exploitation of pairwise class distances for ordinal classification ordinal classification refers to classification problems in which the classes have a natural order imposed on them because of the nature of the concept studied some ordinal classification approaches perform a projection from the input space to one dimensional latent space that is partitioned into a sequence of intervals one for each class class identity of a novel input pattern is then decided based on the interval its projection falls into this projection is trained only indirectly as part of the overall model fitting as with any other latent model fitting direct construction hints one may have about the desired form of the latent model can prove very useful for obtaining high quality models the key idea of this letter is to construct such a projection model directly using insights about the class distribution obtained from pairwise distance calculations the proposed approach is extensively evaluated with 8 nominal and ordinal classifiers methods 10 real world ordinal classification data sets and 4 different performance measures the new methodology obtained the best results in average ranking when considering three of the performance metrics although significant differences are found for only some of the methods also after observing other methods of internal behavior in the latent space we conclude that the internal projections do not fully reflect the intraclass behavior of the patterns our method is intrinsically simple intuitive and easily understandable yet highly competitive with state of the art approaches to ordinal classification

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

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

mathematical modeling of different types of instabilities in time fractional reaction diffusion systems abstract in this article we investigate possible scenarios of pattern formations in reaction diffusion systems with time fractional derivatives linear stability analysis is performed for different values of derivative orders results of qualitative analysis are confirmed by numerical simulations of specific partial differential equations most attention is paid to two models a fractional order reaction diffusion system with bonhoeffer van der pol kinetics and to the brusselator model

adaptive metric learning vector quantization for ordinal classification many pattern analysis problems require classification of examples into naturally ordered classes in such cases nominal classification schemes will ignore the class order relationships which can have a detrimental effect on classification accuracy this article introduces two novel ordinal learning vector quantization lvq schemes with metric learning specifically designed for classifying data items into ordered classes in ordinal lvq unlike in nominal lvq the class order information is used during training in selecting the class prototypes to be adapted as well as in determining the exact manner in which the prototypes get updated prototype based models in general are more amenable to interpretations and can often be constructed at a smaller computational cost than alternative nonlinear classification models experiments demonstrate that the proposed ordinal lvq formulations compare favorably with their nominal counterparts moreover our methods achieve competitive performance against existing benchmark ordinal regression models

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

pattern formation on networks with reactions a continuous time random walk approach we derive the generalized master equation for reaction diffusion on networks from an underlying stochastic process the continuous time random walk ctrw the nontrivial incorporation of the reaction process into the ctrw is achieved by splitting the derivation into two stages the reactions are treated as birth death processes and the first stage of the derivation is at the single particle level taking into account the death process while the second stage considers an ensemble of these particles including the birth process using this model we have investigated different types of pattern formation across the vertices on a range of networks importantly the ctrw defines the laplacian operator on the network in a non ad hoc manner and the pattern formation depends on the structure of this laplacian here we focus attention on ctrws with exponential waiting times for two cases one in which the rate parameter is constant for all vertices and the other where the rate parameter is proportional to the vertex degree this results in nonsymmetric and symmetric ctrw laplacians respectively in the case of symmetric laplacians pattern formation follows from the turing instability however in nonsymmetric laplacians pattern formation may be possible with or without a turing instability

scale free brain functional networks functional magnetic resonance imaging is used to extract functional networks connecting correlated human brain sites analysis of the resulting networks in different tasks shows that a the distribution of functional connections and the probability of finding a link versus distance are both scale free b the characteristic path length is small and comparable with those of equivalent random networks and c the clustering coefficient is orders of magnitude larger than those of equivalent random networks all these properties typical of scale free small world networks reflect important functional information about brain states

power law ansatz in complex systems excessive loss of information the ubiquity of power law relations in empirical data displays physicists love of simple laws and uncovering common causes among seemingly unrelated phenomena however many reported power laws lack statistical support and mechanistic backings not to mention discrepancies with real data are often explained away as corrections due to finite size or other variables we propose a simple experiment and rigorous statistical procedures to look into these issues making use of the fact that the occurrence rate and pulse intensity of crumple sound obey a power law with an exponent that varies with material we simulate a complex system with two driving mechanisms by crumpling two different sheets together the probability function of the crumple sound is found to transit from two power law terms to a bona fide power law as compaction increases in addition to showing the vicinity of these two distributions in the phase space this observation nicely demonstrates the effect of interactions to bring about a subtle change in macroscopic behavior and more information may be retrieved if the data are subject to sorting our analyses are based on the akaike information criterion that is a direct measurement of information loss and emphasizes the need to strike a balance between model simplicity and goodness of fit as a show of force the akaike information criterion also found the gutenberg richter law for earthquakes and the scale free model for a brain functional network a two dimensional sandpile and solar flare intensity to suffer an excessive loss of information they resemble more the crumpled together ball at low compactions in that there appear to be two driving mechanisms that take turns occurring

