented for these equations which can be used to model ow in porousmedia and other physical systems doi 10 1103 physreve 66 060102 pacs number s 05 40 2a 02 50 cw 05 60 2k 47 55 mhi introduction

bounds on information combining when the same data sequence is transmitted over two independent channels or when a data sequence is transmitted twice but independently over the same channel the independent observations can be combined at the receiver side from an information theory point of view the overall mutual information between the data sequence and the received sequences represents a combination of the mutual information of the two channels this concept is termed information combining a lower bound and an upper bound on the combined information is presented and it is proved that these bounds are tight furthermore this principle is extended to the computation of extrinsic information on single code bits for a repetition code and for a single parity check code of length three respectively for illustration of the concept and the bounds on information combining two applications are considered first bounds on the information processing characteristic ipc of a parallel concatenated code are derived from its extrinsic information transfer exit chart second bounds on the exit chart for an outer repetition code and for an outer single parity check code of a serially concatenated coding scheme are computed

on spectrally bounded linear maps on b x abstract let x be a complex banach space and denote by b x the algebra of all linear bounded operators on x for t b x denote by r t its spectral radius we prove that if b x b x is a linear and surjective map such that i given any a b b x there exists a positive real number m a b such that r a b m a b r a b for all complex numbers satisfying 1 ii r t 0 implies r t 0 then there exist a linear functional f b x c a nonzero complex number c and either a bounded invertible linear map u x x such that t c u t u 1 f t i t b x or a bounded invertible linear map u x x such that t c u t u 1 f t i t b x also the linear functional f satisfies the property that given any a b b x there exists n a b 0 such that f a f b n a b r a b for all complex numbers satisfying 1

extremes of information combining extreme densities for information combining are found for two important channel models the binary input symmetric parallel broadcast channel and the parity constrained input symmetric parallel channels following upper and lower mutual information thresholds are stated for per bit maximum a posteriori probability map decoding and low density parity check ldpc code ensembles

implementing the han kobayashi scheme using low density parity check codes over gaussian interference channels we focus on gaussian interference channels gics and study the han kobayashi coding strategy for the two user case with the objective of designing implementable explicit channel codes specifically low density parity check codes are adopted for use over the channel their benefits are studied and suitable codes are designed iterative joint decoding is used at the receivers where independent and identically distributed channel adapters are used to prove that log likelihood ratios exchanged among the nodes of the tanner graph enjoy symmetry when bpsk or qpsk with gray coding is employed this property is exploited in the proposed code optimization algorithm adopting a random perturbation technique code optimization and convergence threshold computations are carried out for different gics employing finite constellations by tracking the average mutual information furthermore stability conditions for the admissible degree distributions under strong and weak interference levels are determined via examples it is observed that the optimized codes using bpsk or qpsk with gray coding operate close to the capacity boundary for strong interference for the case of weak interference it is shown that nontrivial rate pairs are achievable via the newly designed codes which are not possible by single user codes with time sharing performance of the designed codes is also studied for finite block lengths through simulations of specific codes picked with the optimized degree distributions with random constructions where for one instance the results are compared with those of some structured designs

path probability density functions for semi markovian random walks in random walks the path representation of the green s function is an infinite sum over the length of path probability density functions pdfs recently a closed form expression for the green s function of an arbitrarily inhomogeneous semi markovian random walk in a one dimensional 1d chain of l states was obtained by utilizing path pdfs calculations here we derive and solve in laplace space the recursion relation for the n order path pdf for the same system the recursion relation relates the n order path pdf to l 2 round towards zero for an odd l shorter path pdfs and has n independent coefficients that obey a universal formula the z transform of the recursion relation straightforwardly gives the generating function for path pdfs from which we recover the green s function of the random walk but moreover derive an explicit expression for any path pdf of the random walk these expressions give the most detailed description of arbitrarily inhomogeneous semi markovian random walks in 1d

