first passage time densities in resonate and fire models motivated by the dynamics of resonant neurons we discuss the properties of the first passage time fpt densities for non markovian differentiable random processes we start from an exact expression for the fpt density in terms of an infinite series of integrals over joint densities of level crossings and consider different approximations based on truncation or on approximate summation of this series thus the first few terms of the series give good approximations for the fpt density on short times for rapidly decaying correlations the decoupling approximations perform well in the whole time domain as an example we consider resonate and fire neurons representing stochastic underdamped or moderately damped harmonic oscillators driven by white gaussian or by ornstein uhlenbeck noise we show that approximations reproduce all qualitatively different structures of the fpt densities from monomodal to multimodal densities with decaying peaks the approximations work for the systems of whatever dimension and are especially effective for the processes with narrow spectral density exactly when markovian approximations fail

outage probability in η μ η μ and κ μ η μ interference limited scenarios in this paper exact closed form expressions are derived for the outage probability op in scenarios where the signal of interest soi experiences or fading the interfering signals are subject to fading and the background noise can be neglected with the only assumption that the parameter is a positive integer number for the interfering signals the derived expressions when the soi is either or distributed are given in elementary terms for maximal ratio combining mrc with independent branches if the pre combining soi power signals are distributed with their corresponding parameters being positive integer or half integer numbers the previous analysis is extended for spatially correlated mrc at the receiver

first passage times for correlated random walks and some generalizations it is generally difficult to solve fokker planck equations in the presence of absorbing boundaries when both spatial and momentum coordinates appear in the boundary conditions in this note we analyze a simple exactly solvable model of the correlated random walk and its continuum analogue it is shown that one can solve for the moments recursively in one dimension in exact analogy with first passage problems for the fokker planck equation although the boundary conditions are somewhat more complicated further generalizations are suggested to multistate random walks

viscoelastic subdiffusion from anomalous to normal we study viscoelastic subdiffusion in bistable and periodic potentials within the generalized langevin equation approach our results justify the ultra slow fluctuating rate view of the corresponding bistable non markovian dynamics which displays bursting and anticorrelation of the residence times in two potential wells the transition kinetics is asymptotically stretched exponential when the potential barrier v 0 several times exceeds thermal energy k b t v 0 ensuremath sim 2 10 k b t and it cannot be described by the non markovian rate theory nmrt the well known nmrt result approximates however ever better with the increasing barrier height the most probable logarithm of the residence times moreover the rate description is gradually restored when the barrier height exceeds a fuzzy borderline which depends on the power law exponent of free subdiffusion ensuremath alpha such a potential free subdiffusion is ergodic surprisingly in periodic potentials it is not sensitive to the barrier height in the long time asymptotic limit however the transient to this asymptotic regime is extremally slow and it does profoundly depend on the barrier height the time scale of such subdiffusion can exceed the mean residence time in a potential well or in a finite spatial domain by many orders of magnitude all these features are in sharp contrast with an alternative subdiffusion mechanism involving jumps among traps with the divergent mean residence time in these traps

diffusion limited aggregation and iterated conformal maps the creation of fractal clusters by diffusion limited aggregation dla is studied by using iterated stochastic conformal maps following the method proposed recently by hastings and levitov the object of interest is the function phi sup n which conformally maps the exterior of the unit circle to the exterior of an ital n particle dla the map phi sup n is obtained from ital n stochastic iterations of a function phi that maps the unit circle to the unit circle with a bump the scaling properties usually studied in the literature on dla appear in a new light using this language the dimension of the cluster is determined by the linear coefficient in the laurent expansion of phi sup n which asymptotically becomes a deterministic function of ital n we find new relationships between the generalized dimensions of the harmonic measure and the scaling behavior of the laurent coefficients copyright ital 1999 ital the american physical society

levy brownian motion on finite intervals mean first passage time analysis we present the analysis of the first passage time problem on a finite interval for the generalized wiener process that is driven by levy stable noises the complexity of the first passage time statistics mean first passage time cumulative first passage time distribution is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non gaussian noises the validity of the method is tested numerically and compared against analytical formulas when the stability index alpha approaches 2 recovering in this limit the standard results for the fokker planck dynamics driven by gaussian white noise