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

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

the benefits of modeling slack variables in svms in this letter we explore the idea of modeling slack variables in support vector machine svm approaches the study is motivated by svm which models the slacks through a smooth correcting function that is determined by additional privileged information about the training examples not available in the test phase we take a closer look at the meaning and consequences of smooth modeling of slacks as opposed to determining them in an unconstrained manner through the svm optimization program to better understand this difference we only allow the determination and modeling of slack values on the same information that is using the same training input in the original input space we also explore whether it is possible to improve classification performance by combining in a convex combination the original svm slacks with the modeled ones we show experimentally that this approach not only leads to improved generalization performance but also yields more compact lower complexity models finally we extend this idea to the context of ordinal regression where a natural order among the classes exists the experimental results confirm principal findings from the binary case

learning to classify ordinal data the data replication method classification of ordinal data is one of the most important tasks of relation learning this paper introduces a new machine learning paradigm specifically intended for classification problems where the classes have a natural order the technique reduces the problem of classifying ordered classes to the standard two class problem the introduced method is then mapped into support vector machines and neural networks generalization bounds of the proposed ordinal classifier are also provided an experimental study with artificial and real data sets including an application to gene expression analysis verifies the usefulness of the proposed approach

regression and time series model selection in small samples summary a bias correction to the akaike information criterion aic is derived for regression and autoregressive time series models the correction is of particular use when the sample size is small or when the number of fitted parameters is a moderate to large fraction of the sample size the corrected method called aicc is asymptotically efficient if the true model is infinite dimensional furthermore when the true model is of finite dimension aicc is found to provide better model order choices than any other asymptotically efficient method applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed

continuous time random walks with internal dynamics and subdiffusive reaction diffusion equations humboldt university newtonstr 15 d 12489 berlin germany dated september 8 2008 we formulate the generalized master equation for a class of continuous time random walks in thepresence of a prescribed deterministic evolution between successive transitions this formulation isexempli ed by means of an advection di usion and a jump di usion scheme based on this masterequation we also derive reaction di usion equations for subdi usive chemical species using a mean eld approximation

spatial gliding temporal trapping and anomalous transport abstract we introduce and study a model of stochastic dynamics governed by markovian laws of motion in a phase space filled with random traps the resulting non markovian stochastic trapped motion alternates intermittently between periods of spatial gliding where the motion glides the underlying markovian dynamics and periods of temporal trapping where the motion is halted in random traps we investigate the asymptotics and scaling limits of this model we prove that when the random trappings are heavy tailed then mittag leffler functions and probability laws emerge and govern the functional structure and statistics of the system and that the time flow has a random fractal structure whose fractal exponent dimension is determined by the heaviness of the trappings we study the effect of random trapping on general levy dynamics we prove that subjecting levy dynamics to heavy tailed trapping will always result in i sub diffusive behavior when the underlying levy dynamics are of finite variance and ii space time fractal behavior when the underlying levy dynamics are scale invariant furthermore we explore the issue of first exit times to that end a general feynman kac framework for trapped processes is developed and a method of transforming trapped feynman kac equations to standard ones is established the study of first exit times enables us to quantitatively connect macroscopic observations to microscopic behavior in general markovian dynamics subjected to random trapping in the case of levy dynamics first exit times from balls are computed and the relationships between their statistics and the statistics of the trapped motion are derived

ordinal neural networks without iterative tuning ordinal regression or is an important branch of supervised learning in between the multiclass classification and regression in this paper the traditional classification scheme of neural network is adapted to learn ordinal ranks the model proposed imposes monotonicity constraints on the weights connecting the hidden layer with the output layer to do so the weights are transcribed using padding variables this reformulation leads to the so called inequality constrained least squares icls problem its numerical solution can be obtained by several iterative methods for example trust region or line search algorithms in this proposal the optimum is determined analytically according to the closed form solution of the icls problem estimated from the karush kuhn tucker conditions furthermore following the guidelines of the extreme learning machine framework the weights connecting the input and the hidden layers are randomly generated so the final model estimates all its parameters without iterative tuning the model proposed achieves competitive performance compared with the state of the art neural networks methods for or

