We analyze two different confining mechanisms for Lévy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Levy-Schroedinger semigroups which induce so-called topological Levy processes (Levy flights with locally modified jump rates in the master equation). Given a stationary probability function (pdf) associated with the Langevin-based fractional Fokker-Planck equation, we demonstrate that generically there exists a topological Lévy process with the very same invariant pdf and in the reverse.
We analyze the extension of the well known relation between Brownian motion and Schroedinger equation to the family of Levy processes. We consider a Levy-Schroedinger equation where the usual kinetic energy operator - the Laplacian - is generalized by means of a selfadjoint, pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Levy-Khintchin formula shows then how to write down this operator in an integro--differential form. When the underlying Levy process is stable we recover as a particular case the fractional Schroedinger equation. …
We analyze confining mechanisms for Lévy flights. When they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport. Two classes of stochastic jump - type processes are considered: those driven by Langevin equation with Lévy noise and those, named by us topological Lévy processes (occurring in systems with topological complexity like folded polymers or complex networks and generically in inhomogeneous media), whose Langevin representation is unknown and possibly nonexistent. Our major finding is that both above classes of processes stay…
The problem of an arbitrary truncated Levy flight description using the method of cumulant approach has been solved. The set of cumulants of the truncated Levy distribution given the assumption of arbitrary truncation has been found. The influence of truncation shape on the truncated Levy flight properties in the Gaussian and the Levy regimes has been investigated.
The master equation for a probability density function (pdf) driven by Lévy noise, if conditioned to conform with the principle of detailed balance, admits a transformation to a contractive strongly continuous semigroup dynamics. Given a priori a functional form of the semigroup potential, we address the ground-state reconstruction problem for generic Lévy-stable semigroups, for {\em all} values of the stability index $μ\in (0,2)$. That is known to resolve an invariant pdf for confined Lévy flights (e.g. the former jump-type process). Jeopardies of the procedure are discussed, with a focus on:…
We study the long time asymptotics of probability density functions (pdfs) of Lévy flights in different confining potentials. For that we use two models: Langevin - driven and (Lévy - Schrödinger) semigroup - driven dynamics. It turns out that the semigroup modeling provides much stronger confining properties than the standard Langevin one. Since contractive semigroups set a link between Lévy flights and fractional (pseudo-differential) Hamiltonian systems, we can use the latter to control the long - time asymptotics of the pertinent pdfs. To do so, we need to impose suitable restrictions upon…
Levy flights and subdiffusive processes and their properties are discussed. We derive the space- and time-fractional transport equations, and consider their solutions in external potentials. An extensive list of references is included.
We address Lévy-stable stochastic processes in bounded domains, with a focus on a discrimination between inequivalent proposals for what a boundary data-respecting fractional Laplacian (and thence the induced random process) should actually be. Versions considered are: restricted Dirichlet, spectral Dirichlet and regional (censored) fractional Laplacians. The affiliated random processes comprise: killed, reflected and conditioned Lévy flights, in particular those with an infinite life-time. The related concept of quasi-stationary distributions is briefly mentioned.
The paper is devoted to the relationship between the continuous Markovian description of Levy flights developed previously and their equivalent representation in terms of discrete steps of a wandering particle, a certain generalization of continuous time random walks. Our consideration is confined to the one-dimensional model for continuous random motion of a particle with inertia. Its dynamics governed by stochastic self-acceleration is described as motion on the phase plane {x,v} comprising the position x and velocity v=dx/dt of the given particle. A notion of random walks inside a certain n…
Lévy flights and Lévy walks serve as two paradigms of random walks resembling common features but also bearing fundamental differences. One of the main dissimilarities are discontinuity versus continuity of their trajectories and infinite versus finite propagation velocity. In consequence, well developed theory of Lévy flights is associated with their pathological physical properties, which in turn are resolved by the concept of Lévy walks. Here, we explore Lévy flights and Lévy walks models on bounded domains examining their differences and analogies. We investigate analytically and numerical…
We solve a problem of non-convex stochastic optimisation with help of simulated annealing of Levy flights of a variable stability index. The search of the ground state of an unknown potential is non-local due to big jumps of the Levy flights process. The convergence to the ground state is fast due to a polynomial decrease rate of the temperature.
