First passage properties of asymmetric Lévy flights
A. Padash, A. V. Chechkin, B. Dybiec, I. Pavlyukevich, B. Shokri, R. Metzler
FFirst passage properties of asymmetric Lévy flights
Amin Padash † ,(cid:93) , Aleksei V. Chechkin (cid:93), ‡ , Bartlomiej Dybiec (cid:91) ,Ilya Pavlyukevich § , Babak Shokri † , ¶ , & Ralf Metzler (cid:93) † Physics Department of Shahid Beheshti University, 19839-69411 Tehran, Iran (cid:93)
Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm,Germany ‡ Akhiezer Institute for Theoretical Physics, 61108 Kharkov, Ukraine (cid:91)
Marian Smoluchowski Institute of Physics and Mark Kac Center for ComplexSystems Research, Jagiellonian University, ul. St. Lojasiewicza 11, 30-348 Krakow,Poland § Friedrich Schiller University Jena, Faculty of Mathematics and Computer Science,Institute for Mathematics, 07737 Jena, Germany ¶ Laser and Plasma Research Institute, Shahid Beheshti University, 19839-69411Tehran, IranE-mail: [email protected] (Corresponding author)
Abstract.
Lévy Flights are paradigmatic generalised random walk processes, inwhich the independent stationary increments—the "jump lengths"—are drawn froman α -stable jump length distribution with long-tailed, power-law asymptote. As aresult, the variance of Lévy Flights diverges and the trajectory is characterised byoccasional extremely long jumps. Such long jumps significantly decrease the probabilityto revisit previous points of visitation, rendering Lévy Flights efficient search processesin one and two dimensions. To further quantify their precise property as randomsearch strategies we here study the first-passage time properties of Lévy Flights inone-dimensional semi-infinite and bounded domains for symmetric and asymmetricjump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approachis based on the space-fractional diffusion equation for the probability density function,from which the survival probability is obtained for different values of the stable index α and the skewness (asymmetry) parameter β . The other approach is based onthe stochastic Langevin equation with α -stable driving noise. Both methods havetheir advantages and disadvantages for explicit calculations and numerical evaluation,and the complementary approach involving both methods will be profitable forconcrete applications. We also make use of the Skorokhod theorem for processeswith independent increments and demonstrate that the numerical results are in goodagreement with the analytical expressions for the probability density function of thefirst-passage times.
1. Introduction
Normal Brownian motion described by Fick’s second law, the diffusion equation,is characterised by the linear time dependence (cid:104) x ( t ) (cid:105) (cid:39) t of the mean squared a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t irst passage properties of asymmetric Lévy flights (cid:104) x ( t ) (cid:105) (cid:39) t κ of the MSD [2, 3]. Depending on the value of theanomalous diffusion exponent κ we distinguish between subdiffusion for < κ < ,normal, Fickean diffusion for κ = 1 , superdiffusion for < κ < , and ballistic, wave-like motion for κ = 2 . The range κ > is sometimes referred to as hyperdiffusion. Thetheoretical description of anomalous diffusion phenomena of physical particles (passiveor active) often requires a radical departure from the classical formalism for Brownianmotion. Namely, effects of energetic or spatial disorder, collective dynamics, or non-equilibrium conditions need to be addressed with more complex approaches [2, 4]. Forinstance, fractional Brownian motion [5] is a process in which the Langevin equation isdriven by Gaussian yet power-law correlated noise (fractional Gaussian noise) effectingboth sub- and superdiffusion. The generalised Langevin equation [6] includes a memoryintegral with a kernel, that balances the input fractional Gaussian noise and effectsa thermalised process. Processes with explicitly time or position dependent diffusioncoefficients such as scaled Brownian motion [7] or heterogeneous diffusion processes[8], respectively, also lead to sub- and superdiffusion. Diffusion on fractals [9] due tothe fact that the particle in the highly ramified environment often has to back-trackits motion, has similar characteristics as subdiffusive fractional Brownian motion. Wefinally mention the continuous time random walk model, in which the standard Pearsonwalk was generalised to include continuous waiting times [10]. When the distributionof waiting times becomes scale-free, with diverging characteristic waiting time, thecontinuous time random walk process is subdiffusive [11, 12]. Conversely, when thecontinuous time random walk has a finite characteristic waiting time but is equippedwith a scale-free distribution of jump lengths with power-law asymptote λ ( x ) ∼ | x | − − α ( < α < ) the resulting process is a "Lévy flight" (LF). As in this case the variance ofthe process diverges, diffusion can be characterised in terms of rescaled fractional ordermoments (cid:104)| x ( t ) | η (cid:105) /η (cid:39) t /α with < η < α [3, 13, 14, 15, 16, 17]. Mathematically,asymptotic power-law forms of the jump length distribution can be explained by thegeneralised central limit theorem [2, 18, 19, 20], which gives rise to the much higherlikelihood for extremely long jumps [21, 22, 23] in comparison to conventional Pearsonrandom walks.Lévy stable laws play a crucial role in the statistical