kramers escape problem for fractional klein kramers equation with tempered α stable waiting times in this paper we extend the subdiffusive klein kramers model in which the waiting times are modeled by the stable laws to the case of waiting times belonging to the class of tempered stable distributions we introduce a generalized version of the klein kramers equation in which the fractional riemman liouville derivative is replaced with a more general integro differential operator this allows a transition from the initial subdiffusive character of motion to the standard diffusion for long times to be modeled taking advantage of the corresponding langevin equation we study some properties of the tempered dynamics in particular we approximate solutions of the tempered klein kramers equation via monte carlo methods also we study the distribution of the escape time from the potential well and compare it to the classical results in the kramers escape theory finally we derive the analytical formula for the first passage time distribution for the case of free particles we show that the well known sparre andersen scaling holds also for the tempered subdiffusion

a stochastic differential equation model with jumps for fractional advection and dispersion the path of a tracer particle through a porous medium is typically modeled by a stochastic differential equation sde driven by brownian noise this model may not be adequate for highly heterogeneous media this paper extends the model to a general sde driven by a levy noise particle paths follow a markov process with long jumps their transition probability density solves a forward equation derived here via pseudo differential operator theory and fourier analysis in particular the sde with stable driving noise has a space fractional advection dispersion equation fade with variable coefficients as the forward equation this result provides a stochastic solution to anomalous diffusion models and a solid mathematical foundation for particle tracking codes already in use for fractional advection equations

non markovian diffusion equations and processes analysis and simulations abstract in this paper we introduce and analyze a class of diffusion type equations related to certain non markovian stochastic processes we start from the forward drift equation which is made non local in time by the introduction of a suitable chosen memory kernel k t the resulting non markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l t we then consider the subordinated process y t x l t where x t is a markovian diffusion the corresponding time evolution of the marginal density function of y t is governed by a non markovian fokker planck equation which involves the memory kernel k t we develop several applications and derive the exact solutions we consider different stochastic models for the given equations providing path simulations

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

uncoupled continuous time random walk finite jump length probability density function in this work the uncoupled continuous time random walk model with the exponential jump length probability density function pdf is considered exact solutions for the pdf and second moment are obtained and investigated for different kinds of waiting time pdf

polymer translocation through a nanopore a showcase of anomalous diffusion the translocation dynamics of a polymer chain through a nanopore in the absence of an external driving force is analyzed by means of scaling arguments fractional calculus and computer simulations the problem at hand is mapped on a one dimensional anomalous diffusion process in terms of the reaction coordinate s i e the translocated number of segments at time t and shown to be governed by a universal exponent ensuremath alpha 2 2 ensuremath nu 2 ensuremath ensuremath gamma 1 where ensuremath nu is the flory exponent and ensuremath gamma 1 is the surface exponent remarkably it turns out that the value of ensuremath alpha is nearly the same in two and three dimensions the process is described by a fractional diffusion equation which is solved exactly in the interval 0lsln with appropriate boundary and initial conditions the solution gives the probability distribution of translocation times as well as the variation with time of the statistical moments s t and s 2 t ensuremath s t 2 which provide a full description of the diffusion process the comparison of the analytic results with data derived from extensive monte carlo simulations reveals very good agreement and proves that the diffusion dynamics of unbiased translocation through a nanopore is anomalous in its nature

fokker planck type equations associated with subordinated processes controlled by tempered α stable processes in this paper we introduce two models of stochastic processes driven by brownian motion and fractional brownian motion subordinated with tempered stable waiting times by using a new integro differential operator we obtain the generalized fokker planck type equations associated with these subordinated stochastic processes