a fast algorithm for learning a ranking function from large scale data sets we consider the problem of learning a ranking function that maximizes a generalization of the wilcoxon mann whitney statistic on the training data relying on an e accurate approximation for the error function we reduce the computational complexity of each iteration of a conjugate gradient algorithm for learning ranking functions from o m2 to o m where m is the number of training samples experiments on public benchmarks for ordinal regression and collaborative filtering indicate that the proposed algorithm is as accurate as the best available methods in terms of ranking accuracy when the algorithms are trained on the same data however since it is several orders of magnitude faster than the current state of the art approaches it is able to leverage much larger training data sets

memory controlled diffusion memory effects require for their incorporation into random walk models an extension of the conventional equations the linear fokker planck equation for the probability density p stackrel p vec r t is generalized by including nonlinear and nonlocal spatial temporal memory effects the realization of the memory kernel is restricted due the conservation of the basic quantity p a general criteria is given for the existence of stationary solutions in case the memory kernel depends on p polynomially transport may be prevented owing to the delay effects a finite amount of particles remains localized and the further transport is terminated for diffusion with nonlinear memory effects we find an exact solution in the long time limit although the mean square displacement exhibits diffusive behavior higher order cumulants offer differences to diffusion and they depend on the memory strength

overshooting and undershooting subordination scenario for fractional two power law relaxation responses in this paper we propose a transparent subordination approach to anomalous diffusion processes underlying the nonexponential relaxation we investigate properties of a coupled continuous time random walk that follows from modeling the occurrence of jumps with compound counting processes as a result two different diffusion processes corresponding to over and undershooting operational times respectively have been found we show that within the proposed framework all empirical two power law relaxation patterns may be derived this work is motivated by the so called less typical relaxation behavior observed e g for gallium doped cd0 99mn0 01te mixed crystals

reaction subdiffusion equations for the a b reaction we consider a linear reversible isomerization reaction a b under subdiffusion described by continuous time random walks ctrw the reactants transformations take place independently on the motion and are described by constant rates we show that the form of the ensuing system of mesoscopic reaction subdiffusion equations is unusual the equation for time derivative of say a x t contains the terms depending not only on deltaa but also on deltab this mirrors the fact that in subdiffusion the flux of particles at time t is defined by the distributions of the particles concentrations at all previous times since the particles which jump as a at time t could previously be both a or b this flux depends on both a and b concentrations

gaussian processes for ordinal regression we present a probabilistic kernel approach to ordinal regression based on gaussian processes a threshold model that generalizes the probit function is used as the likelihood function for ordinal variables two inference techniques based on the laplace approximation and the expectation propagation algorithm respectively are derived for hyperparameter learning and model selection we compare these two gaussian process approaches with a previous ordinal regression method based on support vector machines on some benchmark and real world data sets including applications of ordinal regression to collaborative filtering and gene expression analysis experimental results on these data sets verify the usefulness of our approach

hopping models of charge transfer in a complex environment coupled memory continuous time random walk approach charge transport processes in disordered complex media are accompanied by anomalously slow relaxation for which usually a broad distribution of relaxation times is adopted to account for those properties of the environment a standard kinetic approach in description of the system is addressed either in the framework of continuous time random walks ctrws or fractional diffusion in this paper the power of the ctrw approach is illustrated by use of the probabilistic formalism and limit theorems that allow one to rigorously predict the limiting distributions of the paths traversed by charges and to derive effective relaxation properties of the entire system of interest in particular the standard ctrw scenario is generalized to a new class of coupled memory ctrws that effectivly can lead to the well known havriliak negami response application of the method is discussed for nonexponential electron transfer processes controlled by dynamics of the surrounding medium

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

online ranking by projecting we discuss the problem of ranking instances in our framework each instance is associated with a rank or a rating which is an integer in 1 to k our goal is to find a rank prediction rule that assigns each instance a rank that is as close as possible to the instance s true rank we discuss a group of closely related online algorithms analyze their performance in the mistake bound model and prove their correctness we describe two sets of experiments with synthetic data and with the eachmovie data set for collaborative filtering in the experiments we performed our algorithms outperform online algorithms for regression and classification applied to ranking