description of scale invariantstochastic processes [21, 24] not only in physical contexts but also in biological, chemical,geophysical, sociological, economical or financial systems, among others. In physicsLévy statistics were demonstrated to explain deviations of complex systems from theGaussian paradigm, inter alia, for the power-law blinking of nano-scale light emitters[25], diffusive transport of light [26], photons in hot atomic vapours [27], tracer particlesin a rotating flow [28], passive scalars in vortices in shear [29], anomalous diffusion indisordered media [2], weakly chaotic and Hamiltonian systems [30, 31], in the divergenceof kinetic energy fluctuations of a single ion in an optical lattice [32], fluctuations in thetransition energy of a single molecule embedded in a solid [33], in the interaction of twolevel systems with single molecules [34], the distribution of random single molecule line irst passage properties of asymmetric Lévy flights irst passage properties of asymmetric Lévy flights ℘ ( t ) . While the mean first-passage time (cid:104) t (cid:105) = (cid:82) ∞ t℘ ( t ) dt can capture some aspects of this dynamics, ‡ the full information encoded in ℘ ( t ) provides significant additional insight [114, 115, 116]. Here we study the first-passageproperties for a general class of α -stable Lévy laws. We go beyond previous approaches[113, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128] focusing on symmetricand one-sided α -stable relocation distributions and consider α -stable laws with arbitraryasymmetry in semi-infinite and bounded domains. Our approach is based on theconvenient formulation of LFs in terms of the space-fractional diffusion equation. Wederive these integro-differential equations for LFs based on general asymmetric α -stabledistributions of relocation lengths in finite domains, and thus go beyond studies of theexit time and escape probability in bounded domain for symmetric LFs [129, 130, 131].An important aspect of this study is that we complement our results with numericalanalyses of the (stochastic) Langevin equation for LFs and show how both approachescomplement each other.The paper is organised as follows. In section 2 we define Lévy stable laws and theassociated fractional diffusion equation. In section 3 we set up our numerical modelfor the fractional diffusion equation and the associated Langevin equation. Moreover,a comparison between the numerical method and α -stable distributions for symmetricand asymmetric density functions is presented. Section 4 then presents the numericalresults for the survival probability and first-passage time density for both symmetricand asymmetric probability density functions. Our findings are compared with resultsderived from the Skorokhod theorem for symmetric, one-sided, and extremal two-sidedstable distributions. We draw our conclusion in section 5. In the Appendices, we presentdetails of several derivations. α -stable processes and space-fractional diffusion equations An α -stable Lévy process X ( t ) with X (0) = 0 and probability density function (PDF) (cid:96) α,β ( x, t ) is fully specified by its characteristic function in the Fourier domain as [18, 132] ˆ (cid:96) α,β ( k, t ) = ∞ (cid:90) −∞ (cid:96) α,β ( x, t ) exp(i kx )d x = exp( − tK α | k | α [1 − i β sgn( k ) ω ( k, α )] + i µkt ) , (1)where the index α with < α ≤ is called the index of stability or Lévy index, andthe skewness parameter β is allowed to vary within the limits − ≤ β ≤ . Further,the generalised diffusion coefficient K α > is a scale parameter, the shift parameter µ ‡ Often, a better choice is the mean inverse first-passage time (cid:104) /t (cid:105) [113]. irst passage properties of asymmetric Lévy flights ω is defined as ω ( k, α ) = (cid:40) tan( πα ) , α (cid:54) = 1 − π ln | k | , α = 1 . (2)In the real space-time domain the PDF of the α -stable distribution can be expressedvia elementary functions for the following three cases:(a) Gaussian distribution, α = 2 , β irrelevant: (cid:96) ( x, t ) = 1 √ πK t exp (cid:18) − ( x − µt ) K t (cid:19) , −∞ < x < ∞ . (3 a )(b) Cauchy distribution, α = 1 , β = 0 : (cid:96) , ( x, t ) = 1 π K t ( x − µt ) + ( K t ) , −∞ < x < ∞ . (3 b )In physics, the Cauchy distribution is also often called a Lorentz distribution.(c) Lévy-Smirnov distribution, α = 1 / , β = 1 : (cid:96) / , ( x, t ) = K / t (cid:112) π ( x − µt ) exp (cid:18) − ( K / t ) x − µt ) (cid:19) , x ≥ . (3 c )Schneider reported the representation of general Lévy stable densities in terms ofFox H -functions [133, 134, 135]. Somewhat simpler representations for rational indices α and β are given in [136, 137]. More information on Lévy stable densities and theirparametrisation are provided in Appendix A.Physically, the parameter µ accounts for a constant drift in the system. In this paperwe consider the first-passage process in the absence of a drift, thus in what follows weset µ = 0 . Let us first consider the case α (cid:54) = 1 and − ≤ β ≤ . The correspondingspace-fractional diffusion equation for the PDF (cid:96) α,β ( x, t ) then reads ∂(cid:96) α,β ( x, t ) ∂t = K α D αx (cid:96) α,β ( x, t ) , (cid:96) α,β ( x,