markovian embedding of non markovian superdiffusion we consider different markovian embedding schemes of non markovian stochastic processes that are described by generalized langevin equations and obey thermal detailed balance under equilibrium conditions at thermal equilibrium superdiffusive behavior can emerge if the total integral of the memory kernel vanishes such a situation of vanishing static friction is caused by a super ohmic thermal bath one of the simplest models of ballistic superdiffusion is determined by a biexponential memory kernel that was proposed by bao j stat phys 114 503 2004 we show that this non markovian model has infinitely many different four dimensional markovian embeddings implementing numerically the simplest one we demonstrate that i the presence of a periodic potential with arbitrarily low barriers changes the asymptotic large time behavior from free ballistic superdiffusion into normal diffusion ii an additional biasing force renders the asymptotic dynamics superdiffusive again the development of transients that display a qualitatively different behavior compared to the true large time asymptotics presents a general feature of this non markovian dynamics these transients though may be extremely long as a consequence they can be even mistaken as the true asymptotics we find that such intermediate asymptotics exhibit a giant enhancement of superdiffusion in tilted washboard potentials and it is accompanied by a giant transient superballistic current growing proportional to t ensuremath alpha text eff with an exponent ensuremath alpha text eff that can exceed the ballistic value of 2

laplacian growth as one dimensional turbulence abstract a new model of laplacian stochastic growth is formulated using conformal mappings the model describes two growth regimes stable and turbulent separated by a sharp phase transition the first few fourier components of the mapping define the web an envelope of the cluster the web is used to study the transition and the dynamics of large scale features of the cluster characterized by evolution from macro to micro scales also we derive scaling laws for the cluster size

phase transitions in optimal search times how random walkers should combine resetting and flight scales recent works have explored the properties of levy flights with resetting in one dimensional domains and have reported the existence of phase transitions in the phase space of parameters which minimizes the mean first passage time mfpt through the origin l kusmierz et al phys rev lett 113 220602 2014 here we show how actually an interesting dynamics including also phase transitions for the minimization of the mfpt can also be obtained without invoking the use of levy statistics but for the simpler case of random walks with exponentially distributed flights of constant speed we explore this dynamics both in the case of finite and infinite domains and for different implementations of the resetting mechanism to show that different ways to introduce resetting consistently lead to a quite similar dynamics the use of exponential flights has the strong advantage that exact solutions can be obtained easily for the mfpt through the origin so a complete analytical characterization of the system dynamics can be provided furthermore we discuss in detail how the phase transitions observed in random walks with resetting are closely related to several ideas recurrently used in the field of random search theory in particular to other mechanisms proposed to understand random search in space as mortal random walks or multiscale random walks as a whole we corroborate that one of the essential ingredients behind mfpt minimization lies in the combination of multiple movement scales regardless of their specific origin

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

levy walks in random scattering and growth of waves random spatial wave scattering and stochastic wave growth are studied where one or both of the random processes can be described by a levy walk this analysis extends previous work on randomly growing and scattering waves where both the random processes are modeled by gaussian diffusive statistics both random spatial scattering and stochastic wave growth modeled by levy walks are studied separately together and in combination with gaussian processes transmission coefficients lasing thresholds and energy densities in the medium are obtained for the different permutations

finite size effect of harmonic measure estimation in a dla model variable size of probe particles abstract a finite size effect in the probing of the harmonic measure in simulation of diffusion limited aggregation dla growth is investigated we introduce a variable size of probe particles to estimate harmonic measure and extract the fractal dimension of dla clusters taking two limits of vanishingly small probe particle size and of infinitely large size of a dla cluster we generate 1000 dla clusters consisting of 50 million particles each using an off lattice killing free algorithm developed in the early work the introduced method leads to unprecedented accuracy in the estimation of the fractal dimension we discuss the variation of the probability distribution function with the size of probing particles

random walks with self similar clusters abstract we construct a random walk on a lattice having a hierarchy of self similar clusters built into the distribution function of allowed jumps the random walk is a discrete analog of a levy flight and coincides with the levy flight in the continum limit the fourier transform of the jump distribution function is the continuous nondifferentiable function of weierstrass we show that for cluster formation it is necessary that the mean squared displacement per jump be infinite and that the random walk be transient we interpret our random walk as having an effective dimension higher than the spatial dimension available to the walker the difference in dimensions is related to the fractal hausdorff besicovitch dimension of the self similar clusters

