on a random walk with memory and its relation with markovian processes we study a one dimensional random walk with memory in which the step lengths to the left and to the right evolve at each step in order to reduce the wandering of the walker the feedback is quite efficient and leads to a non diffusive walk the time evolution of the displacement is given by an equivalent markovian dynamical process the probability density for the position of the walker is the same at any time as for a random walk with shrinking steps although the two time correlation functions are quite different

random walk in chemical space of cantor dust as a paradigm of superdiffusion we point out that the chemical space of a totally disconnected cantor dust k n ensuremath subset e n is a compact metric space c n with the spectral dimension d s d ensuremath ell ngd where d and d ensuremath ell n are the fractal and chemical dimensions of k n respectively hence we can define a random walk in the chemical space as a markovian gaussian process the mapping of a random walk in c n into k n ensuremath subset e n defines the quenched l evy flight on the cantor dust with a single step duration independent of the step length the equations describing the superdiffusion and diffusion reaction front propagation ruled by the local quenched l evy flight on k n ensuremath subset e n are derived the use of these equations to model superdiffusive phenomena observed in some physical systems in which propagators decay faster than algebraically is discussed

binary n step markov chains and long range correlated systems a theory of systems with long range correlations based on the consideration of binary n step markov chains is developed in our model the conditional probability that the i th symbol in the chain equals zero or unity is a linear function of the number of unities among the preceding n symbols the correlation and distribution functions as well as the variance of number of symbols in the words of arbitrary length l are obtained analytically and numerically if the persistent correlations are not extremely strong the variance is shown to be nonlinearly dependent on l a self similarity of the studied stochastic process is revealed the applicability of the developed theory to the coarse grained written and dna texts is discussed

memory effects in a nonequilibrium growth model we study memory effects in a kinetic roughening model for d 1 a different dynamic scaling is uncovered in the memory dominated phases the kardar parisi zhang scaling is restored in the absence of noise dc 2 represents the critical dimension where memory is shown to smoothen the roughening front a 0 studies on a discrete atomistic model in the same universality class reconfirm the analytical results in the large time limit while a different scaling behavior shows up for t t with t being the memory characteristic of the atomistic model results can be generalized for other nonconservative systems

variational bayesian strategies for high dimensional stochastic design problems this paper is concerned with a lesser studied problem in the context of model based uncertainty quantification uq that of optimization design control under uncertainty the solution of such problems is hindered not only by the usual difficulties encountered in uq tasks e g the high computational cost of each forward simulation the large number of random variables but also by the need to solve a nonlinear optimization problem involving large numbers of design variables and potentially constraints we propose a framework that is suitable for a class of such problems and is based on the idea of recasting them as probabilistic inference tasks to that end we propose a variational bayesian vb formulation and an iterative vb expectation maximization scheme that is capable of identifying a local maximum as well as a low dimensional set of directions in the design space along which the objective exhibits the largest sensitivity we demonstrate the validity of the proposed approach in the context of two numerical examples involving thousands of random and design variables in all cases considered the cost of the computations in terms of calls to the forward model was of the order of 100 or less the accuracy of the approximations provided is assessed by information theoretic metrics

recurrence time correlations in random walks with preferential relocation to visited places random walks with memory usually involve rules where a preference for either revisiting or avoiding those sites visited in the past are introduced somehow such effects have a direct consequence on the transport properties as well as on the statistics of first passage and subsequent recurrence times through a site a preference for revisiting sites is thus expected to result in a positive correlation between consecutive recurrence times here we derive a continuous time generalization of the random walk model with preferential relocation to visited sites proposed in phys rev lett 112 240601 2014 prltao0031 900710 1103 physrevlett 112 240601 to explore this effect together with the main transport properties induced by the long range memory despite the long range memory effects governing the process our analytical treatment allows us to i observe the existence of an asymptotic logarithmic ultraslow growth for the mean square displacement in accordance to the results found for the original discrete time model and ii confirm the existence of positive correlations between first passage and subsequent recurrence times this analysis is completed with a comprehensive numerical study which reveals among other results that these correlations between first passage and recurrence times also exhibit clear signatures of this ultraslow relaxation dynamics

discrete stochastic evolution rules and continuum evolution equations abstract the continuum evolution equations are derived from updating rules for three classes of stochastic models the first class corresponds to models whose stochastic continuum equations are of the langevin type obtained after carrying out a local average known as coarse graining the second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations this average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables in which case a hierarchy of ordinary differential equations is obtained the third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice this double average generates a hierarchy of deterministic ordinary differential equations the second and third classes of equations are truncated using a mean field m n closure approximation in order to obtain a finite set of equations illustrative examples of every class are given

fractional brownian motion and motion governed by the fractional langevin equation in confined geometries motivated by subdiffusive motion of biomolecules observed in living cells we study the stochastic properties of a non brownian particle whose motion is governed by either fractional brownian motion or the fractional langevin equation and restricted to a finite domain we investigate by analytic calculations and simulations how time averaged observables e g the time averaged mean squared displacement and displacement correlation are affected by spatial confinement and dimensionality in particular we study the degree of weak ergodicity breaking and scatter between different single trajectories for this confined motion in the subdiffusive domain the general trend is that deviations from ergodicity are decreased with decreasing size of the movement volume and with increasing dimensionality we define the displacement correlation function and find that this quantity shows distinct features for fractional brownian motion fractional langevin equation and continuous time subdiffusion such that it appears an efficient measure to distinguish these different processes based on single particle trajectory data