representation of solutions and large time behavior for fully nonlocal diffusion equations abstract we study the cauchy problem for a nonlocal heat equation which is of fractional order both in space and time we prove four main theorems i a representation formula for classical solutions ii a quantitative decay rate at which the solution tends to the fundamental solution iii optimal l 2 decay of mild solutions in all dimensions iv l 2 decay of weak solutions via energy methods the first result relies on a delicate analysis of the definition of classical solutions after proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of ii next we use fourier analysis techniques to obtain the optimal decay rate for mild solutions here we encounter the critical dimension phenomenon where the decay rate attains the decay rate of that in a bounded domain for large enough dimensions consequently the decay rate does not anymore improve when the dimension increases the theory is markedly different from that of the standard caloric functions and this substantially complicates the analysis finally we use energy estimates and a comparison principle to prove a quantitative decay rate for weak solutions defined via a variational formulation our main idea is to show that the l 2 norm is actually a subsolution to a purely time fractional problem which allows us to use the known theory to obtain the result

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

stochastic calculus in physics the relationship of the ito stratonovich stochastic calculus to studies of weakly colored noise is explained a functional calculus approach is used to obtain an effective fokker planck equation for the weakly colored noise regime in a smooth limit this representation produces the stratonovich version of the ito stratonovich calculus for white noise it also provides an approach to steady state behavior for strongly colored noise numerical simulation algorithms are explored and a novel suggestion is made for efficient and accurate simulation of white noise equations

uncoupled continuous time random walks solution and limiting behavior of the master equation a detailed study is presented for a large class of uncoupled continuous time random walks ctrws the master equation is solved for the mittag leffler survival probability the properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed finally some common objections found in the literature are thoroughly reviewed

asymptotic solutions of decoupled continuous time random walks with superheavy tailed waiting time and heavy tailed jump length distributions we study the long time behavior of decoupled continuous time random walks characterized by superheavy tailed distributions of waiting times and symmetric heavy tailed distributions of jump lengths our main quantity of interest is the limiting probability density of the position of the walker multiplied by a scaling function of time we show that the probability density of the scaled walker position converges in the long time limit to a nondegenerate one only if the scaling function behaves in a certain way this function as well as the limiting probability density are determined in explicit form also we express the limiting probability density which has heavy tails in terms of the fox h function and find its behavior for small and large distances

a chebyshev pseudospectral method to solve the space time tempered fractional diffusion equation a chebyshev pseudospectral method to solve the space time tempered fractional diffusion equation

stretched gaussian asymptotics of the truncated levy flights for the diffusion in nonhomogeneous media abstract the levy jumping process defined in terms of the jumping size distribution and the waiting time distribution is considered the jumping rate depends on the process value the fractional diffusion equation which contains the variable diffusion coefficient is solved in the diffusion limit that solution resolves itself to the stretched gaussian when the order parameter 2 the truncation of the levy flights in the exponential and power law form is introduced and the corresponding random walk process is simulated by the monte carlo method the stretched gaussian tails are found in both cases the time which is needed to reach the limiting distribution strongly depends on the jumping rate parameter when the cutoff function falls slowly the tail of the distribution appears to be algebraic

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

approximate performance analysis for linear codes in superposition schemes over gaussian broadcast channels unequal error protection schemes obtained by means of two level superposition coding are considered their performance over gaussian broadcast channels gbcs is investigated with optimal maximum likelihood decoding as well as with a suboptimal decoding strategy based on interference cancellation we focus on gbcs without fading and assuming that linear codes are used we evaluate for both decoding strategies analytical approximations of the word error rate based on a suitable application of the union bound as in the case of turbo codes and turbo coded modulations in schemes without superposition the derivation of the approximations exploits the concept of uniform interleaving the analytical expressions obtained are in excellent agreement with the simulation results and thus provide a useful tool for analysis and design of practical superposition coding schemes unlike the existing design tools which rely on the assumption of infinite length superposition codes the proposed approach allows us to study the effectiveness of finite length coding schemes with known distance spectrum