the κ μ phase envelope joint distribution this paper derives an exact closed form expression for the phase envelope joint distribution of the fading environment a general fading model that includes the rice and the nakagami m models as special cases the derived joint statistics are obtained so that compatibility with both rice as well as nakagami m cases are kept monte carlo simulation plots are presented that are in accordance with the formulations developed here as a consequence of the derivation of the required pdf an interesting solution to a very general probability problem is proposed namely given the pdf of the modulus of a variate find the pdf of the variate additionally several new integral relations of the modified bessel function are given

optimization of random searches on defective lattice networks we study the general problem of how to search efficiently for targets randomly located on defective lattice networks i e regular lattices which have some fraction of its nodes randomly removed we consider large but finite triangular lattices and assume for the search dynamics that the walker chooses steps lengths lj from the power law distribution p lj approximately lj mu with the exponent mu regulating the strategy of the search process at each step lj the searcher moves in straight lines and constantly looks within a detection radius of vision rv for the targets along the way if there is contact with a defect the movement stops and a new step length is chosen hence the presence of defects decreases the efficiency of the overall process we study numerically how three different aspects of the lattice influence the optimization of the search efficiency i the type of boundary conditions ii the concentration of targets and defects and iii the category or class of search destructive nondestructive or regenerative motivated by the results we develop a type of mean field model for the problem and obtain an analytical approximation for the search efficiency function finally we discuss in the context of searches how defective lattices compare with perfect lattices and with continuous environments

inverse square levy walks are not optimal search strategies for d 2 the levy hypothesis states that inverse square levy walks are optimal search strategies because they maximize the encounter rate with sparse randomly distributed replenishable targets it has served as a theoretical basis to interpret a wealth of experimental data at various scales from molecular motors to animals looking for resources putting forward the conclusion that many living organisms perform levy walks to explore space because of their optimal efficiency here we provide analytically the dependence on target density of the encounter rate of levy walks for any space dimension d in particular this scaling is shown to be independent of the levy exponent for the biologically relevant case d 2 which proves that the founding result of the levy hypothesis is incorrect as a consequence we show that optimizing the encounter rate with respect to is irrelevant it does not change the scaling with density and can lead virtually to any optimal value of depending on system dependent modeling choices the conclusion that observed inverse square levy patterns are the result of a common selection process based purely on the kinetics of the search behavior is therefore unfounded

scaling laws in the diffusion limited aggregation of persistent random walkers abstract we investigate the diffusion limited aggregation of particles executing persistent random walks the scaling properties of both random walks and large aggregates are presented the aggregates exhibit a crossover between ballistic and diffusion limited aggregation models a non trivial scaling relation l 1 25 between the characteristic size in which the cluster undergoes a morphological transition and the persistence length l between ballistic and diffusive regimes of the random walk is observed

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

optimal search strategies of space time coupled random walkers with finite lifetimes we present a simple paradigm for detection of an immobile target by a space time coupled random walker with a finite lifetime the motion of the walker is characterized by linear displacements at a fixed speed and exponentially distributed duration interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed we call these walkers mortal creepers a mortal creeper may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate m while still alive the creeper has a finite mean frequency of change of the direction of motion in particular we consider the efficiency of the target search process characterized by the probability that the creeper will eventually detect the target analytic results confirmed by numerical results show that there is an m dependent optimal frequency opt that maximizes the probability of eventual target detection we work primarily in one dimensional d 1 domains and examine the role of initial conditions and of finite domain sizes numerical results in d 2 domains confirm the existence of an optimal frequency of change of direction thereby suggesting that the observed effects are robust to changes in dimensionality in the d 1 case explicit expressions for the probability of target detection in the long time limit are given in the case of an infinite domain we compute the detection probability for arbitrary times and study its early and late time behavior we further consider the survival probability of the target in the presence of many independent creepers beginning their motion at the same location and at the same time we also consider a version of the standard target problem in which many creepers start at random locations at the same time