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

limit theorems for the laziest minimal random walk model of elephant type we consider a minimal model of one dimensional discrete time random walk with step reinforcement introduced by harbola kumar and lindenberg 2014 the walker can move forward never backward or remain at rest for each n 1 2 ldots a random time u n between 1 and n is chosen uniformly and if the walker moved forward resp remained at rest at time u n then at time n 1 it can move forward with probability p resp q or with probability 1 p resp 1 q it remains at its present position for the case q 0 several limit theorems are obtained by coletti gava and de lima 2019 in this paper we prove limit theorems for the case q 0 where the walker can exhibit all three forms of asymptotic behavior as p is varied as a byproduct we obtain limit theorems for the cluster size of the root in percolation on uniform random recursive trees

fermionic path integration and grassmann brownian motion a grassmann probability theory with anticommuting random variables and stochastic processes is developed using an extension of berezin integration to infinite dimensional spaces a kolmogorov type consistency condition allows integration on spaces of paths in anticommuting space one particular stochastic process grassmann brownian motion is described and the associated measure used to give a path integral formula for the kernel of the evolution operator in fermionic quantum mechanics the fourier mode expansion of grassmann brownian motion is derived

analytical results for long time behavior in anomalous diffusion we investigate through a generalized langevin formalism the phenomenon of anomalous diffusion for asymptotic times and we generalized the concept of the diffusion exponent a method is proposed to obtain the diffusion coefficient analytically through the introduction of a time scaling factor ensuremath lambda we obtain as well an exact expression for ensuremath lambda for all kinds of diffusion moreover we show that ensuremath lambda is a universal parameter determined by the diffusion exponent the results are then compared with numerical calculations and very good agreement is observed the method is general and may be applied to many types of stochastic problem

elephant random walks and their connection to polya type urns in this paper we explain the connection between the elephant random walk erw and an urn model a la polya and derive functional limit theorems for the former the erw model was introduced in phys rev e 70 045101 2004 10 1103 physreve 70 045101 to study memory effects in a highly non markovian setting more specifically the erw is a one dimensional discrete time random walk with a complete memory of its past the influence of the memory is measured in terms of a memory parameter p between zero and one in the past years a considerable effort has been undertaken to understand the large scale behavior of the erw depending on the choice of p here we use known results on urns to explicitly solve the erw in all memory regimes the method works as well for erws in higher dimensions and is widely applicable to related models

a generalization of euler s hypergeometric transformation euler s transformation formula for the gauss hypergeometric function 2 f 1 is extended to hypergeometric functions of higher order unusually the generalized transformation constrains the hypergeometric function parameters algebraically but not linearly its consequences for hypergeometric summation are explored it has as a corollary a summation formula of slater from this formula new one term evaluations of 2 f 1 1 and 3 f 2 1 are derived by applying transformations in the thomae group their parameters are also constrained nonlinearly several new one term evaluations of 2 f 1 1 with linearly constrained parameters are derived as well

reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation we present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations our approach is based on empirical interpolation of nonlinear differential operators and their frechet derivatives efficient offline online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals an a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically we introduce a new algorithm the podei greedy algorithm which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way the approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations we show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions in case of symmetries of the problem the approach realizes an automatic and intuitive space compression or even space dimensionality reduction we perform empirical investigations of the error convergence and run times in all cases we obtain a good run time acceleration

the diffusion of opposite opinions in a randomly biased environment we propose a model for diffusion of two opposite opinions here the decision to be taken by each individual is a random variable which depends on the tendency of the population as well as on its own trend characteristic the influence of the population trend can be positive negative or nonexistent in a random form we prove a phase transition in the behavior of the proportion of each opinion specifically the mean square proportions are linear functions of time in the diffusive case but are given by a power law in the superdiffusive regime

anomalous diffusion in non markovian walks having amnestically induced persistence we report numerically and analytically estimated values for the hurst exponent for a recently proposed non markovian walk characterized by amnestically induced persistence these results are consistent with earlier studies showing that log periodic oscillations arise only for large memory losses of the recent past we also report numerical estimates of the hurst exponent for non markovian walks with diluted memory finally we study walks with a fractal memory of the past for a thue morse and fibonacci memory patterns these results are interpreted and discussed in the context of the necessary and sufficient conditions for the central limit theorem to hold

current fluctuations in stochastic systems with long range memory we propose a method to calculate the large deviations of current fluctuations in a class of stochastic particle systems with history dependent rates long range temporal correlations are seen to alter the speed of the large deviation function in analogy with long range spatial correlations in equilibrium systems we give some illuminating examples and discuss the applicability of the gallavotti cohen fluctuation theorem