fluid limit of the continuous time random walk with general levy jump distribution functions the continuous time random walk ctrw is a natural generalization of the brownian random walk that allows the incorporation of waiting time distributions psi t and general jump distribution functions eta x there are two well known fluid limits of this model in the uncoupled case for exponential decaying waiting times and gaussian jump distribution functions the fluid limit leads to the diffusion equation on the other hand for algebraic decaying waiting times psi approximately t 1 beta and algebraic decaying jump distributions eta approximately x 1 alpha corresponding to levy stable processes the fluid limit leads to the fractional diffusion equation of order alpha in space and order beta in time however these are two special cases of a wider class of models here we consider the ctrw for the most general levy stochastic processes in the levy khintchine representation for the jump distribution function and obtain an integrodifferential equation describing the dynamics in the fluid limit the resulting equation contains as special cases the regular and the fractional diffusion equations as an application we consider the case of ctrws with exponentially truncated levy jump distribution functions in this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory long jumps and truncation effects in the intermediate asymptotic regime the dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as tauc approximately lambda alpha beta where 1 lambda is the truncation length scale the asymptotic behavior of the propagator green s function of the truncated fractional equation exhibits a transition from algebraic decay for t tauc

martingale integrals over poissonian processes and the ito type equations with white shot noise the construction of the ito type stochastic integrals and differential equations for compound poisson processes is provided the general martingale and nonanticipating properties of the ordinary gaussian ito theory are conserved these properties appear particularly important if the stochastic description has to be proposed according to game theory or the linear relaxation or the exponential growth requirements in contrast to the ordinary ito theory the uncorrelated parametric fluctuation of a definite sign can be still modeled by asymmetric white shot noise so the general scope of applications is not restricted by the positivity requirements the possible use of the developed formalism in econophysics is addressed

limiting distributions of continuous time random walks with superheavy tailed waiting times we study the long time behavior of the scaled walker particle position associated with decoupled continuous time random walks which is characterized by superheavy tailed distribution of waiting times and asymmetric heavy tailed distribution of jump lengths both the scaling function and the corresponding limiting probability density are determined for all admissible values of tail indexes describing the jump distribution to analytically investigate the limiting density function we derive a number of different representations of this function and in this way establish its main properties we also develop an efficient numerical method for computing the limiting probability density and compare our analytical and numerical results

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

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

option pricing in subdiffusive bachelier model the earliest model of stock prices based on brownian diffusion is the bachelier model in this paper we propose an extension of the bachelier model which reflects the subdiffusive nature of the underlying asset dynamics the subdiffusive property is manifested by the random infinitely divisible periods of time during which the asset price does not change we introduce a subdiffusive arithmetic brownian motion as a model of stock prices with such characteristics the structure of this process agrees with two stage scenario underlying the anomalous diffusion mechanism in which trapping random events are superimposed on the langevin dynamics we find the corresponding fractional fokker planck equation governing the probability density function of the introduced process we construct the corresponding martingale measure and show that the model is incomplete we derive the formulas for european put and call option prices we describe explicit algorithms and present some monte carlo simulations for the particular cases of stable and tempered stable distributions of waiting times

moments for tempered fractional advection diffusion equations this paper develops moment formulas for exponentially tempered fractional advection diffusion equations tfades that transition from anomalous to asymptotic diffusion limits over time exact analytical expressions or series representations for spatial moments up to the fourth order are derived by integral transform or asymptotic expansion approach a fully lagrangian solver cross verified by an implicit eulerian approach is also developed to calculate numerically the complete evolution of moments for the tfades with complex initial and boundary conditions moment analysis identifies the diffusion equation that attracts the tempered anomalous diffusion in the long time limit fitting of moments measured at two end members of alluvial systems checks the applicability of moment analysis in understanding real diffusion

regional gradient controllability of sub diffusion processes abstract by sub diffusion systems we mean to replace the first order time derivative of normal diffusion system with a riemann liouville time fractional derivative of order 0 1 in this paper we first introduce a new definition of regional gradient controllability for the sub diffusion system with the control inputs appearing in the differential equations as distributed inputs which recovers the usual definition of regional gradient controllability as 1 the sufficient and necessary conditions on actuators to achieve the gradient controllability in a given subregion of the whole domain are presented an approach to guarantee the regional gradient controllability of the problems studied within the considered subregion using minimum energy control effort is presented several examples are presented in the end to illustrate the effectiveness of our results where zone actuators pointwise actuators or filament actuators are respectively considered