novel dynamical scaling in kinetic growth phenomena abstract a novel dynamical scaling characterized by infinitely many growth exponents is presented this new behavior is manifested in the time dependent pair correlation function and therefore is amenable to direct experimental observation an exact analytical solution on the time dependent ginzburg landau model is obtained in the large n limit for a quench at zero temperature which exhibits such multiscaling behavior this new form of behavior reflects a morphological pattern in the growing domains and is expected in other growth phenomena as indicated by preliminary analysis on dla clusters a connection between multiscaling and multifractality is discussed

the influence of the environment on levy random search efficiency fractality and memory effects an open problem in the field of random searches relates to optimizing the search efficiency in fractal environments here we address this issue through a systematic study of levy searches in landscapes encompassing several degrees of target aggregation and fractality for scarce resources non destructive searches with unrestricted revisits to targets are shown to present universal optimal behavior irrespective of the general scaling properties of the spatial distribution of targets in contrast no such universal behavior occurs in the destructive case with forbidden revisits in which the optimal strategy strongly depends on the degree of target aggregation we also investigate how the presence of memory and learning skills of the searcher affect the search efficiency by considering a limiting model in which the searcher learns through recent experience to recognize food rich areas we find that a statistical memory of previous encounters does not necessarily increase the rate of target findings in random searches instead there is an optimal extent of memory dependent on specific details of the search space and stochastic dynamics which maximizes the search efficiency this finding suggests a more general result namely that in some instances there are actual advantages to ignoring certain pieces of partial information while searching for objects

levy flights in an infinite potential well as a hypersingular fredholm problem we study levy flights with arbitrary index 0 2 inside a potential well of infinite depth such a problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems the major technical tool is a transformation of the eigenvalue problem for initial fractional schrodinger equation into that for fredholm integral equation with hypersingular kernel the latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain d reducing the problem to the spectrum of a matrix of infinite dimensions the eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum

optimization of random searches on regular lattices we investigate random searches on isotropic and topologically regular square and triangular lattices with periodic boundary conditions and study the efficiency of search strategies based on a power law distribution p ensuremath ell ensuremath sim ensuremath ell ensuremath ensuremath mu of step lengths ensuremath ell we consider both destructive searches in which a target can be visited only once and nondestructive searches when a target site is always available for future visits we discuss i the dependence of the search efficiency on the choice of the lattice topology ii the relevance of the periodic boundary conditions iii the behavior of the optimal power law exponent ensuremath mu mathrm opt as a function of target site density iv the differences between destructive and nondestructive environments and finally v how the results for the discrete searches differ from the continuous cases previously studied

levy flight random searches in biological phenomena abstract there has been growing interest in the study of levy flights observed in the movements of biological organisms performing random walks while searching for other organisms here we approach the problem of what is the best statistical strategy for optimizing the encounter rate between searcher and target organisms either of the same or of different species in terms of a limiting generalized searcher target model e g predator prey mating partner pollinator flower in this context we discuss known results showing that for fixed targets an inverse square density distribution of step lengths can optimize the encounter rate for moving targets we review how the encounter rate depends on whether organisms move in levy or brownian random walks we discuss recent findings indicating that levy walks confer a significant advantage for increasing encounter rates only when the searcher is larger or moves rapidly relative to the target and when the target density is low

levy flights and random searches in this work we discuss some recent contributions to the random search problem our analysis includes superdiffusive levy processes and correlated random walks in several regimes of target site density mobility and revisitability we present results in the context of mean field like and closed form average calculations as well as numerical simulations we then consider random searches performed in regular lattices and lattices with defects and we discuss a necessary criterion for distinguishing true superdiffusion from correlated random walk processes we invoke energy considerations in relation to critical survival states on the edge of extinction and we analyze the emergence of levy behavior in deterministic search walks finally we comment on the random search problem in the context of biological foraging

average time spent by levy flights and walks on an interval with absorbing boundaries we consider a levy flyer of order alpha that starts from a point x 0 on an interval o l with absorbing boundaries we find a closed form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed these two quantities are equivalent to the mean first passage times for levy flights and levy walks respectively using fractional differential equations with a riesz kernel we find exact analytical expressions for both quantities in the continuous limit we show that numerical solutions for the discrete levy processes converge to the continuous approximations in all cases except the case of alpha 2 and the cases of x 0 0 and x 0 l for alpha 2 when the second moment of the flight length distribution exists our result is replaced by known results of classical diffusion we show that if x 0 is placed in the vicinity of absorbing boundaries the average total length has a minimum at alpha 1 corresponding to the cauchy distribution we discuss the relevance of this result to the problem of foraging which has received recent attention in the statistical physics literature