0) = δ ( x ) , (4)where D αx is the space-fractional derivative operator, D αx (cid:96) α,β ( x, t ) = L α,β −∞ D αx (cid:96) α,β ( x, t ) + R α,β x D α ∞ (cid:96) α,β ( x, t ) , (5)that we compose of −∞ D αx and x D α ∞ , the left and right hand side space-fractionaloperators, respectively. We use the Caputo form of the operators defined by ( n − <α < n ) [138] −∞ D αx f ( x ) = 1Γ( n − α ) x (cid:90) −∞ f ( n ) ( ζ )( x − ζ ) α − n +1 d ζ (6 a ) x D α ∞ f ( x ) = ( − n Γ( n − α ) ∞ (cid:90) x f ( n ) ( ζ )( ζ − x ) α − n +1 d ζ (6 b )and L α,β and R α,β are the left and right weight coefficients, defined as [52, 53] L α,β = − β απ ) , R α,β = − − β απ ) . (7) irst passage properties of asymmetric Lévy flights a ) and (6 b ) have the forms [138] −∞ D αx f ( x ) ÷ ( − i k ) α ˆ f ( k ) = | k | α exp (cid:18) − απ i2 sgn( k ) (cid:19) ˆ f ( k ) , (8 a ) x D α ∞ f ( x ) ÷ (i k ) α ˆ f ( k ) = | k | α exp (cid:18) απ i2 sgn( k ) (cid:19) ˆ f ( k ) , (8 b )where ÷ defines the Fourier transform pairs.For the case α = 1 and β = 0 we have L , = R , = 1 /π [139], and instead ofequation (5) we find ∂(cid:96) , ( x, t ) ∂t = − K α ∂∂x H (cid:110) (cid:96) , ( x, t ) (cid:111) , (9)where H is the Hilbert transform H (cid:110) f ( x ) (cid:111) = 1 π − ∞ (cid:90) −∞ f ( ζ ) x − ζ d ζ, (10)in terms of the principle value integral − (cid:82) . In Fourier space the Hilbert transform hasthe simple form H (cid:110) f ( x ) (cid:111) ÷ isgn( k ) ˆ f ( k ) . (11)In what follows we do not consider the particular case α = 1 , β (cid:54) = 0 since it cannotbe described in terms of a space-fractional operator. For all other choices of theparameters by substitution of relations (8 a ), (8 b ), and (11) into equation (4) we recoverthe characteristic function (1) of the α -stable process after Fourier transform.
3. Numerical scheme
To determine the first-passage properties of α -stable processes we will employ differentnumerical schemes based on the space-fractional diffusion equation and the Langevinequation for LFs. We here detail their implementation. There are several numerical methods to solve space-fractional diffusion equations, suchas the finite difference [140, 141] and finite element [142, 143, 144] methods as well as thespectral method [145, 146]. In this paper we use the finite difference method that usesdifferential quotients to replace the derivatives in the differential equations. The domainis partitioned in space and time, and approximations of the solution are computed. Dueto causality we use forward differences in time on the left hand side of equation (4), ∂∂t f ( x i , t j ) ≈ f j +1 i − f ji ∆ t , (12)where f ji = f ( x i , t j ) , x i = ( i − I/ x , and t j = j ∆ t , where ∆ x and ∆ t are step sizes inposition and time, respectively. The i and j are non-negative integers, i = 0 , , , . . . , I ,and further x = − L , x I = L , and ∆ x = 2 L/I . Similarly, j = 0 , , , . . . , J − , t = 0 , irst passage properties of asymmetric Lévy flights t J = t , and ∆ t = t/J . Absorbing boundary conditions for the determination of thefirst-passage event imply f j = f jI = 0 for all j . The integrals on the right hand side ofequation (4) are discretised as follows. For < α < , x i (cid:90) − L f (1) ( ζ, t j )( x i − ζ ) α d ζ ≈ i (cid:88) k =1 f jk − f jk − ∆ x x k (cid:90) x k − x i − ζ ) α d ζ, (13 a )for the left side derivative, and L (cid:90) x i f (1) ( ζ, t j )( ζ − x i ) α d ζ ≈ I − (cid:88) k = i f jk +1 − f jk ∆ x x k +1 (cid:90) x k ζ − x i ) α d ζ, (13 b )for the right side derivative. Thus the idea is to approximate only the derivative by thedifferences. The integral kernel is then calculated explicitly. For the estimation of theerror in this scheme we refer to Appendix C. For the case < α < we use the centraldifference approximation, namely, x i (cid:90) − L f (2) ( ζ, t j )( x i − ζ ) α − d ζ ≈ i (cid:88) k =1 f jk +1 − f jk + f jk − (∆ x ) x k (cid:90) x k − x i − ζ ) α − d ζ, (14 a )for the left side, and L (cid:90) x i f (2) ( ζ, t j )( ζ − x i ) α − d ζ ≈ I − (cid:88) k = i f jk +1 − f jk + f jk − (∆ x ) x k +1 (cid:90) x k ζ − x i ) α − d ζ, (14 b )for the right side. For the special case α = 1 , β = 0 by using the discrete Hilberttransform [147] we approximate the derivative in space, namely − ddx ( H { f ( x, t ) } ) ≈ − π i (cid:88) k =1 f jk − f jk − ∆ x i − k ) + 1 − π I − (cid:88) k = i f jk − f jk +1 ∆ x k − i ) + 1 . (15)For further details of the numerical scheme we refer the reader to Appendix B. Toimprove the stability we use the Crank-Nicolson method. By substitution of equations(12) to (15) into equation (4) we obtain A f j +1 = B f j , (16)in which the coefficients A and B have matrix form with dimension ( I + 1) × ( I + 1) and j = 0 , , , . . . , J − . In the numerical scheme the initial condition f ( x,
0) = δ ( x ) is approximated as f ( x i , t ) = (cid:40) x , i = L/ ∆ x , otherwise . (17 a )For the setup used in our numerical simulations, see section 4 below, the initial pointis f ( x,