We study Lévy flights {with arbitrary index $0< μ\leq 2$} inside a potential well of infinite depth. Such 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 Schrödinger 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…
Truncated Lévy flights are random walks in which the arbitrarily large steps of a Lévy flight are eliminated. Since this makes the variance finite, the central limit theorem applies, and as time increases the probability distribution of the increments becomes Gaussian. Here, truncated Lévy flights with correlated fluctuations of the variance (heteroskedasticity) are considered. What makes these processes interesting is the fact that the crossover to the Gaussian regime may occur for times considerably larger than for uncorrelated (or no) variance fluctuations. These processes may find direct a…
Multi-scaling properties of one-dimensional truncated Levy flights are studied. Due to the broken self-similarity of the distribution of jumps, they are expected to possess multi-scaling properties in contrast to the ordinary Levy flights. We argue this fact based on a smoothly truncated Levy distribution, and derive the functional form of the scaling exponents. Specifically, they exhibit bi-fractal behavior, which is the simplest case of multi-scaling.
We employed the method of virial expansion in order to compute the retarded density correlation function (generalized diffusion propagator) in the critical random matrix ensemble in the limit of strong multifractality. We found that the long-range nature of the Hamiltonian is a common root of both multifractality and Levy flights which show up in the power-law intermediate- and long-distance behavior, respectively, of the density correlation function. We review certain models of classical random walks on fractals and show the similarity of the density correlation function in them to that for t…
How to simulate Lévy flights in a steep potential: An explicit splitting numerical scheme
Friedrich Schiller University Jena（耶拿弗里德里希·席勒大学）；Institute of Geophysics, National Academy of Sciences of Ukraine（乌克兰国家科学院地球物理研究所）；Igor Sikorsky Kyiv Polytechnic Institute（伊戈尔·西科尔斯基基辅理工学院）；Max Planck Institute of Microstructure Physics（马克斯·普朗克微观结构物理研究所）；Wroclaw University of Science and Technology（弗罗茨瓦夫科技大学）；Akhiezer Institute for Theoretical Physics, National Science Center “Kharkiv Institute of Physics and Technology”（阿赫谢泽理论物理研究所，国家科学中心“哈尔科夫物理与技术研究所”）
We derive the generalized Fokker-Planck equation associated with a Langevin equation driven by arbitrary additive white noise. We apply our result to study the distribution of symmetric and asymmetric Lévy flights in an infinitely deep potential well. The fractional Fokker-Planck equation for Lévy flights is derived and solved analytically in the steady state. It is shown that Lévy flights are distributed according to the beta distribution, whose probability density becomes singular at the boundaries of the well. The origin of the preferred concentration of flying objects near the boundaries i…
On the basis of multivariate Langevin processes we present a realization of Levy flights as a continuous process. For the simple case of a particle moving under the influence of friction and a velocity dependent stochastic force we explicitly derive the generalized Langevin equation and the corresponding generalized Fokker-Planck equation describing Levy flights. Our procedure is similar to the treatment of the Kramers-Fokker Planck equation in the Smoluchowski limit. The proposed approach forms a feasible way of tackling Levy flights in inhomogeneous media or systems with boundaries what is u…
Starting from the forward and backward infinitesimal generators of bilateral, time-homogeneous Markov processes, the self-adjoint Hamiltonians of the generalized Schroedinger equations are first introduced by means of suitable Doob transformations. Then, by broadening with the aid of the Dirichlet forms the results of the Nelson stochastic mechanics, we prove that it is possible to associate bilateral, and time-homogeneous Markov processes to the wave functions stationary solutions of our generalized Schroedinger equations. Particular attention is then paid to the special case of the Levy-Schr…
We generalise the Langevin equation with Gaussian white noise by replacing the velocity term by a local fractional derivative. The solution of this equation is a Levy process. We further consider the Brownian motion of a fractal particle, for example, a colloidal aggregate or a biological molecule and argue that it leads to a Levy flight. This effect can also be described using the local fractional Langevin equation. The implications of this development to other complex data series are discussed.