the alpha eta kappa mu fading model new fundamental results a number of new results aiming at facilitating the use of the alpha eta kappa mu fading model are presented these results include 1 fast convergent with no recursions series representations for the envelope probability density function pdf and cumulative distribution function cdf 2 higher order moments of the envelope 3 moment generating function mgf of the envelope 4 asymptotic behavior of the envelope pdf and also of the envelope cdf 5 a procedure for parameter estimation 6 new closed form expressions for particular cases of the envelope distribution 7 new mathematical identity for the generalized laguerre polynomial and the generalized hypergeometric function and 8 approximate but tight series representation for the phase pdf as application examples the following are given 1 outage probability 2 outage capacity 3 amount of fading and 4 bit error rate for which the asymptotic behavior is also presented

correction to scaling analysis of diffusion limited aggregation diffusion limited aggregation is consistent with simple scaling however strong subdominant terms are present and these can account for various earlier claims of anomalous scaling we show this in detail for the case of multiscaling

first passage times in multiscale random walks the impact of movement scales on search efficiency an efficient searcher needs to balance properly the trade off between the exploration of new spatial areas and the exploitation of nearby resources an idea which is at the core of scale free levy search strategies here we study multiscale random walks as an approximation to the scale free case and derive the exact expressions for their mean first passage times in a one dimensional finite domain this allows us to provide a complete analytical description of the dynamics driving the situation in which both nearby and faraway targets are available to the searcher so the exploration exploitation trade off does not have a trivial solution for this situation we prove that the combination of only two movement scales is able to outperform both ballistic and levy strategies this two scale strategy involves an optimal discrimination between the nearby and faraway targets which is only possible by adjusting the range of values of the two movement scales to the typical distances between encounters so this optimization necessarily requires some prior information albeit crude about target distances or distributions furthermore we found that the incorporation of additional three four movement scales and its adjustment to target distances does not improve further the search efficiency this allows us to claim that optimal random search strategies arise through the informed combination of only two walk scales related to the exploitative and the explorative scales respectively expanding on the well known result that optimal strategies in strictly uninformed scenarios are achieved through levy paths or equivalently through a hierarchical combination of multiple scales

fractional kinetics in kac zwanzig heat bath models we study a variant of the kac zwanzig model of a particle in a heat bath the heat bath consists of n particles which interact with a distinguished particle via springs and have random initial data as n the trajectories of the distinguished particle weakly converge to the solution of a stochastic integro differential equation a generalized langevin equation gle with power law memory kernel and driven by 1 f noise the limiting process exhibits fractional sub diffusive behaviour we further consider the approximation of non markovian processes by higher dimensional markovian processes via the introduction of auxiliary variables and use this method to approximate the limiting gle in contrast we show the inadequacy of a so called fractional fokker planck equation in the present context all results are supported by direct numerical experiments

the spectrum of large powers of the laplacian in bounded domains we present exact results for the spectrum of the nth power of the laplacian in a bounded domain we begin with the one dimensional case and show that the whole spectrum can be obtained in the limit of large n we also show that it is a useful numerical approach valid for any n finally we discuss implications of this work and present its possible extensions for non integer n and for 3d laplacian problems

continuous time multidimensional markovian description of levy walks the paper presents a multidimensional model for nonlinear markovian random walks that generalizes the one we developed previously i lubashevsky r friedrich and a heuer phys rev e 79 011110 2009 in order to describe the levy type stochastic processes in terms of continuous trajectories of walker motion this approach may open a way to treat levy flights or levy random walks in inhomogeneous media or systems with boundaries in the future the proposed model assumes the velocity of a wandering particle to be affected by a linear friction and a nonlinear langevin force whose intensity is proportional to the magnitude of the velocity for its large values based on the singular perturbation technique the corresponding fokker planck equation is analyzed and the relationship between the system parameters and the levy exponent is found following actually the previous paper we demonstrate also that anomalously long displacements of the wandering particle are caused by extremely large fluctuations in the particle velocity whose duration is determined by the system parameters rather than the duration of the observation interval in this way we overcome the problem of ascribing to levy random walk non markov properties