0) = δ ( x − x (0)) at x (0) = L − d , where d < L is the distance of x (0) from the irst passage properties of asymmetric Lévy flights f ( x i , t ) = (cid:40) x , i = (2 L − d ) / ∆ x , otherwise . (17 b )In the next step the time evolution of the PDF is obtained by using the absorbingboundary conditions f j = f jI = 0 for all j and applying to the matrix coefficients A and B . The fractional diffusion equation (4) can be related to the Langevin equation [14, 123,148] dd t x ( t ) = K /αα ζ ( t ) , (18)where ζ ( t ) is white Lévy noise characterised by the same α and β parameters as thespace-fractional operator (5) and with unit scale parameter. The Langevin equation (18)provides a microscopic (trajectory-wise) representation of the space-fractional diffusionequation (4). Therefore, from an ensemble of trajectories generated from equation (18)it is possible to estimate the time dependent PDF whose evolution is described byequation (4). In numerical simulations Lévy flights can be described by the discretisedform of the Langevin equation x ( t + ∆ t ) = x ( t ) + K /αα (∆ t ) /α ζ, (19)where ζ stands for the α -stable random variable with a unit scale parameter [18, 149] andthe same index of stability α and skewness β parameters as in equation (18). Relation(19) is exactly the Euler-Maruyama approximation [150, 151, 152] to the general α -stable Lévy process. From the trajectories x ( t ) , see equations (18) and (19), it is alsopossible to estimate the first-passage time τ as τ = min { t : | x ( t ) | ≥ L } . (20)From the ensemble of first-passage times it is also possible to estimate the survivalprobability S ( t ) , the complementary cumulative density of first-passage times. Moreprecisely, the initial condition is S (0) = 1 , and at every recorded first-passage eventat time τ i , S ( t ) is decreased by the amount /N , where N is the number of records offirst-passage times. If a given estimation of the first-passage time is recorded k timesthe survival probability is decreased by k/N . For a finite set of first-passage timesthere exists a small fraction of very large first-passage times. Therefore, this estimationbecomes poorer with increasing time t . In the next section we present a comparisonbetween the numerical solution of equation (4) and the α -stable probability laws withcharacteristic function (1). irst passage properties of asymmetric Lévy flights -1 t = 0.5t = 1.0t = 3.0t = 0.5t = 1.0t = 3.0 -10 -5 0 5 1010 Figure 1.
Probability density function of symmetric α -stable probability law with β = 0 and interval length L = 100 . Left: for different sets of α (see figure legend) fortime t = 1 . Right: for α = 0 . at different times. In both figures we use ∆ t = 0 . as time step and ∆ x = 0 . for the step length. In the insets we show a zoom intothe central part of the PDF on a log-linear scale. The symbols show the solution ofthe diffusion equation (4) while the lines show the α -stable distributions obtained fromFourier inversion of the characteristic function, displaying excellent agreement. Theblack solid line shows the asymptotic behaviour of the PDF (main panels). Effects ofthe absorbing boundary condition at L = 100 can be seen as fairly sudden drops of thePDF while at the times shown the central part of the PDF remains hardly affected. α -stable distributions We now show that the difference scheme for the space-fractional diffusion equationprovides excellent agreement with the theoretical results for the shapes of α -stableprobability densities. To this end we use a MATLAB code to obtain the inverse Fouriertransform of the characteristic function [153]. This programme employs Zolotarev’sso-called M-form for the parametrisation of α -stable distributions with parameters α , β M , µ M , and K Mα , while in the main text we use the A-form with parameters α , β A , µ A and K Aα [154]. Thus, along with the code we use the corresponding change of thedistribution’s parameters, see Appendix A and, in particular, equation (A.8) for details. α -stable distributions. In this section we show a comparison between α -stable distributions obtained by inverse Fourier transform of the characteristicfunction with the numerical solution of the space-fractional diffusion equation in abounded domain [ − L, L ] for skewness parameter β = 0 .We use absorbing boundary conditions and a finite domain with half length L = 100 in one dimension, the initial condition is a Dirac δ -function located at x (0) = 0 . Theprobability density function for β = 0 and different sets of the index of stability α at t = 1 is displayed in figure 1 on the left. The tails display the correct power-law scaling.In the right panel of figure 1 we show the PDF for α = 0 . and β = 0 at different times.The insets focus on the central part of the PDFs. In all cases and over the entire plotted irst passage properties of asymmetric Lévy flights α -stable distributions. Asymmetric stable distributions with non-zero values of the skewness β may occur in various situations, for instance, when in arandom walk the left and right diffusion coefficients are different. In figure 2 (top) weplot the PDF with α = 1 / at t = 1 for different values of the skewness parameter β .On the left side in the main panel the negative side of the tails is shown, on the rightside we display the positive side of the tails. Figure 2 (bottom) analogously shows thePDF for α = 1 . and different β at time t = 1 . Note that for β = 1 and < α < thePDF is completely one-sided on the positive axis and does not possess a left tail.By comparison we see that again the numerical scheme for solving the space-fractional diffusion equation produces solutions that are in very good agreement withthe numerical results for the α -stable laws. In the following we study the first-passageproperties of random walk processes with α -stable jump length distribution obtainedfrom two numerical methods: the space-fractional diffusion equation and the Langevinapproach. We also compare the numerical method for solving the space-fractionaldiffusion equation with results following from Skorokhod’s theorem.