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

on fractional levy processes tempering sample path properties and stochastic integration we define two new classes of stochastic processes called tempered fractional levy process of the first and second kinds tflp and tflp ii respectively tflp and tflp ii make up very broad finite variance generally non gaussian families of transient anomalous diffusion models that are constructed by exponentially tempering the power law kernel in the moving average representation of a fractional levy process accordingly the increment processes of tflp and tflp ii display semi long range dependence we establish the sample path properties of tflp and tflp ii we further use a flexible framework of tempered fractional derivatives and integrals to develop the theory of stochastic integration with respect to tflp and tflp ii which may not be semimartingales depending on the value of the memory parameter and choice of marginal distribution

first passage time moments of asymmetric levy flights we investigate the first passage dynamics of symmetric and asymmetric l evy flights in a semi infinite and bounded intervals by solving the space fractional diffusion equation we analyse the fractional order moments of the first passage time probability density function for different values of the index of stability and the skewness parameter a comparison with results using the langevin approach to l evy flights is presented for the semi infinite domain in certain special cases analytic results are derived explicitly and in bounded intervals a general analytical expression for the mean first passage time of l evy flights with arbitrary skewness is presented these results are complemented with extensive numerical analyses

coupled continuous time random walks in finance continuous time random walks ctrws are used in physics to model anomalous diffusion by incorporating a random waiting time between particle jumps in finance the particle jumps are log returns and the waiting times measure delay between transactions these two random variables log return and waiting time are typically not independent for these coupled ctrw models we can now compute the limiting stochastic process just like brownian motion is the limit of a simple random walk even in the case of heavy tailed power law price jumps and or waiting times the probability density functions for this limit process solve fractional partial differential equations in some cases these equations can be explicitly solved to yield descriptions of long term price changes based on a high resolution model of individual trades that includes the statistical dependence between waiting times and the subsequent log returns in the heavy tailed case this involves operator stable space time random vectors that generalize the familiar stable models in this paper we will review the fundamental theory and present two applications with tick by tick stock and futures data

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

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

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

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

actuator characterisations to achieve approximate controllability for a class of fractional sub diffusion equations abstractthis paper is devoted to analysing the actuator characterisations for the fractional sub diffusion equation under consideration to become approximately controllable two different cases are considered where the control inputs emerge in the differential equation as distributed inputs and as boundary inputs in the boundary conditions the dual system for fractional sub diffusion equation is solved and the necessary and sufficient conditions for the approximate controllability of the system are established several examples are worked out in the end to illustrate our results

approximation of heavy tailed fractional pearson diffusions in skorokhod topology abstract continuous time random walks ctrws have random waiting times between particle jumps we establish fractional diffusion approximation via correlated ctrws instead of a random walk modeling particle jumps in the classical ctrw model we use discrete time markov chain with correlated steps the waiting times are selected from the domain of attraction of a stable law

on fractional tempered stable processes and their governing differential equations we derive the governing equation of the tempered stable subordinator hereafter tss which generalizes the space fractional differential equation satisfied by the law of the stable subordinator itself this equation is expressed in terms of the shifted fractional derivative of order 0 1 coinciding with the stability parameter we then generalize this equation by introducing a time fractional derivative of order 0 1 resp 1 1 and we prove that it is satisfied by the law of a tss time changed by the inverse of a stable subordinator resp by the stable subordinator itself the corresponding processes can therefore be called fractional ts processes finally we provide fractional extensions of the relativistic stable processes which we define as a brownian motion with a random time argument represented by independent fractional ts processes of order resp 1