random recursive trees and the elephant random walk one class of random walks with infinite memory so called elephant random walks are simple models describing anomalous diffusion we present a surprising connection between these models and bond percolation on random recursive trees we use a coupling between the two models to translate results from elephant random walks to the percolation process we calculate besides other quantities exact expressions for the first and the second moment of the root cluster size and of the number of nodes in child clusters of the first generation we further introduce another model the skew elephant random walk and calculate the first and second moment of this process

memory induced anomalous dynamics emergence of diffusion subdiffusion and superdiffusion from a single random walk model we present a random walk model that exhibits asymptotic subdiffusive diffusive or superdiffusive behavior in different parameter regimes this appears to be an instance of a single random walk model leading to all three forms of behavior by simply changing parameter values furthermore the model offers the great advantage of analytical tractability our model is non markovian in that the next jump of the walker is probabilistically determined by the history of past jumps it also has elements of intermittency in that one possibility at each step is that the walker does not move at all this rich encompassing scenario arising from a single model provides useful insights into the source of different types of asymptotic behavior

limit theorems for a random walk with memory perturbed by a dynamical system we introduce a new random walk with unbounded memory obtained as a mixture of the elephant random walk and the dynamic random walk which we call the dynamic elephant random walk derw as a consequence of this mixture the distribution of the increments of the resulting random process is time dependent we prove a strong law of large numbers for the derw and in a particular case we provide an explicit expression for its speed finally we give sufficient conditions for the central limit theorem and the law of the iterated logarithm to hold

interpolatory projection methods for parameterized model reduction we provide a unifying projection based framework for structure preserving interpolatory model reduction of parameterized linear dynamical systems i e systems having a structured dependence on parameters that we wish to retain in the reduced order model the parameter dependence may be linear or nonlinear and is retained in the reduced order model moreover we are able to give conditions under which the gradient and hessian of the system response with respect to the system parameters is matched in the reduced order model we provide a systematic approach built on established interpolatory mathcal h 2 optimal model reduction methods that will produce parameterized reduced order models having high fidelity throughout a parameter range of interest for single input single output systems with parameters in the input output maps we provide reduced order models that are optimal with respect to an mathcal h 2 otimes mathcal l 2 joint error measure the capabilities of these approaches are illustrated by several numerical examples from technical applications

stabilization of projection based reduced order models summary a rigorous method for stabilizing projection based linear reduced order models without significantly affecting their accuracy is proposed unlike alternative approaches this method is computationally efficient it requires primarily the solution of a small scale convex optimization problem furthermore it is nonintrusive in the sense that it operates directly on readily available reduced order operators these can be precomputed using any data compression technique including balanced truncation balanced proper orthogonal decomposition proper orthogonal decomposition or moment matching the proposed method is illustrated with three applications the stabilization of the reduction of the computational fluid dynamics based model of a linearized unsteady supersonic flow the reduction of a computational structural dynamics system and the stabilization of the reduction of a coupled computational fluid dynamics computational structural dynamics model of a linearized aeroelastic system in the transonic flow regime copyright 2012 john wiley sons ltd

model reduction for large scale systems with high dimensional parametric input space a model constrained adaptive sampling methodology is proposed for the reduction of large scale systems with high dimensional parametric input spaces our model reduction method uses a reduced basis approach which requires the computation of high fidelity solutions at a number of sample points throughout the parametric input space a key challenge that must be addressed in the optimization control and probabilistic settings is the need for the reduced models to capture variation over this parametric input space which for many applications will be of high dimension we pose the task of determining appropriate sample points as a pde constrained optimization problem which is implemented using an efficient adaptive algorithm that scales well to systems with a large number of parameters the methodology is demonstrated using examples with parametric input spaces of dimension 11 and 21 which describe thermal analysis and design of a heat conduction fin and compared with statistically based sampling methods for these examples the model constrained adaptive sampling leads to reduced models that for a given basis size have error several orders of magnitude smaller than that obtained using the other methods

supersymmetric quantum mechanics defined as sesquilinear forms a formulation of supersymmetric quantum mechanics in terms of sesquilinear forms is presented otherwise occurring rather complicated domain questions are solved in a simple and natural way the authors formulate the axioms and describe a number of models supersymmetry transformations are implemented as superunitary transformations in a space defined over a grassmann algebra generalised bogoliubov transformations acting in that space are also formulated

some summation formulas for the series 3f2 1 summation formulas contiguous to watson s and whipple s theorems in the theory of the generalized hypergeometric series are obtained certain limiting cases of these results are given 1 two results contiguous to watson s theorem the two following summation formulas for series 3f2 1 are useful interesting easily established and probably new they are

on the multi dimensional elephant random walk the purpose of this paper is to investigate the asymptotic behavior of the multi dimensional elephant random walk merw it is a non markovian random walk which has a complete memory of its entire history a wide range of literature is available on the one dimensional erw surprisingly no references are available on the merw the goal of this paper is to fill the gap by extending the results on the one dimensional erw to the merw in the diffusive and critical regimes we establish the almost sure convergence the law of iterated logarithm and the quadratic strong law for the merw the asymptotic normality of the merw properly normalized is also provided in the superdiffusive regime we prove the almost sure convergence as well as the mean square convergence of the merw all our analysis relies on asymptotic results for multi dimensional martingales