4. Survival probability and first-passage properties
The survival probability and the first-passage time are observable statistical quantitiescharacterising the stochastic motion in bounded domains with absorbing boundaryconditions. In the following we investigate the properties of the survival probabilityand the first-passage time density in a finite interval for symmetric and asymmetric α -stable laws underlying the space-fractional diffusion equation. To this end we use thesetup shown in figure 3, in which absorbing boundaries are located at − L and L , andthe initial point of the initial δ -distribution is located the distance d away from the rightboundary.The probability that at time t the random walker is still present in the interval [ − L, L ] is defined as the survival probability [155] S ( t ) = L (cid:90) − L (cid:96) α,β ( x, t )d x, (21)and the first-passage time PDF is given by the negative time derivative, ℘ ( t ) = − d S ( t )d t . (22)Therefore, in Laplace domain with initial condition S (0) = 1 the relation between thesurvival probability and the first-passage time reads ℘ ( u ) = 1 − uS ( u ) . (23) irst passage properties of asymmetric Lévy flights -10 -10 -10 -2 -4 -3 -2 -1 = 0.0 = 0.5 = 1.0 = 0.0 = 0.5 = 1.0 -10 -5 0 5 1010 -2 -4 -3 -2 -1 = 0.0 = 0.5 = 1.0 = 0.0 = 0.5 = 1.0 -10 -10 -10 -10 -1 -6 -4 -2 = 0.0 = 0.5 = 1.0 = 0.0 = 0.5 = 1.0 -1 -6 -4 -2 = 0.0 = 0.5 = 1.0 = 0.0 = 0.5 = 1.0 -10 -5 0 5 1000.10.20.3 Figure 2.
Probability density function of α -stable probability law for differentskewness parameters β at time t = 1 . Top left: negative side of the PDF for α = 0 . (the inset focuses on the central part of the PDF on the log-linear scale). Top right:positive side of the PDF for α = 0 . . Bottom left: negative side of the PDF for α = 1 . .Bottom right: positive side of the PDF for α = 1 . (the inset shows the central partof the PDF on log-linear scale). For each panel we use L = 100 , ∆ t = 0 . , and ∆ x = 0 . . The symbols show the solution of the diffusion equation (4) and the linesshow α -stable distributions obtained from Fourier inversion. The short black solid linesshow the asymptotic behaviour of the PDF (main panels). We now first consider the survival probability for symmetric α -stable laws in a semi-infinite and finite interval and demonstrate how the asymptotic properties change withthe length of the interval. Furthermore, we compare the results obtained from thenumerical difference scheme in section 3.1 with the Langevin equation approach, beforeembarking for the study of asymmetric α -stable laws. α -stable laws Here we study the properties of α -stable processes in domains restricted by one ortwo boundaries. In figure 4 we show the survival probability for different α and two irst passage properties of asymmetric Lévy flights d (0) x L L (cid:0) Figure 3.
Schematic of the setup used in our approach: in the interval of length L the initial probability density function is given by a δ -distribution located at x (0) = L − d , where d is the distance from the right boundary. At both intervalboundaries we implement absorbing boundary conditions, that is, when the particlehits the boundaries or attempts to move beyond them, it is absorbed. -3 -2 -1 = 0.5 LE = 1.0 LE = 1.5 LE = 2.0 LE = 0.5 DE = 1.0 DE = 1.5 DE = 2.0 DE -3 -2 -1 = 0.5 LE = 1.0 LE = 1.5 LE = 2.0 LE = 0.5 DE = 1.0 DE = 1.5 DE = 2.0 DE Figure 4.