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

anomalous diffusion in stochastic systems with nonhomogeneously distributed traps the stochastic motion in a nonhomogeneous medium with traps is studied and diffusion properties of that system are discussed the particle is subjected to a stochastic stimulation obeying a general levy stable statistics and experiences long rests due to nonhomogeneously distributed traps the memory is taken into account by subordination of that process to a random time then the subordination equation is position dependent the problem is approximated by a decoupling of the medium structure and memory and exactly solved for a power law position dependence of the memory in the case of the gaussian statistics the density distribution and moments are derived depending on geometry and memory parameters the system may reveal both the subdiffusion and enhanced diffusion the similar analysis is performed for the levy flights where the finiteness of the variance follows from a variable noise intensity near a boundary two diffusion regimes are found in the bulk and near the surface the anomalous diffusion exponent as a function of the system parameters is derived

tempered fractional calculus fractional derivatives and integrals are convolutions with a power law multiplying by an exponential factor leads to tempered fractional derivatives and integrals tempered fractional diffusion equations where the usual second derivative in space is replaced by a tempered fractional derivative govern the limits of random walk models with an exponentially tempered power law jump distribution the limiting tempered stable probability densities exhibit semi heavy tails which are commonly observed in finance tempered power law waiting times lead to tempered fractional time derivatives which have proven useful in geophysics the tempered fractional derivative or integral of a brownian motion called a tempered fractional brownian motion can exhibit semi long range dependence the increments of this process called tempered fractional gaussian noise provide a useful new stochastic model for wind speed data a tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations and it also provides a useful new correlation model in time series

stochastic calculus for uncoupled continuous time random walks the continuous time random walk ctrw is a pure jump stochastic process with several applications not only in physics but also in insurance finance and economics a definition is given for a class of stochastic integrals driven by a ctrw which includes the it and stratonovich cases an uncoupled ctrw with zero mean jumps is a martingale it is proved that as a consequence of the martingale transform theorem if the ctrw is a martingale the it integral is a martingale too it is shown how the definition of the stochastic integrals can be used to easily compute them by monte carlo simulation the relations between a ctrw its quadratic variation its stratonovich integral and its it integral are highlighted by numerical calculations when the jumps in space of the ctrw have a symmetric levy alpha stable distribution and its waiting times have a one parameter mittag leffler distribution remarkably these distributions have fat tails and an unbounded quadratic variation in the diffusive limit of vanishing scale parameters the probability density of this kind of ctrw satisfies the space time fractional diffusion equation fde or more in general the fractional fokker planck equation which generalizes the standard diffusion equation solved by the probability density of the wiener process and thus provides a phenomenologic model of anomalous diffusion we also provide an analytic expression for the quadratic variation of the stochastic process described by the fde and check it by monte carlo

random walks with infinite spatial and temporal moments the continuous time random walk of montroll and weiss has been modified by scher and lax to include a coupled spatial temporal memory we treat novel cases for the random walk and the corresponding generalized master equation when combinations of both spatial and temporal moments of the memory are infinite the asymptotic properties of the probability distribution for being at any lattice site as a function of time and its variance are calculated the resulting behavior includes localized diffusive wavelike and levy s stable laws for the appropriate scaled variable we show that an infinite mean waiting time can lead to long time diffusive behavior while a finite mean waiting time is not sufficient to ensure the same

stochastic algorithms for discontinuous multiplicative white noise stochastic differential equations with multiplicative noise need a mathematical prescription due to different interpretations of the stochastic integral this fact implies specific algorithms to perform numerical integrations or simulations of the stochastic trajectories moreover if the multiplicative noise function is not continuous then the standard algorithms cannot be used we present an explicit algorithm to avoid this problem and we apply it to a well controlled example finally we discuss on the existence of higher order algorithms for this specific situation

numerical schemes of the time tempered fractional feynmankac equation this paper focuses on providing the computation methods for the backward time tempered fractional feynmankac equation being one of the models recently proposed in wu etal 2016 the discretization for the tempered fractional substantial derivative is derived and the corresponding finite difference and finite element schemes are designed with well established stability and convergence the performed numerical experiments show the effectiveness of the presented schemes