exact solution of an anisotropic 2d random walk model with strong memory correlations over the last decade there has been progress in understanding one dimensional non markovian processes via analytic sometimes exact solutions the extension of these ideas and methods to two and higher dimensions is challenging we report the first exactly solvable two dimensional 2d non markovian random walk model belonging to the family of the elephant random walk model in contrast to levy walks or fractional brownian motion such models incorporate memory effects by keeping an explicit history of the random walk trajectory we study a memory driven 2d random walk with correlated memory and stops i e pauses in motion the model has an inherent anisotropy with consequences for its diffusive properties thereby mixing the dominant regime along one dimension with a subdiffusive walk along a perpendicular dimension the anomalous diffusion regimes are fully characterized by an exact determination of the hurst exponent we discuss the remarkably rich phase diagram as well as several possible combinations of the independent walks in both directions the relationship between the exponents of the first and second moments is also unveiled

limit theorems for triangular urn schemes we study a generalized polya urn with balls of two colours and a triangular replacement matrix the urn is not required to be balanced we prove limit theorems describing the asymptotic distribution of the composition of the urn after a long time several different types of asymptotics appear depending on the ratio of the diagonal elements in the replacement matrix the limit laws include normal stable and mittag leffler distributions as well as some less familiar ones the results are in some cases similar to but in other cases strikingly different from the results for irreducible replacement matrices

an efficient output error estimation for model order reduction of parametrized evolution equations in this work we present an a posteriori output error bound for model order reduction of parametrized evolution equations with the help of the dual system and a simple representation of the relationship between the field variable error and the residual of the primal system a sharp output error bound is derived such an error bound successfully avoids the accumulation of the residual over time which is a common drawback in the existing error estimations for time stepping schemes an estimation needs to be performed for practical computation of the error bound and as a result the output error bound reduces to an output error estimation the proposed error estimation is applied to four kinds of problems the first one is a linear convection diffusion equation which is used to compare the performance of the new error estimation and an existing primal dual error bound the second one is the unsteady viscous burgers equation an academic benchmark of nonlinear evolution equations in fluid dynamics often u

n dimensional nonlinear fokker planck equation with time dependent coefficients an n dimensional nonlinear fokker planck equation is investigated here by considering the time dependence of the coefficients where drift controlled and source terms are present we exhibit the exact solution based on the generalized gaussian function related to the tsallis statistics furthermore we show that a rich class of diffusive processes including normal and anomalous ones can be obtained by changing the time dependence of the coefficients

accelerating pde constrained optimization by the reduced basis method application to batch chromatography summary in this work we show that the reduced basis method accelerates a partial differential equation constrained optimization problem where a nonlinear discretized system with a large number of degrees of freedom must be repeatedly solved during optimization such an optimization problem arises for example from batch chromatography to reduce the computational burden of repeatedly solving the large scale system under parameter variations a parametric reduced order model with a small number of equations is derived by using the reduced basis method as a result the small reduced order model rather than the full system is solved at each step of the optimization process an adaptive technique for selecting the snapshots is proposed so that the complexity and runtime for generating the reduced basis are largely reduced an output oriented error bound is derived in the vector space whereby the construction of the reduced model is managed automatically an early stop criterion is proposed to circumvent the stagnation of the error and to make the construction of the reduced model more efficient numerical examples show that the adaptive technique is very efficient in reducing the offline time the optimization based on the reduced model is successful in terms of the accuracy and the runtime for acquiring the optimal solution copyright 2015 john wiley sons ltd

a martingale approach for the elephant random walk the purpose of this paper is to establish via a martingale approach some refinements on the asymptotic behavior of the one dimensional elephant random walk erw the asymptotic behavior of the erw mainly depends on a memory parameter p which lies between zero and one this behavior is totally different in the diffusive regime the critical regime and the superdiffusive regime in the diffusive and critical regimes we establish some new results on the almost sure asymptotic behavior of the erw such as the quadratic strong law and the law of the iterated logarithm in the superdiffusive regime we provide the first rigorous mathematical proof that the limiting distribution of the erw is not gaussian

hypergeometric identities arising from the elephant random walk abstract a probabilistic approach is provided to establish new hypergeometric identities it is based on the calculation of moments of the limiting distribution of the position of the elephant random walk in the superdiffusive regime

inhomogeneous diffusion and ergodicity breaking induced by global memory effects fil budini adrian adolfo comision nacional de energia atomica gerencia del area de investigaciones y aplicaciones no nucleares gerencia de fisica centro atomico constituyentes argentina consejo nacional de investigaciones cientificas y tecnicas centro cientifico tecnologico conicet patagonia norte argentina