Survival probability for symmetric α -stable laws ( β = 0 ) in log-linear scalewith distance d = 0 . of the initial point from the right boundary and interval halflength L = 1 (left) and L = 10 (right), for different indices α . Symbols show resultsfrom numerical solution of the space-fractional diffusion equation and lines correspondto the Langevin equation simulation. Results constructed for the diffusion equationuse time step ∆ t = 0 . and space increment ∆ x = 0 . . In the Langevin dynamicssimulations the time step is ∆ t = 0 . , data were averaged over N = 10 runs. different interval lengths L based on the difference scheme solution of the space-fractionaldiffusion equation and simulations of the Langevin dynamics. The results constructedwith both methods are in very good agreement. The data in figure 4 clearly show anexponential decay (in analogy to the escape dynamics of Lévy flights from a confiningpotential [156, 157]). For the short interval the exponential behaviour sets in almostimmediately on the linear time axis in the plot, while for the longer interval a crossoverbehaviour is visible, as we will see below. irst passage properties of asymmetric Lévy flights -3 -2 -1 = 0.5 DE = 1.0 DE = 1.5 DE = 2.0 DE < x ( ) > - L = 1L = 10
Figure 5.
Left: Survival probability for symmetric α -stable processes ( β = 0 ) inlog-linear scale with d = 0 . and L = 1 , for different α . Symbols show results from thenumerical solution of the space-fractional diffusion equation and solid lines correspondto the exponential approximation (24). Right: inverse mean first-passage time (cid:104) τ x (0) (cid:105) − versus stable index α for two values of the interval half length L = 1 and L = 10 , for d = 0 . . Interestingly, we see in figure 4 that the trend of decay reverses with respect tothe stable index α . To understand this behaviour of the survival probability we use thefollowing approximation of the survival probability of symmetric Lévy flights in finiteintervals, S ( t | x (0)) ≈ exp (cid:0) −(cid:104) τ x (0) (cid:105) − t (cid:1) , (24)where the mean first-passage time (cid:104) τ x (0) (cid:105) = (cid:90) ∞ S ( t | x (0))d t (25)is given by [158, 159] (cid:104) τ x (0) (cid:105) = ( L − | x (0) | ) α/ Γ(1 + α ) K α . (26)Figure 5 compares this approximation with the numerical solution of the space-fractionaldiffusion equation. As we can see from the left panel, relations (24) and (26) agree verywell with the numerical results for L = 1 : an increase of the interval length L leads toa decrease of (cid:104) τ x (0) (cid:105) − for larger α (right panel of figure 5).For a semi-infinite domain with absorbing boundary condition it is well knownthat the first-passage time density for any symmetric jump length distribution in aMarkovian setting has the universal Sparre Andersen asymptotic ℘ ( t ) (cid:39) t − / (and thus S ( t ) (cid:39) t − / [155, 160, 161]. This is exactly our setting here for the symmetric casewith β = 0 , and the Sparre Andersen universality was consistently corroborated forsymmetric LFs in a number of works, inter alia, [117, 123, 124, 125].In figure 6 we study what happens at intermediate times in the case of a finiteinterval, before the terminal exponential shoulder cuts off the survival probability, as irst passage properties of asymmetric Lévy flights / for the survivalprobability. The onset of the hard exponential cutoff is shifted to longer times withincreasing interval size L , in which, on average, it takes the particles longer to explorethe full extent of the domain. This, of course, is fully consistent with results for normaldiffusion as well as continuous time random walk subdiffusion subordinated to regularrandom walks [114, 115, 116, 155, 162], compare also the discussion of the area fillingdynamics of LFs [163]. Moreover, we see that the results from numerical solution of thefractional diffusion equation and simulations of the Langevin equation almost perfectlyagree with each other. The lines without symbols in the top left of figure 6 correspondto cases when the numerical approach based on the fractional diffusion equation did notconverge well. We note that one has to increase the value of L with decreasing α in orderto meet the Sparre-Andersen scaling for a semi-infinite interval. This is intuitively clear,as smaller α enhances the likelihood of longer jumps and thus effects a higher probabilityof interaction with the interval boundaries at fixed L .In figure 7 for the interval size L = 100 we show the survival probability in the leftpanel along with the the first-passage time density in the right panel, for different α and β = 0 . Consistently, the transient Sparre Andersen scaling is passed on from the power-law exponent / for the survival probability to the exponent / of the first-passagedensity.In the theory of a general class of Lévy processes, that is, homogeneous randomprocesses with independent increments, there exists a theorem, that provides ananalytical expression for the PDF of first passage times in a semi-infinite interval, oftenreferred to as the Skorokhod theorem [132, 164]. Based on this theorem the asymptoticexpression for symmetric α -stable laws, the first-passage time PDF is (Appendix D)[125] ℘ ( t ) ∼ d α/ α √ πK α Γ( α/ t − / , (27)which specifies an exact expression for the prefactor of the Sparre Andersen power-law.For Brownian motion ( α = 2 ), the PDF for the first-passage time has the well knownLévy-Smirnov form [165] ℘ ( t ) = d √ πK t exp (cid:18) − d K t (cid:19) , (28)that also emerges from the Skorokhod theorem in the limit α = 2 . Equation (28) isexact for all times [165, 166], and apart from the Sparre Andersen law ℘ ( t ) (cid:39) t − / it includes the hard short time exponential cutoff reflecting the fact that it takes theparticle a typical time ∝ d /K to reach the absorbing boundary. α -stable laws The case of asymmetric α -stable laws is mathematically more involved and also has beenless well studied numerically. We now present results for the survival probability and irst passage properties of asymmetric Lévy flights -3 -2 -1 L = 10 LEL = 10 LEL = 10 LEL = 10 LEL = 10 LEL = 10 LEL = 10 LEL = 10 LEL = 10 LE -3 -2 -1 L = 10 LEL = 10 LEL = 10 LEL = 10 LEL = 10 LE -3 -2 -1 L = 10 LEL = 10 LEL = 10 LEL = 10 LE -3 -2 -1 L = 10 LEL = 10 LEL = 10 LEL = 10 LE Figure 6.