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

generalized mittag leffler relaxation clustering jump continuous time random walk approach a stochastic generalization of renormalization group transformation for continuous time random walk processes is proposed the renormalization consists in replacing the jump events from a randomly sized cluster by a single renormalized i e overall jump the clustering of the jumps followed by the corresponding transformation of the interjump time intervals yields a new class of coupled continuous time random walks which applied to modeling of relaxation lead to the general power law properties usually fitted with the empirical havriliak negami function

black scholes formula in subdiffusive regime in the classical approach the price of an asset is described by the celebrated black scholes model in this paper we consider a generalization of this model which captures the subdiffusive characteristics of financial markets we introduce a subdiffusive geometric brownian motion as a model of asset prices exhibiting subdiffusive dynamics we find the corresponding fractional fokker planck equation governing the dynamics of the probability density function of the introduced process we prove that the considered model is arbitrage free and incomplete we find the corresponding subdiffusive black scholes formula for the fair prices of european options and show how these prices can be evaluated using monte carlo methods we compare the obtained results with the classical ones

on the first passage of one sided levy motions abstract we study the first passage problem for one sided levy motions how does such a motion started at the origin cross a barrier positioned at the point x x 0 since one sided levy motions are pure jump processes they always leap over barriers rather than crossing them continuously we hence explore the following issues i first passage times fpts how long would it take the motion to cross the barrier ii first passage leapovers fpls how far would the motion leap over the barrier and iii what is dependence between the fpts and the fpls formulae for the joint laplace transforms of the fpts and fpls are derived and the scaling limits of the fpts and fpls are computed the scaling limits turn out to display dramatically different behavior when i the underlying levy motion has finite cumulants and when ii the underlying levy motion is heavy tailed special attention is devoted to the investigation of the first passage problem for self similar one sided levy motions

cattaneo type subdiffusion reaction equation subdiffusion in a system in which mobile particles a can chemically react with static particles b according to the rule a b b is considered within a persistent random walk model this model which assumes a correlation between successive steps of particles provides hyperbolic cattaneo normal diffusion or fractional subdiffusion equations starting with the difference equation which describes a persistent random walk in a system with chemical reactions using the generating function method and the continuous time random walk formalism we will derive the cattaneo type subdiffusion differential equation with fractional time derivatives in which the chemical reactions mentioned above are taken into account we will also find its solution over a long time limit based on the obtained results we will find the cattaneo type subdiffusion reaction equation in the case in which mobile particles of species a and b can chemically react according to a more complicated rule

non linear shot noise levy noah joseph abstract we introduce and study a generic non linear shot noise system model shots of random magnitudes arrive to the system stochastically following an arbitrary time homogeneous poisson point process after hitting the system the magnitude of an arriving shot decays to zero the decay is governed by an arbitrary differential equation dynamics shots are independent and their overall effect on the system is additive the system s noise level at time t equals the sum of the magnitudes at time t of all the shots arriving to the system prior to time t the resulting shot noise is i a levy process when the decay dynamics are degenerate ii a levy driven ornstein uhlenbeck process when the decay dynamics are linear and iii a stationary non markov process when the decay dynamics are non linear the resulting shot noise admits an underlying levy structure which we explicitly compute and can yield both the noah effect and the joseph effect closed form analytic formulae for various statistics are derived including the log laplace transform and cumulants of the stationary noise level the process auto covariance function and the process range of dependence

support vector ordinal regression in this letter we propose two new support vector approaches for ordinal regression which optimize multiple thresholds to define parallel discriminant hyperplanes for the ordinal scales both approaches guarantee that the thresholds are properly ordered at the optimal solution the size of these optimization problems is linear in the number of training samples the sequential minimal optimization algorithm is adapted for the resulting optimization problems it is extremely easy to implement and scales efficiently as a quadratic function of the number of examples the results of numerical experiments on some benchmark and real world data sets including applications of ordinal regression to information retrieval verify the usefulness of these approaches

reduction from cost sensitive ordinal ranking to weighted binary classification we present a reduction framework from ordinal ranking to binary classification the framework consists of three steps extracting extended examples from the original examples learning a binary classifier on the extended examples with any binary classification algorithm and constructing a ranker from the binary classifier based on the framework we show that a weighted 0 1 loss of the binary classifier upper bounds the mislabeling cost of the ranker both error wise and regret wise our framework allows not only the design of good ordinal ranking algorithms based on well tuned binary classification approaches but also the derivation of new generalization bounds for ordinal ranking from known bounds for binary classification in addition our framework unifies many existing ordinal ranking algorithms such as perceptron ranking and support vector ordinal regression when compared empirically on benchmark data sets some of our newly designed algorithms enjoy advantages in terms of both training speed and generalization performance over existing algorithms in addition the newly designed algorithms lead to better cost sensitive ordinal ranking performance as well as improved listwise ranking performance