The search for hidden targets is a fundamental problem in many areas of science, engineering, and other fields. Studies of search processes often adopt a probabilistic framework, in which a searcher randomly explores a spatial domain for a randomly located target. There has been significant interest and controversy regarding optimal search strategies, especially for superdiffusive processes. The optimal search strategy is typically defined as the strategy that minimizes the time it takes a given single searcher to find a target, which is called a first hitting time (FHT). However, many systems…
We consider the A + A --> emptyset reaction, where the transport of the particles is given by Levy flights in a quenched random potential. With a common literature model of the disorder, the random potential can only increase the rate of reaction. With a model of the disorder that obeys detailed balance, however, the rate of reaction initially increases and then decreases as a function of the disorder strength. The physical behavior obtained with this second model is in accord with that for reactive turbulent flow, indicating that Levy flight statistics can model aspects of turbulent fluid tra…
After a short excursion from discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the $α$-stable distributions theory and fractional order derivatives. The further development of the idea lies on the generalization of the Langevin equation wit…
We investigate the impact of external periodic potentials on superdiffusive random walks known as Levy flights and show that even strongly superdiffusive transport is substantially affected by the external field. Unlike ordinary random walks, Levy flights are surprisingly sensitive to the shape of the potential while their asymptotic behavior ceases to depend on the Levy index $μ$. Our analysis is based on a novel generalization of the Fokker-Planck equation suitable for systems in thermal equilibrium. Thus, the results presented are applicable to the large class of situations in which superdi…
Optimal random foraging strategy has gained increasing concentrations. It is shown that Lévy flight is more efficient compared with the Brownian motion when the targets are sparse. However, standard Lévy flight generally cannot be followed in practice. In this paper, we assume that each flight of the forager is possibly interrupted by some uncertain factors, such as obstacles on the flight direction, natural enemies in the vision distance, and restrictions in the energy storage for each flight, and introduce the tempered Lévy distribution $p(l)\sim {\rm e}^{-ρl}l^{-μ}$. It is validated by both…
We investigated the relationships between search efficiency, movement strategy, and non-local communication in the biological context of animal foraging. We considered situations where the members of a population of foragers perform either Gaussian jumps or Levy flights, and show that the search time is minimized when communication among individuals occurs at intermediate ranges, independently of the type of movement. Additionally, while Brownian strategies are more strongly influenced by the communication mechanism, Levy flights still result in shorter overall search durations.
We explore the statistical behavior of the order statistics of the flights of One-sided Levy Processes (OLPs). We begin with the study of the extreme flights of general OLPs,and then focus on the class of selfsimilar processes,investigating the following issues:(i)the inner hierarchy of the extreme flights - for example:how big is the 7th largest flight relative to the 2nd largest one?; and,(ii)the relative contribution of the extreme flights to the entire 'flight aggregate' - for example: how big is the 3rd largest flight relative to the OLP's value?. Furthermore, we show that all 'hierarchic…
In the last ten years, the generalized central limit theorem established by Paul Levy in the thirties has been found more and more relevant in physics. Physicists call 'Levy flights' random walks for which the probability density of the jump lenghts $x$ decays as $1/x^{1+α}$ with $α<2$ for large $x$. We give here a glimpse of Levy flights in physics through two examples, without going into technical details. We first introduce a simple toy model, the Arrhenius cascade. We then present an important physical process, subrecoil laser cooling of atomic gases, in which Levy flights play an essentia…
Rock-paper-scissors (RPS) game is a nice model to study the biodiversity in ecosystem. However, the previous studies only consider the nearest- neighbor- interaction among the species. In this paper, taking the long range migration into account, the effects of the interplay between nearest-neighbor-interaction and long-range-interaction of Levy flight obey the power law distance distribution with the exponent h (-0.3<h<-0.1) in spatial RPS game is investigated. Taking the probability of long range Levy flight and the power exponent as parameters, the coexistence conditions of three species are…
The functional method to derive the fractional Fokker-Planck equation for probability distribution from the Langevin equation with Levy stable noise is proposed. For the Cauchy stable noise we obtain the exact stationary probability density function of Levy flights in different smooth potential profiles. We find confinement of the particle in the superdiffusion motion with a bimodal stationary distribution for all the anharmonic symmetric monostable potentials investigated. The stationary probability density functions show power-law tails, which ensure finiteness of the variance. By reviewing …
Transport of overdamped particles in an asymmetrically periodic potential is investigated in the presence of Lévy noise and ac-driving forces. The group velocity is used to measure the transport driven by the nonthermal noise. It is found that the Lévy flights and ac-driving forces are the two different driving factors that can break thermodynamical equilibrium. The competition between the two factors will induce some peculiar phenomena. For example, multiple transport reversals occur on changing the noise intensity. Additionally, we also find that the group velocity as a function of Lévy inde…
By using the backward fractional Fokker-Planck equation we investigate the barrier crossing event in the presence of Levy noise. After shortly review recent results obtained with different approaches on the time characteristics of the barrier crossing, we derive a general differential equation useful to calculate the nonlinear relaxation time. We obtain analytically the nonlinear relaxation time for free Levy flights and a closed expression in quadrature of the same characteristics for cubic potential.