Survival probability for different symmetric ( β = 0 ) α -stable densities withindex α = 0 . (top left), α = 1 . (top right), α = 1 . (bottom left), and α = 2 . (bottom right) for different L (see figure legends) with d = 0 . . The lines representsimulation results of the Langevin equation, while the symbols correspond to numericalresults based on the space-fractional diffusion equation. The black solid lines representthe universal Sparre-Andersen scaling S ( t ) (cid:39) t − / , which in the finite interval iseventually cut off by an exponential shoulder. first-passage time PDF for different skewness parameters β , addressing first the specialcases of completely one-sided and extremal two-sided laws. α -stable laws. One-sided α -stable laws with α ∈ (0 , , β = 1 satisfythe non-negativity of their increments. Physically, such laws are appropriate for jumpprocesses that always move in the same direction, for instance, as a generalisation ofshot noise. Applying the Skorokhod theorem to this case one can show that the first-passage time PDF in the permitted interval α ∈ (0 , has the exact analytical solution(Appendix D) [125] ℘ ( t ) = ξd α M α (cid:18) ξtd α (cid:19) , (29) irst passage properties of asymmetric Lévy flights -2 -3 -2 -1 = 1.1 = 1.3 = 1.5 = 1.8 = 2.0 -2 -5 = 1.1 = 1.3 = 1.5 = 1.8 = 2.0asymp Figure 7.
Left: Survival probabilities for d = 0 . , L = 100 and the skewness parameter β = 0 for different sets of the index of stability α obtained from solving the space-fractional diffusion equation for ∆ t = 0 . and ∆ x = 0 . . The solid short black lineshows the Sparre Andersen scaling. Right: First passage time probability densityfunction. The solid black line shows equation (27) with the prefactor. with ξ = K α | cos( απ/ | , (30)and where M α is the Wright M -function (also called Mainardi function) [138, 167] withseries representation (E.9) and asymptotic exponential decay (E.11). At sufficiently longtimes the first-passage time PDF reads ℘ ( t ) ∼ A ( α ) t ( α − / / (1 − α ) exp ( − B ( α ) t / (1 − α ) ) , (31)where we used the coefficients A ( α ) = ( αξ/d α ) / (2 − α ) α (cid:112) π (1 − α ) , B ( α ) = 1 − αα ( αξ/d α ) / (1 − α ) . (32)From equation (31) we see that for smaller α the first-passage time density decays fasterwhich is intuitively clear since Lévy flights with smaller α feature longer jumps in thepositive direction. The value of ℘ ( t ) at t = 0 , ℘ ( t = 0) = ξ Γ(1 − α ) d α , (33)demonstrates that, in contrast to the Gaussian case, the probability density does notvanish at t = 0 , indicating the possibility of immediate escape.Using equation (22) the survival probability for < α < , β = 1 can be expressedexactly in terms of the Wright function W a,b (see equation (E.7)) or its series expansion, S ( t ) = W − α, ( − ξt/d α ) = ∞ (cid:88) n =0 ( − ξt/d α ) n n !Γ(1 − αn ) . (34) irst passage properties of asymmetric Lévy flights = 0.25 = 0.50 = 0.75 = 0.25 = 0.50 = 0.75 = 0.25 = 0.50 = 0.75 = 0.25 = 0.50 = 0.75 = 0.25 = 0.50 = 0.75 Figure 8.
