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First exit time of a Lévy flight from a bounded region in RN
Abstract
In this paper we characterize the mean and the distribution of the first exit time of a Lévy flight from a bounded region in
N
-dimensional spaces. We characterize tight upper and lower bounds on the tail distribution of the first exit time, and provide the exact asymptotics of the mean first exit time for a given range of step-length distribution parameters.
... Moreover, noteworthy are simulations of radial LFs in two dimensions [7], the effect of Lévy noise on a gene transcriptional regulatory system [87], the study of the mean exit time and the escape probability of oneand two-dimensional stochastic dynamical systems with non-Gaussian noises [88][89][90]. The tail distribution of the first-exit time of LFs from a closed N-ball of radius R in a recursive manner was constructed in [91]. Very recently, extensive simulations of the space-fractional diffusion equation and the Langevin equation were used to investigate the first-passage properties of asymmetric LFs in a semi-infinite domain in [60]. ...
... For completely asymmetric LFs the first-passage of the two-sided exit problem was addressed in [68][69][70][71][72][73]. A different expression (instead of d α−1 in equation (91) it is d α ) for the MFPT of completely asymmetric LFs with 1 < α < 2 and β = 1 in dimensionless form was derived with the help of the Green's function method in [68] (see equation (1.8)). In [71] the distribution of the first-exit time from a finite interval for extremal two-sided α-stable probability laws with 1 < α < 2 and β = −1 was reported in the Laplace domain. ...
... In the right panels of figure 7 we show the MFPT for extremal α-stable processes with skewness β = 1 for two different interval lengths as function of the initial distance d from the right boundary. To compare the MFPT of extremal two-sided LFs with arbitrary α ∈ (1, 2) and β = 1 with that of Brownian motion, we employ equation (91) and obtain ...
We investigate the first-passage dynamics of symmetric and asymmetric L\'evy flights in a semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A comparison with results using the Langevin approach to L\'evy flights is presented. For the semi-infinite domain, in certain special cases analytic results are derived explicitly, and in bounded intervals a general analytical expression for the mean first-passage time of L\'evy flights with arbitrary skewness is presented. These results are complemented with extensive numerical analyses.
... Moreover, noteworthy are simulations of radial LFs in two dimensions [7], the effect of Lévy noise on a gene transcriptional regulatory system [87], the study of the mean exit time and the escape probability of oneand two-dimensional stochastic dynamical systems with non-Gaussian noises [88][89][90]. The tail distribution of the first-exit time of LFs from a closed N-ball of radius R in a recursive manner was constructed in [91]. Very recently, extensive simulations of the space-fractional diffusion equation and the Langevin equation were used to investigate the first-passage properties of asymmetric LFs in a semi-infinite domain in [60]. ...
... For completely asymmetric LFs the first-passage of the two-sided exit problem was addressed in [68][69][70][71][72][73]. A different expression (instead of d α−1 in equation (91) it is d α ) for the MFPT of completely asymmetric LFs with 1 < α < 2 and β = 1 in dimensionless form was derived with the help of the Green's function method in [68] (see equation (1.8)). In [71] the distribution of the first-exit time from a finite interval for extremal two-sided α-stable probability laws with 1 < α < 2 and β = −1 was reported in the Laplace domain. ...
... In the right panels of figure 7 we show the MFPT for extremal α-stable processes with skewness β = 1 for two different interval lengths as function of the initial distance d from the right boundary. To compare the MFPT of extremal two-sided LFs with arbitrary α ∈ (1, 2) and β = 1 with that of Brownian motion, we employ equation (91) and obtain 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 A c c e p t e d M a n u s c r i p t with ξ defined in equation (50). By solving for d, we find d = 2 cos (απ(1/2 − 1/α)) Γ(1 + α) ...
We investigate the first-passage dynamics of symmetric and asymmetric
Lévy flights in a semi-infinite and bounded intervals. By solving the space-fractional
diffusion equation, we analyse the fractional-order moments of the first-passage time
probability density function for different values of the index of stability and the
skewness parameter. A comparison with results using the Langevin approach to Lévy
flights is presented. For the semi-infinite domain, in certain special cases analytic
results are derived explicitly, and in bounded intervals a general analytical expression
for the mean first-passage time of Lévy flights with arbitrary skewness is presented.
These results are complemented with extensive numerical analyses.
... The influence of heavy tailed modeling methods has spread to many fields. Application areas for the modeling and statistical methods include finance [27], insurance [12], social networks and random graphs [11,4,25,26], mobility modeling for wireless phone users [15], parallel processing queueing models of cloud computing [14], models to optimize power usage when a mobile user changes between wifi and mobile networks [16]. ...
Regular variation of a multivariate measure with a Lebesgue density implies the regular variation of its density provided the density satisfies some regularity conditions. Unlike the univariate case, the converse also requires regularity conditions. We extend these arguments to discrete mass functions and their associated measures using the concept that the the mass function can be embedded in a continuous density function. We give two different conditions, monotonicity and convergence on the unit sphere, both of which can make the discrete function embeddable. Our results are then applied to the preferential attachment network model, and we conclude that the joint mass function of in- and out-degree is embeddable and thus regularly varying.
... derived with the corresponding PDF p(t|x 0 ) and depicted in Fig. 2. Clearly, the survival probability denotes the probability that a process starting at x(0) = x 0 = 0 has not reached or crossed up to time t the levels ±L. Note that, by construction, the process described by Eq. (1) is Markovian, which remains in line with the observation of exponential asymptotics in Fig. 2. The behavior is well documented in simulations of Lévy flights [15,20] and can be inferred by an estimation of the lower and upper bounds [21][22][23] for tails of S(t|x 0 ) or from the master equation [18,24,25]. It can also be deduced by a separation of variables [16,20], ...