non markovian random walks with memory lapses we propose an approach to construct bernoulli trials xi i 1 combining dependence and independence periods and we call it the bernoulli sequence with random dependence bsrd the structure of dependence in the past si x1 xi defines a class of non markovian random walks of recent interest in the literature in this paper the dependence is activated by an auxiliary collection of bernoulli trials yi i 1 called memory switch sequence we introduce the concept of memory lapse property which is characterized by intervals of consecutive independent steps in bsrd the main results include classical limit theorems for a class of linear bsrd in particular we obtain a central limit theorem for a class of bsrd which generalizes some previous results in the literature along the paper several examples of potential applications are provided

automatic generation of hypergeometric identities by the beta integral method in this article hypergeometric identities or transformations for p 1fp series and for kampe de feriet series of unit arguments are derived systematically from known transformations of hypergeometric series and products of hypergeometric series respectively using the beta integral method in an automated manner based on the mathematica package hyp as a result we obtain some known and some identities which seem to not have been recorded before in literature

path integrals in n 2 supersymmetric quantum mechanics by use of path integrals in n 2 supersymmetric quantum mechanics a formula for the kernel of the supersymmetric time evolution operator is found structurally it is similar to the feynman kac formula used in the path integrals of quantum mechanics crucial to the development of this formula is a generalization of the wiener measure based upon the free supercharges rather than the free hamiltonian

a stochastic model of human gait dynamics abstract we present a stochastic model of gait rhythm dynamics based on transitions between different neural centers that reproduces distinctive statistical properties of normal human walking by tuning one model parameter the transition hopping range the model can describe alterations in gait dynamics from childhood to adulthood including a decrease in the correlation and volatility exponents with maturation the model also generates time series with multifractal spectra whose broadness depends only on this parameter moreover we find that the volatility exponent increases monotonically as a function of the width of the multifractal spectrum suggesting the possibility of a change in multifractality with maturation

exact moments in a continuous time random walk with complete memory of its history we present a continuous time generalization of a random walk with complete memory of its history phys rev e 70 045101 r 2004 and derive exact expressions for the first four moments of the distribution of displacement when the number of steps is poisson distributed we analyze the asymptotic behavior of the normalized third and fourth cumulants and identify new transitions in a parameter regime where the random walk exhibits superdiffusion these transitions which are also present in the discrete time case arise from the memory of the process and are not reproduced by fokker planck approximations to the evolution equation of this random walk

phase transition in random walks with long range correlations motivated by recent results in the theory of correlated sequences we analyze the dynamics of random walks with long term memory binary chains with long range correlations in our model the probability for a unit bit in a binary string depends on the fraction of unities preceding it we show that the system undergoes a dynamical phase transition from normal diffusion in which the variance d l scales as the string s length l into a superdiffusion phase d l approximately lalpha alpha 1 when the correlation strength exceeds a critical value we demonstrate the generality of our results with respect to alternative models and discuss their applicability to various data such as coarse grained dna sequences written texts and financial data

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

on a comparative study between dependence scales determined by linear and non linear measures in this manuscript we present a comparative study about the determination of the relaxation i e independence time scales obtained from the correlation function the mutual information and a criterion based on the evaluation of a nonextensive generalisation of mutual entropy our results show that for systems with a small degree of complexity standard mutual information and the criterion based on its nonextensive generalisation provide the same scale whereas for systems with a higher complex dynamics the standard mutual information presents a time scale consistently smaller

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

discrete time non markovian random walks the effect of memory limitations on scaling abstract an important class of random walks includes those in which the random increment at time step t depends on the complete history of the process we consider a recently proposed discrete time non markovian random walk process characterized by a memory parameter p we numerically calculate the first and second moments of the position distribution and relate our results to known analytic results for special cases we obtain data collapse for the position distribution we study the effect of reducing the memory by considering a modified model in which only a fraction f of the previous steps are remembered surprisingly the behavior becomes markovian for small f even though the correlation time diverges for f 0 we also study the transient effects near the memory edge by considering a markovian limit of the original model we observe a crossover to markovian behavior for times much larger than the range of the memory

bayesian analysis for reversible markov chains we introduce a natural conjugate prior for the transition matrix of a reversible markov chain this allows estimation and testing the prior arises from random walk with reinforcement in the same way the dirichlet prior arises from polya s urn we give closed form normalizing constants a simple method of simulation from the posterior and a characterization along the lines of w e johnson s characterization of the dirichlet prior

central limit theorem and related results for the elephant random walk we study the so called elephant random walk erw which is a non markovian discrete time random walk on with unbounded memory which exhibits a phase transition from a diffusive to superdiffusive behavior we prove a law of large numbers and a central limit theorem remarkably the central limit theorem applies not only to the diffusive regime but also to the phase transition point which is superdiffusive inside the superdiffusive regime the erw converges to a non degenerate random variable which is not normal we also obtain explicit expressions for the correlations of increments of the erw