Numerical evidence of directed transport driven by symmetric Levy noise in time-independent ratchet potentials in the absence of an external tilting force is presented. The results are based on the numerical solution of the fractional Fokker-Planck equation in a periodic potential and the corresponding Langevin equation with Levy noise. The Levy noise drives the system out of thermodynamic equilibrium and an up-hill net current is generated. For small values of the noise intensity there is an optimal value of the Levy noise index yielding the maximum current. The direction and magnitude of the…
We consider the time evolution of two-dimensional Levy flights in a finite area with periodic boundary conditions. From simulations we show that the fractal path dimension d_f and thus the degree of area coverage grows in time until it reaches the saturation value d_f=2 at sufficiently long times. We also investigate the time evolution of the probability density function and associated moments in these boundary conditions. Finally we consider the mean first passage time as function of the stable index. Our findings are of interest to assess the ergodic behavior of Levy flights, to estimate the…
Let L(t) be a Levy flights process with a stability index α\in(0,2), and U be an external multi-well potential. A jump-diffusion Z satisfying a stochastic differential equation dZ(t)=-U'(Z(t-))dt+σ(t)dL(t) describes an evolution of a Levy particle of an `instant temperature' σ(t) in an external force field. The temperature is supposed to decrease polynomially fast, i.e. σ(t)\approx t^{-θ} for some θ>0. We discover two different cooling regimes. If θ<1/α(slow cooling), the jump diffusion Z(t) has a non-trivial limiting distribution as t\to \infty, which is concentrated at the potential's local …
We develop a fully fledged theory of quantum dynamical patterns of behavior that are nonlocally induced. To this end we generalize the standard Laplacian-based framework of the Schrödinger picture quantum evolution to that employing nonlocal (pseudodifferential) operators. Special attention is paid to the Salpeter (here, $m\geq 0$) quasirelativistic equation and the evolution of various wave packets, in particular to their radial expansion in 3D. Foldy's synthesis of "covariant particle equations" is extended to encompass free Maxwell theory, which however is devoid of any "particle" content. …
We consider Levy flights subject to external force fields. This anomalous transport process is described by two approaches, a Langevin equation with Levy noise and the corresponding generalized Fokker-Planck equation containing a fractional derivative in space. The cases of free flights, constant force and linear Hookean force are analyzed in detail, and we corroborate our findings with results from numerical simulations. We discuss the non-Gibbsian character of the stationary solution for the case of the Hookean force, i.e. the deviation from Boltzmann equilibrium for long times. The possible…
The Levy-flight dynamics can stem from simple random walks in a system whose operational time (number of steps n) typically grows superlinearly with physical time t. Thus, this processes is a kind of continuous-time random walks (CTRW), dual to usual Scher-Montroll model, in which $n$ grows sublinearly with t. The models in which Levy-flights emerge due to a temporal subordination let easily discuss the response of a random walker to a weak outer force, which is shown to be nonlinear. On the other hand, the relaxation of en ensemble of such walkers in a harmonic potential follows a simple expo…
We discuss dual time evolution scenarios which, albeit running according to the same real time clock, in each considered case may be mapped among each other by means of an analytic continuation in time. This dynamical duality is a generic feature of diffusion-type processes. Technically that involves a familiar transformation from a non-Hermitian Fokker-Planck operator to the Hermitian operator (e.g. Schroedinger Hamiltonian), whose negative is known to generate a dynamical semigroup. Under suitable restrictions upon the generator, the semigroup admits an analytic continuation in time and ulti…
Complex dynamical systems which are governed by anomalous diffusion often can be described by Langevin equations driven by Lévy stable noise. In this article we generalize nonlinear stochastic differential equations driven by Gaussian noise and generating signals with 1/f power spectral density by replacing the Gaussian noise with a more general Lévy stable noise. The equations with the Gaussian noise arise as a special case when the index of stability alpha=2. We expect that this generalization may be useful for describing 1/f fluctuations in the systems subjected to Lévy stable noise.