L\'evy flights and L\'evy 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 L\'evy flights is associated with their pathological physical properties, which in turn are resolved by the concept of L\'evy walks. Here, we explore L\'evy flights and L\'evy walks models on bounded domains examining their differences and analogies. We investigate analytically and numerically whether and under which conditions both approaches yield similar results in terms of selected statistical observables characterizing the motion: the survival probability, mean first passage time and stationary PDFs. It is demonstrated that similarity of models is affected by the type of boundary conditions and value of the stability index defining asymptotics of the jump length distribution.
... Tiandong Wang tw398@cornell.edu 1 wireless phone users (Kim et al. 2015), parallel processing queueing models of cloud computing (Jiang et al. 2013), models to optimize power usage when a mobile user changes between wifi and mobile networks (Kim et al. 2014). ...
Regular variation of a multivariate measure with a Lebesgue density implies
the regular variation of its density provided the density satisfies some
regularity conditions. Unlike the univariate case, the converse also requires
regularity conditions. We extend these arguments to discrete mass functions and
their associated measures using the concept that the the mass function can be
embedded in a continuous density function. We give two different conditions,
monotonicity and convergence on the unit sphere, both of which can make the
discrete function embeddable. Our results are then applied to the preferential
attachment network model, and we conclude that the joint mass function of in-
and out-degree is embeddable and thus regularly varying.
The term 'Lévy flights' was coined by Benoit Mandelbrot, who thus poeticized α-stable Lévy random motion, a Markovian process with stationary independent increments distributed according to the α-stable Lévy probability law. Contrary to the Brownian motion, the trajectories of the α-stable Lévy motion are discontinous, that is exhibit jumps. This feature implies that the process of first passage through the boundary of a given space domain, or the first escape, is different from the process of first arrival (hit) at the boundary. Here we investigate the properties of first escapes and first arrivals for Lévy flights and explore how the asymptotic behavior of the corresponding (passage and hit) probabilities is sensitive to the size of the domain. In particular, we find that the survival probability to stay in a large enough, finite domain has a universal Sparre Andersen temporal scaling , which is transient and changes to an exponential non-universal decay at longer times. Also, the probability to arrive at a finite domain possesses a similar transient Sparre Andersen universality that turns into a non-universal and slower power-law decay in course of time. Finally, we demonstrate that the probability density of the leapover length ℓ over the boundary, related to overshooting events, has an intermediate asymptotics () which is inherent for the escape from a semi-infinite domain. However, for larger leapovers the probability density decays faster according to the law. Thus, we find that the laws derived for the α-stable processes on the semi-infinite domain, manifest themselves as transients for Lévy flights on the finite domain.
In this paper, we consider the fractional Laplacian ( ) = 2 on an open subset in Rd with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such Dirichlet fractional Laplacian in C1;1 open sets. This heat kernel is also the transition density of a rotationally symmetric -stable process killed upon leaving a C1;1 open set. Our results are the rst sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.
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 their
efficiency as stochastic search mechanisms and to discriminate them from other
types of search processes.
The central result of this paper consists in proving that all stable densities are bell-shaped (i.e. its kth derivative has exactly k zeros and they are simple) thereby generalizing the well-known property of the normal distribution and the associated Hermite polynomials.
We consider a Lévy 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 Lévy flights and Lévy 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 Lévy 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.
We present the analysis of the first passage time problem on a finite interval for the generalized Wiener process that is driven by Lévy 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.
The fractional Laplacian operator -(-delta)(alpha/2) appears in a wide class of physical systems, including Lévy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely, hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalue spectrum are also obtained.
We present exact results for the spectrum of the fractional Laplacian in a bounded domain and apply them to First Passage Time (FPT) Statistics of L\'evy flights. We specifically show that the average is insufficient to describe the distribution of FPT, although it is the only quantity available in the existing literature. In particular, we show that the FPT distribution is not peaked around the average, and that knowledge of the whole distribution is necessary to describe this phenomenon. For this purpose, we provide an efficient method to calculate higher order cumulants and the whole distribution.
1. In a paper [1] I have proved some theorems concerning the signs of sums of symmetrically distributed random variables. Later in a note [2] I have stated some results on the signs of sums of independent random variables. It was promised that complete proofs should be publisbed elsewhere. In this paper the results in [1] are generalized. Furthermore the results proved by D. A. Darling in [4] for independent variables are extended to symmetrically dependent variables. A complete proof of one of the results stated in [2] is also given. This result, however, is restricted to independent variables. It is the idea in the original proof of this result, which now forms the basis of the proof of Theorem 1 below.
We consider a Lévy flyer of order α that starts from a point x0 on an interval [0,L] with absorbing boundaries. We find a closed-form expression for an arbitrary average quantity, characterizing the trajectory of the flyer, such as mean first passage time, average total path length, probability to be absorbed by one of the boundaries. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for these quantities in the continuous limit. We find numerically the eigenfunctions and the eigenvalues of these equations. We study how the results of Monte-Carlo simulations of the Lévy flights with different flight length distributions converge to the continuous approximations. We show that if x0 is placed in the vicinity of absorbing boundaries, the average total path length has a minimum near α=1, corresponding to the Cauchy distribution. We discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere.
Let be a sequence of independent, identically distributed non-degenerate random variables taking values in and . Define for and . Then if , it is proved that for for , while the is 0 and the is for . Some alternative characterizations of the indices are obtained as well as the analogous results for Levy processes.