edge and vertex reinforced random walks with super linear reinforcement on infinite graphs in this paper we introduce a new simple but powerful general technique for the study of edge and vertex reinforced processes with super linear reinforcement based on the use of order statistics for the number of edge respectively of vertex traversals the technique relies on upper bound estimates for the number of edge traversals proved in a different context by cotar and limic ann appl probab 2009 for finite graphs with edge reinforcement we apply our new method both to edge and to vertex reinforced random walks with super linear reinforcement on arbitrary infinite connected graphs of bounded degree we stress that unlike all previous results for processes with super linear reinforcement we make no other assumption on the graphs for edge reinforced random walks we complete the results of limic and tarr es ann probab 2007 and we settle a conjecture of sellke technical report 94 26 purdue university 1994 by showing that for any reciprocally summable reinforcement weight function w the walk traverses a random attracting edge at all large times for vertex reinforced random walks we extend results previously obtained on z by volkov ann probab 2001 and by basdevant schapira and singh ann probab 2014 and on complete graphs by benaim raimond and schapira alea 2013 we show that on any infinite connected graph of bounded degree with reinforcement weight function w taken from a general class of reciprocally summable reinforcement weight functions the walk traverses two random neighbouring attracting vertices at all large times

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

slow levy flights among markovian processes the hallmark of levy flights is superdiffusion or faster than brownian dynamics here we show that levy laws as well as gaussian distributions can also be the limit distributions of processes with long range memory that exhibit very slow diffusion logarithmic in time these processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites we show how the central limit theorem is modified in this context keeping the usual distinction between analytic and nonanalytic characteristic functions a fluctuation dissipation relation is also derived our results may have important applications in the study of animal and human displacements

memory effects in recurrent and extreme events a dynamical approach to recurrent and extreme events is developed focusing on the role of correlations and memory in the structure of the probability distributions and their low order moments the procedure is illustrated on homogeneous first and second order markov chains non markovian and nonhomogeneous processes and deterministic dynamical systems substantial differences with classical statistical theory as applied to independent identically distributed random variables are identified

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

how linear reinforcement affects donsker s theorem for empirical processes a reinforcement algorithm introduced by simon biometrika 42 3 4 425 440 1955 produces a sequence of uniform random variables with long range memory as follows at each step with a fixed probability p in 0 1 hat u n 1 is sampled uniformly from hat u 1 ldots hat u n and with complementary probability 1 p hat u n 1 is a new independent uniform variable the glivenko cantelli theorem remains valid for the reinforced empirical measure but not the donsker theorem specifically we show that the sequence of empirical processes converges in law to a brownian bridge only up to a constant factor when p 1 2 and that a further rescaling is needed when p 1 2 and the limit is then a bridge with exchangeable increments and discontinuous paths this is related to earlier limit theorems for correlated bernoulli processes the so called elephant random walk and more generally step reinforced random walks

diffusion and memory effects for stochastic processes and fractional langevin equations we consider the diffusion processes defined by stochastic differential equations when the noise is correlated a functional method based on the dyson expansion for the evolution operator associated to the stochastic continuity equation is proposed to obtain the fokker planck equation after averaging over the stochastic process in the white noise limit the standard result corresponding to the stratonovich interpretation of the non linear langevin equation is recovered when the noise is correlated the averaged operator series cannot be summed unless a family of time dependent operators commutes in the case of a linear equation the constraints are easily worked out the process defined by a linear langevin equation with additive noise is gaussian and the probability density function of its fluctuating component satisfies a fokker planck equation with a time dependent diffusion coefficient the same result holds for a linear langevin equation with a fractional time derivative defined according to caputo 1 in the generic linear or nonlinear case approximate equations for small noise amplitude are obtained for small correlation time the evolution equations further simplify in agreement with some previous alternative derivations the results are illustrated by the linear oscillator with coloured noise and the fractional wiener process where the numerical simulation for the probability density and its moments is compared with the analytical solution

multidimensional elephant random walk with coupled memory the elephant random walk erw is a microscopic one dimensional discrete time non markovian random walk which can lead to anomalous diffusion due to memory effects in this study i propose a multidimensional generalization in which the probability of taking a step in a certain direction depends on the previous steps in other directions the original model is generalized in a straightforward manner by introducing coefficients that couple the probability of moving in one direction with the previous steps in all directions i motivate the model by first introducing a two elephant system and then elucidating it with a specific coupling with the explicit calculation of the first moments i show the existence of two newsworthy relative movement behaviors one in which one elephant follows the other and another in which they go in opposite directions with the aid of a fokker planck equation the second moment is evaluated and two superdiffusion regimes appear not found in other erws then i reinterpret the equations as a bidimensional elephant random walk model and further generalize it to n dimensions i argue that the introduction of coupling coefficients is a way of extending any one dimensional erw to many dimensions

elephants can always remember exact long range memory effects in a non markovian random walk we consider a discrete time random walk where the random increment at time step t depends on the full history of the process we calculate exactly the mean and variance of the position and discuss its dependence on the initial condition and on the memory parameter p at a critical value p 1 c 1 2 where memory effects vanish there is a transition from a weakly localized regime where the walker elephant returns to its starting point to an escape regime inside the escape regime there is a second critical value where the random walk becomes superdiffusive the probability distribution is shown to be governed by a non markovian fokker planck equation with hopping rates that depend both on time and on the starting position of the walk on large scales the memory organizes itself into an effective harmonic oscillator potential for the random walker with a time dependent spring constant k 2p 1 t the solution of this problem is a gaussian distribution with time dependent mean and variance which both depend on the initiation of the process