The fractional Fokker-Planck equation, which contains a variable diffusion coefficient, is discussed and solved. It corresponds to the Lévy flights in a nonhomogeneous medium. For the case with the linear drift, the solution is stationary in the long-time limit and it represents the Lévy process with a simple scaling. The solution for the drift term in the form $λ\hbox{sgn}(x)$ possesses two different scales which correspond to the Lévy indexes $μ$ and $μ+1$ $(μ<1)$. The former component of the solution prevails at large distances but it diminishes with time for a given $x$. The fractional mom…
The generalized correlation approach, which has been successfully used in statistical radio physics to describe non-Gaussian random processes, is proposed to describe stochastic financial processes. The generalized correlation approach has been used to describe a non-Gaussian random walk with independent, identically distributed increments in the general case, and high-order correlations have been investigated. The cumulants of an asymmetrically truncated Levy distribution have been found. The behaviors of asymmetrically truncated Levy flight, as a particular case of a random walk, are conside…
For both Levy flight and Levy walk search processes we analyse the full distribution of first-passage and first-hitting (or first-arrival) times. These are, respectively, the times when the particle moves across a point at some given distance from its initial position for the first time, or when it lands at a given point for the first time. For Levy motions with their propensity for long relocation events and thus the possibility to jump across a given point in space without actually hitting it ("leapovers"), these two definitions lead to significantly different results. We study the first-pas…
We investigate the non-Langevin relative of the Lévy-driven Langevin random system, under an assumption that both systems share a common (asymptotic, stationary, steady-state) target pdf. The relaxation to equilibrium in the fractional Langevin-Fokker-Planck scenario results from an impact of confining conservative force fields on the random motion. A non-Langevin alternative has a built-in direct response of jump intensities to energy (potential) landscapes in which the process takes place. We revisit the problem of Lévy flights in superharmonic potential wells, with a focus on the extremally…
We consider different generalizations of the Fokker-Planck-equation devised to describe Levy processes in potential force fields. We show that such generalizations can proceed along different lines. On one hand, Levy statistics can emerge from the fractal temporal nature of the underlying process, i.e. a high variability in the rate of microscopic events. On the other hand, they may be a direct consequence of the scale-free spatial structure on which the process evolves. Although both forms considered lead to Boltzmann equilibrium, the relaxation patterns are quite different. As an example, ge…
We study the probability distribution function (pdf) of the position of a Lévy flight of index 0<α<2 in presence of an absorbing wall at the origin. The solution of the associated fractional Fokker-Planck equation can be constructed using a perturbation scheme around the Brownian solution (corresponding to α= 2), as an expansion in ε= 2 - α. We obtain an explicit analytical solution, exact at the first order in ε, which allows us to conjecture the precise asymptotic behavior of this pdf, including the first subleading corrections, for any α. Careful numerical simulations, as well as an exact c…
We study the photoluminescence spectra excited at an edge side of n-InP slabs and observed from the broadside. In a moderately doped sample the intensity drops off as a power-law function of the distance from the excitation - up to several millimeters - with no change in the spectral shape.The hole distribution is described by a stationary Levy-flight process over more than two orders of magnitude in both the distance and hole concentration. For heavily-doped samples, the power law is truncated by free-carrier absorption. Our experiments are near-perfectly described by the Biberman-Holstein tr…
Random walk simulation of the Levy flight shows a linear relation between the mean square displacement <r2> and time. We have analyzed different aspects of this linearity. It is shown that the restriction of jump length to a maximum value (lm) affects the diffusion coefficient, even though it remains constant for lm greater than 1464. So, this factor has no effect on the linearity. In addition, it is shown that the number of samples does not affect the results. We have demonstrated that the relation between the mean square displacement and time remains linear in a continuous space, while conti…
We consider a system of particles undergoing the branching and annihilating reactions A -> (m+1)A and A + A -> 0, with m even. The particles move via long-range Levy flights, where the probability of moving a distance r decays as r^{-d-sigma}. We analyze this system of branching and annihilating Levy flights (BALF) using field theoretic renormalization group techniques close to the upper critical dimension d_c=sigma, with sigma<2. These results are then compared with Monte-Carlo simulations in d=1. For sigma close to unity in d=1, the critical point for the transition from an absorbing to an a…
We consider the statistical properties of primary sequences of two-letter HP copolymers (H for hydrophobic and P for polar) designed to have water soluble globular conformations with H monomers shielded from water inside the shell of P monomers. We show, both by computer simulations and by exact analytical calculation, that for large globules and flexible polymers such sequences exhibit long-range correlations which can be described by Levy-flight statistics.