extremal behavior of a coupled continuous time random walk abstract coupled continuous time random walks ctrws model normal and anomalous diffusion of random walkers by taking the sum of random jump lengths dependent on the random waiting times immediately preceding each jump they are used to simulate diffusion like processes in econophysics such as stock market fluctuations where jumps represent financial market microstructure like log returns in this and many other applications the magnitude of the largest observations e g a stock market crash is of considerable importance in quantifying risk we use a stochastic process called a coupled continuous time random maxima ctrm to determine the density governing the maximum jump length of a particle undergoing a ctrw ctrm are similar to continuous time random walks but track maxima instead of sums the many ways in which observations can depend on waiting times can produce an equally large number of ctrm governing density shapes we compare densities governing coupled ctrm with their uncoupled counterparts for three simple observation wait dependence structures

high order numerical approximation formulas for riemann liouville riesz tempered fractional derivatives construction and application i abstract in this paper we develop a new numerical algorithm for solving the riesz tempered space fractional diffusion equation the stability and convergence of the numerical scheme are discussed via the technique of matrix analysis finally numerical experiments are performed to confirm the effectiveness of our numerical algorithm

a hereditarily indecomposable mathcal l _ infty space that solves the scalar plus compact problem we construct a hereditarily indecomposable banach space with dual space isomorphic to l1 every bounded linear operator on this space is expressible as i k with a scalar and k compact

generalized master equations for random walks with time dependent jump sizes in this work we develop a unified generalized master equation gme framework that extends the theory of continuous time random walks to include the cases when the jump sizes may have a delayed dependence on time and are not restricted to any particular class of distributions we compare and contrast analytical and numerical behavior of the corresponding master equations including the instantaneous vs delayed jump dependence on time and exponential vs mittag leffler interarrival times with the latter leading to fractional evolution equation we provide existence and uniqueness proofs for the resulting gmes

monte carlo simulation of uncoupled continuous time random walks yielding a stochastic solution of the space time fractional diffusion equation we present a numerical method for the monte carlo simulation of uncoupled continuous time random walks with a levy stable distribution of jumps in space and a mittag leffler distribution of waiting times and apply it to the stochastic solution of the cauchy problem for a partial differential equation with fractional derivatives both in space and in time the one parameter mittag leffler function is the natural survival probability leading to time fractional diffusion equations transformation methods for mittag leffler random variables were found later than the well known transformation method by chambers mallows and stuck for levy stable random variables and so far have not received as much attention nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the mittag leffler distribution combining the two methods we obtain an accurate approximation of space and time fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes

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

design of low density parity check codes for modulation and detection a coding and modulation technique is studied where the coded bits of an irregular low density parity check ldpc code are passed directly to a modulator at the receiver the variable nodes of the ldpc decoder graph are connected to detector nodes and iterative decoding is accomplished by viewing the variable and detector nodes as one decoder the code is optimized by performing a curve fitting on extrinsic information transfer charts design examples are given for additive white gaussian noise channels as well as multiple input multiple output mimo fading channels where the receiver but not the transmitter knows the channel for the mimo channels the technique operates within 1 25 db of capacity for various antenna configurations and thereby outperforms a scheme employing a parallel concatenated turbo code by wide margins when there are more transmit than receive antennas

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

first passage and first exit times of a bessel like stochastic process we study a stochastic process x t which is a particular case of the rayleigh process and whose square is the bessel process with various applications in physics chemistry biology economics finance and other fields the stochastic differential equation is dx t nd x t dt 2d dw t where w t is the wiener process the drift term can arise from a logarithmic potential or from taking x t as the norm of a multidimensional random walk due to the singularity of the drift term for x t 0 different natures of boundary at the origin arise depending on the real parameter n entrance exit and regular for each of them we calculate analytically and numerically the probability density functions of first passage times or first exit times nontrivial behavior is observed in the case of a regular boundary