generalized langevin equations anomalous diffusion and probability distributions we study the motion of a particle governed by a generalized langevin equation we show that when no fluctuation dissipation relation holds the long time behavior of the particle may be from stationary to superdiffusive along with subdiffusive and diffusive when the random force is gaussian we derive the exact equations for the joint and marginal probability density functions for the position and velocity of the particle and find their solutions s1063 651x 96 10706 6

goal oriented model constrained optimization for reduction of large scale systems optimization oriented reduced order models should target a particular output functional span an applicable range of dynamic and parametric inputs and respect the underlying governing equations of the system to achieve this goal we present an approach for determining a projection basis that uses a goal oriented model constrained optimization framework the mathematical framework permits consideration of general dynamical systems with general parametric variations and is applicable to both linear and nonlinear systems results for a simple linear model problem of the two dimensional heat equation demonstrate the ability of the goal oriented approach to target a particular output functional of interest application of the methodology to a more challenging example of a subsonic blade row governed by the unsteady euler flow equations shows a significant advantage of the new method over the proper orthogonal decomposition

transient superdiffusion in random walks with a q exponentially decaying memory profile abstract we propose a random walk model with q exponentially decaying memory profile the q exponential function is a generalization of the ordinary exponential function in the limit q 1 the q exponential becomes the ordinary exponential function this model presents a markovian diffusive regime that is characterized by finite memory correlations it is well known that central limit theorems prohibit superdiffusion for markovian walks with finite variance of step sizes in this problem we report the outcome of a transient superdiffusion for finite sized walks

a formalism for studying long range correlations in many alphabets sequences abstract we formulate a mean field like theory of long range correlated l alphabets sequences which are actually systems with l 1 independent parameters depending on the values of these parameters the variance on the average number of any given symbol in the sequence shows a linear or a superlinear dependence on the total length of the sequence we present exact solution to the four alphabets and three alphabets sequences we also demonstrate that a mapping of the given sequence into a smaller alphabets sequence namely a coarse graining process does not necessarily imply that long range correlations found in the latter would correspond to those of the former

non levy stable random walk propagators for a non markovian walk with both superdiffusive and subdiffusive regimes abstract one of the most important quantities of interest in the theory of diffusion and transport is the random walk propagator for markovian processes such as the standard brownian random walk and levy flights the functional form of the random walk propagator is well understood similarly for certain kinds of simple non markovian processes such as levy walks the problem can be mapped to a solvable markovian model however more complicated non markovian walks pose a challenge here we study a non markovian model that is rich enough to exhibit superdiffusion normal diffusion and subdiffusion regimes kumar harbola and lindenberg 2010 we numerically estimate propagators for this model and obtain good fits with a family of non levy propagators based on the tsallis q exponential function we conclude that stops and restrictions play similar roles in the long time limit of the propagator

on a class of random walks with reinforced memory this paper deals with different models of random walks with a reinforced memory of preferential attachment type we consider extensions of the elephant random walk introduced by schutz and trimper phys rev e 70 044510 r 2004 with stronger reinforcement mechanisms where roughly speaking a step from the past is remembered proportional to some weight and then repeated with probability p with probability 1 p the random walk performs a step independent of the past the weight of the remembered step is increased by an additive factor b ge 0 making it likelier to repeat the step again in the future a combination of techniques from the theory of urns branching processes and alpha stable processes enables us to discuss the limit behavior of reinforced versions of both the elephant random walk and its alpha stable counterpart the so called shark random swim introduced by businger j stat phys 172 3 701 717 2004 we establish phase transitions separating subcritical from supercritical regimes

a two dimensional non markovian random walk leading to anomalous diffusion abstract exact solutions are rare for non markovian random walk models even in 1d and much more so in 2d here we propose a 2d genuinely non markovian random walk model with a very rich phase diagram such that the motion in each dimension can belong to one of 3 categories i subdiffusive ii superdiffusive or iii normally diffusive the main advance reported here is a different method and the consequent physical insight for analytically solving this model simpler non markovian models such as levy walks have been solved in 2d but it is not clear if the method of solution could be made to work for more complicated models such as the one studied here we also report the exact solutions for the first two moments of the random walk propagator along with the complete phase diagram the latter is surprisingly rich and admits diverse diffusion regimes finally we discuss these results in the context of theoretical underpinnings as well as possible applications

enablers for robust pod models this paper focuses on improving the stability as well as the approximation properties of reduced order models rom based on proper orthogonal decomposition pod the rom is obtained by seeking a solution belonging to the pod subspace and that at the same time minimizes the navier stokes residuals we propose a modified rom that directly incorporates the pressure term in the model the rom is then stabilized making use of a method based on the fine scale equations an improvement of the pod solution subspace is performed thanks to an hybrid method that couples direct numerical simulations and reduced order model simulations the methods proposed are tested on the two dimensional confined square cylinder wake flow in laminar regime

critical polya urn we propose a variant model of the polya urn process where the dynamics consist of two competing elements suppression of growth and enhancement of dormant character here the level of such features is controlled by an internal parameter in a random manner so that the system self organizes and characteristic observables exhibit scale invariance suggesting the existence of criticality varying the internal control parameter one can explain interesting universality classes for avalanche type events we also discuss different versions of the model it is pointed out that such an underlying mechanism has earlier been noted to operate in a neural network