subordination to periodic processes and synchronization abstract we study the subordination to a process that is periodic in the natural time scale and equivalent to a clock with n states the rationale for this investigation is given by a set of many interacting clocks with n states the natural time scale representation corresponds to the dynamics of an individual clock with no interaction with the other clocks of this set we argue that the cooperation among the clocks of this set has the effect of generating a global clock whose times of sojourn in each of its n states are described by a distribution density with an inverse power law form and power index 2 this is equivalent to extending the widely used subordination method from fluctuation dissipation processes to periodic processes thereby raising the question of whether special conditions exist of perfect synchronization signaled by regular oscillations and especially by oscillations with no damping we study first the case of a poisson subordination function we show that in spite of the random nature of the subordination method the procedure has the effect of creating damped oscillations whose damping vanishes in the limiting case of n 1 thereby suggesting a condition of perfect synchronization in this limit the bateman s mathematical arguments h bateman higher transcendental functions vol iii robert k krieger publishing company inc krim fr drive malabar fl copyright 1953 by mcgraw hill book company inc indicate that the condition of perfect synchronization is possible also in the non poisson case with 2 although it may lie beyond the range of computer simulation to make the theoretical predictions accessible to numerical simulation we use a subordination function whose survival probability is a mittag leffler exponential function this method prevents us from directly establishing the macroscopic coherence emerging from 2 which generates a perfect form of 1 f noise however it affords indirect evidence that perfect synchronization signaled by undamped regular oscillations may be produced in this case furthermore we explore a condition characterized by an excellent agreement between theory and numerical simulation where the long time region relaxation with a perfect inverse power law decay emerging from the subordination to ordinary fluctuation dissipation processes is replaced by exponentially damped regular oscillations

different microscopic interpretations of the reaction telegrapher equation in this paper we provide some new insights into the microscopic interpretation of the telegrapher s and the reaction telegrapher equations we use the framework of continuous time random walks to derive the telegrapher s equation from two different perspectives reported before the kinetic derivation kd and the delayed random walk derivation drwd we analyze the similarities and the differences between both derivations paying special attention to the case when a reaction process is also present in the system as a result we are able to show that the equivalence between the kd and the drwd can break down when transport and reaction are coupled processes also this analysis allows us to elaborate on the specific role of relaxation effects in reaction diffusion processes

non markovian levy diffusion in nonhomogeneous media we study the diffusion equation with a position dependent power law diffusion coefficient the equation possesses the riesz weyl fractional operator and includes a memory kernel it is solved in the diffusion limit of small wave numbers two kernels are considered in detail the exponential kernel for which the problem resolves itself to the telegrapher s equation and the power law one the resulting distributions have the form of the l evy process for any kernel the renormalized fractional moment is introduced to compare different cases with respect to the diffusion properties of the system

exact solution of a generalized kramers fokker planck equation retaining retardation effects in order to describe non gaussian kinetics in weakly damped systems the concept of continuous time random walks is extended to particles with finite inertia one thus obtains a generalized kramers fokker planck equation which retains retardation effects i e nonlocal couplings in time and space it is shown that despite this complexity exact solutions of this equation can be given in terms of superpositions of gaussian distributions with varying variances in particular the long time behavior of the respective low order moments is calculated

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

fokker planck equation and subdiffusive fractional fokker planck equation of bistable systems with sinks abstract we study the dynamics of stochastic systems obeying the one dimensional diffusive fokker planck equation fpe as well as systems described by the subdiffusive fractional fokker planck equation sffpe with a confining potential u x and in the presence of delta function sinks for the one sink and two sink problems we obtain exact expressions for the laplace transform of the propagator p and derive several asymptotic results for the survival probability s p and the survival time distribution f the decay rate constants of the diffusive system are also analyzed we apply our method to a bistable system with a quartic double well potential and calculate p s p and f of the corresponding fpe and sffpe for different strengths and positions of the sink s finally we derive asymptotic expressions of s p and f for the subdiffusive system which are valid for a general u x and a general sink distribution

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