The non-Markovian stochastic dynamics involving Levy flights and a potential in the form of a harmonic and non-linear oscillator is discussed. The subordination technique is applied and the memory effects, which are nonhomogeneous, are taken into account by a position-dependent subordinator. In the non-linear case, the asymptotic stationary states are found. The relaxation pattern to the stationary state is derived for the quadratic potential: the density decays like a linear combination of the Mittag-Leffler functions. It is demonstrated that in the latter case the density distribution satisf…
We consider a Levy flyer of order alpha that starts from a point x0 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…
We study the photoluminescence spectra of n-doped InP bulk wafers, both in the reflection and the transmission geometries relative to the excitation beam. From the observed spectra we estimate the spatial distribution of minority carriers allowing for the spectral filtering due to re-absorption of luminescence in the wafer. This distribution unambiguously demonstrates a non-exponential drop-off with distance from the excitation region. Such a behavior evidences an anomalous photon-assisted transport of minority carriers enhanced owing to the high quantum efficiency of emission. It is shown tha…
The Topology of Rayleigh-Levy Flights in Two Dimensions
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Department of Physics, POSTECH（浦项科技大学物理系）；Université Paris-Saclay, CNRS, CEA, Institut de physique théorique（巴黎萨克雷大学、法国国家科学研究中心、法国原子能和替代能源委员会、理论物理研究所）；Institut d’Astrophysique de Paris（巴黎天文物理研究所）；Kyung Hee University, Dept. of Astronomy & Space Science（庆熙大学天文学与空间科学系）
Levy flights are known to be optimal search strategies in the particular case of revisitable targets. In the relevant situation of non revisitable targets, we propose an alternative model of bidimensional search processes, which explicitly relies on the widely observed intermittent behavior of foraging animals. We show analytically that intermittent strategies can minimize the search time, and therefore do constitute real optimal strategies. We study two representative modes of target detection, and determine which features of the search time are robust and do not depend on the specific charac…
We compare our results on empirical analysis of financial data with simulations of two stochastic models of the dynamics of stock market prices. The two models are (i) the truncated Lévy flight recently introduced by us and (ii) the ARCH(1) and GARCH(1,1) processes. We find that the TLF well describes the scaling and its breakdown observed in empirical data, while it is not able to properly describe the fluctuations of volatility empirically detected. The ARCH(1) and GARCH(1,1) models are able to describe the probability density function of price changes at a given time horizon, but both fail …
Intriguing phenomena such as subrecoil laser cooling of atoms, or aging phenomenon in glasses, have in common that the systems considered do not reach a steady-state during the experiments, although the experimental time scales are very large compared to the microscopic ones. We revisit some standard models describing these phenomena, and reformulate them in a unified framework in terms of lifetimes of the microscopic states of the system. A universal dynamical mechanism emerges, leading to a generic time-dependent distribution of lifetimes, independently of the physical situation considered.
We analyze two-dimensional (2D) random systems driven by a symmetric Lévy stable noise which, under the sole influence of external (force) potentials $Φ(x) $, asymptotically set down at Boltzmann-type thermal equilibria. Such behavior is excluded within standard ramifications of the Langevin approach to Lévy flights. In the present paper we address the response of Lévy noise not to an external conservative force field, but directly to its potential $Φ(x)$. We prescribe a priori the target pdf $ρ_*$ in the Boltzmann form $\sim \exp[- Φ(x)]$ and next select the Lévy noise of interest. Given suit…
A biological competition model where the individuals of the same species perform a two-dimensional Markovian continuous-time random walk and undergo reproduction and death is studied. The competition is introduced through the assumption that the reproduction rate depends on the crowding in the neighborhood. The spatial dynamics corresponds either to normal diffusion characterized by Gaussian jumps or to superdiffusion characterized by Lévy flights. It is observed that in both cases periodic patterns occur for appropriate parameters of the model, indicating that the general macroscopic collecti…