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

nonstationary markovian replication process causing diverse diffusions we introduce a single generative mechanism that can be used to describe diverse nonstationary diffusions a nonstationary markovian replication process for steps is considered for which we derive analytically the time evolution of the probability distribution of the walker s displacement and the generalized telegrapher equation with time varying coefficients and we find that diffusivity can be determined by temporal changes of replication of an immediate step by controlling the replications we realize diverse diffusions such as alternating diffusion superdiffusion subdiffusion and marginal diffusion which originate from oscillating increasing decreasing and slowly increasing or decreasing replications with time respectively

scaling behavior for random walks with memory of the largest distance from the origin we study a one dimensional random walk with memory the behavior of the walker is modified with respect to the simple symmetric random walk only when he or she is at the maximum distance ever reached from his or her starting point home in this case having the choice to move farther or to move closer the walker decides with different probabilities if the probability of a forward step is higher then the probability of a backward step the walker is bold otherwise he or she is timorous we investigate the asymptotic properties of this bold timorous random walk showing that the scaling behavior varies continuously from subdiffusive timorous to superdiffusive bold the scaling exponents are fully determined with a new mathematical approach based on a decomposition of the dynamics in active journeys the walker is at the maximum distance and lazy journeys the walker is not at the maximum distance

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

interacting elephant random walks the elephant random walk is a history dependent random walk we study a class of interacting elephant random walks our model includes the exclusion process as a special case by means of monte carlo simulations and mean field arguments we find that it exhibits a condensation phenomenon with a phase transition that is manifestly of first order the transition point depends on the initial configuration

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

time evolution kernels uniform asymptotic expansions for a wide class of self adjoint schrodinger hamiltonians a detailed description of the time evolution kernel is obtained in a setting of a d dimensional euclidean space without boundaries the schrodinger hamiltonian h is the sum of the negative laplacian plus a real valued local potential v x the class of potentials studied is the family of bounded and continuous functions that are formed from the fourier transforms of complex bounded measures these potentials are suitable for the n body problem since they do not necessarily decrease as x an asymptotic expansion in the complex parameter z around z 0 is derived for the family of kernels uz x y corresponding to the analytic semigroup e zh re z 0 which is uniform in the coordinate variables x and y the asymptotic expansion has a simple semiclassical interpretation furthermore an explicit bound for the remainder term in the asymptotic expansion is found the expansion and the remainder term bound continue to the time axis boundary z it

spontaneous symmetry breaking in amnestically induced persistence we investigate a recently proposed non markovian random walk model characterized by loss of memories of the recent past and amnestically induced persistence we report numerical and analytical results showing the complete phase diagram consisting of four phases for this system i classical nonpersistence ii classical persistence iii log periodic nonpersistence and iv log periodic persistence driven by negative feedback the first two phases possess continuous scale invariance symmetry however log periodicity breaks this symmetry instead log periodic motion satisfies discrete scale invariance symmetry with complex rather than real fractal dimensions we find for log periodic persistence evidence not only of statistical but also of geometric self similarity

solvable random walk model with memory and its relations with markovian models of anomalous diffusion motivated by studies on the recurrent properties of animal and human mobility we introduce a path dependent random walk model with long range memory for which not only the mean square displacement msd but also the propagator can be obtained exactly in the asymptotic limit the model consists of a random walker on a lattice which at a constant rate stochastically relocates at a site occupied at some earlier time this time in the past is chosen randomly according to a memory kernel whose temporal decay can be varied via an exponent parameter in the weakly non markovian regime memory reduces the diffusion coefficient from the bare value when the mean backward jump in time diverges the diffusion coefficient vanishes and a transition to an anomalous subdiffusive regime occurs paradoxically at the transition the process is an anticorrelated l evy flight although in the subdiffusive regime the model exhibits some features of the continuous time random walk with infinite mean waiting time it belongs to another universality class if memory is very long ranged a second transition takes place to a regime characterized by a logarithmic growth of the msd with time in this case the process is asymptotically gaussian and effectively described as a scaled brownian motion with a diffusion coefficient decaying as 1 t

a version of herbert a simon s model with slowly fading memory and its connections to branching processes construct recursively a long string of words w 1 ldots w n such that at each step k w k 1 is a new word with a fixed probability p in 0 1 and repeats some preceding word with complementary probability 1 p more precisely given a repetition occurs w k 1 repeats the jth word with probability proportional to j alpha for j 1 ldots k we show that the proportion of distinct words occurring exactly ell times converges as the length n of the string goes to infinity to some probability mass function in the variable ell ge 1 whose tail decays as a power function when p 1 1 alpha