Left: First passage time PDF for one-sided α -stable laws with < α < and β = 1 with interval half length L = 100 and x (0) = 0 . The dashed linesrepresent numerical evaluations using the exact analytic expression (29), the dottedlines represent the asymptotic behaviour (31), and the symbols show simulation resultsbased on the space fractional diffusion equation. Right: Asymptotic first-passage timePDF for L = 1 and x (0) = 0 . The lines represent (32) including the prefactors, and thesymbols show the simulation results based on the space-fractional diffusion equation.Note the specific choice of the ordinate such that in this figure we see a log-log plotof the power-law t / (1 − α ) in the exponential of equation (31). We used the time step ∆ t = 0 . and space increment ∆ x = 0 . . In the limit α = 1 / this expression can be reduced to the simple form S ( t ) = 1 − erf (cid:18) K / t √ d (cid:19) , (35)and the corresponding first-passage time density is the half-sided Gaussian [124, 168] ℘ ( t ) = K / (cid:114) πd exp (cid:18) − ( K / t ) d (cid:19) . (36)Figure 8 shows numerical and simulations results for ℘ ( t ) , lending excellent support forresult (31). α -stable probability laws. Stable probability laws withstability index < α < and skewness β = 1 or β = − are called extremal two-sided skewed α -stable laws. When β = 1 the PDF of an α -stable random variable has apositive power-law tail x − − α , and a negative tail which is lighter than Gaussian [169],see figure 2. For β = − the properties of the tails are exchanged. In Appendix D(see equation (D.54)) by applying the Skorokhod theorem it is shown that for β = − the PDF of the first-passage time for extremal two-sided stable probability laws has theexact form ℘ ( t ) = t − − /α dαξ /α M /α (cid:18) d ( ξt ) /α (cid:19) (37) irst passage properties of asymmetric Lévy flights M -function M /α . In the limit α = 2 we recover the PDF (28)for a Gaussian process. Moreover by using equation (22) the survival probability canbe transformed to S ( t ) = 1 − W − /α, (cid:18) − d ( ξt ) /α (cid:19) = ∞ (cid:88) n =1 ( − n − d n ( ξt ) − n/α n !Γ(1 − n/α ) . (38)Equation (37) has the following limiting behaviours: at short times t → correspondingto the asymptotic of large argument in the Wright function, by using the asymptoticexpression (E.11) we find ℘ ( t ) ∼ A ( α ) t − α − α − exp ( − B ( α ) t − α − ) , (39)where A ( α ) = ( αξ/d α ) − / (2 α − (cid:112) π ( α − , B ( α ) = ( α − α α ξ/d α ) − / ( α − . (40)At long times, t → ∞ , ℘ ( t ) ∼ ξ − /α dα Γ(1 − /α ) t − − /α , (41)consistent with the result in [124].For the extremal two-sided α -stable probability laws with index < α < andskewness β = 1 , by using the Skorokhod theorem the PDF of first-passage times hasthe following series representation (see Appendix D, equation (D.72)) ℘ ( t ) = t − /α d α − αξ − /α ∞ (cid:88) n =1 ( ξt/d α ) − n +1 Γ( αn − /α − n ) . (42)For α = 2 we again consistently recover result (28). The asymptotic behaviour ofequation (42) at long times is ℘ ( t ) ∼ ξ − /α d α − α Γ( α − /α ) t − /α , (43)and in the limit t → , we find the finite value ℘ (0) = − ξd − α Γ(1 − α ) . (44)By using the relation between the survival probability and the first-passage timePDF in Laplace space (equation (23)) and applying the inverse Laplace transform weobtain a series representation for the survival probability for extremal two-sided α -stableprobability laws ( < α < , β = 1 ) in the form S ( t ) = ∞ (cid:88) n =1 ( ξt ) /α − n d nα − Γ( αn )Γ(1 − n + 1 /α ) . (45)The first-passage time PDFs for extremal two-sided α -stable probability laws aredisplayed in figure 9. irst passage properties of asymmetric Lévy flights = 1.3 = 1.5 = 1.8 = 2.0 = 1.3 = 1.5 = 1.8 = 2.0 = 1.3 = 1.5 = 1.8 = 2.0 -5 = 1.3 = 1.5 = 1.8 = 2.0 = 1.3 = 1.5 = 1.8 = 2.0 = 1.3 = 1.5 = 1.8 = 2.0 = 1.3 = 1.5 = 1.8 = 2.0 -2 -6 -4 -2 = 1.3 = 1.5 = 1.8 = 2.0 = 1.3 = 1.5 = 1.8 = 2.0 Figure 9.
First passage time PDF for two-sided α -stable laws with < α ≤ and parameters L = 100 , d = 1 , ∆ t = 0 . , and ∆ x = 0 . . Top left (skewness β = − ): The dashed lines represent numerical evaluations using the exact analyticexpression (37), the dotted lines are for the asymptotic behaviour (39), and the symbolsshow simulations results based on the space-fractional diffusion equation. Top right( β = − ): Asymptotic behaviour in log-log scale. The lines represent expression(37), symbols show the simulation results, and the black lines show the power-law(41). Bottom left ( β = 1 ): The lines represent numerical evaluations using theexact analytic expression (42), and the symbols show simulation results. Bottomright ( β = 1 ): Asymptotic behaviour of the first-passage time on log-log scale. Linesrepresent equation (42), symbols show simulations, and the black line is the power-law(43). irst passage properties of asymmetric Lévy flights -4 -3 -2 -1 = 1.3 DE = 1.5 DE = 1.8 DE = 2.0 DE = 1.3 LE = 1.5 LE = 1.8 LE = 2.0 LE -4 -3 -2 -1
0 = 1.3 DE = 1.5 DE = 1.8 DE = 2.0 DE = 1.3 LE = 1.5 LE = 1.8 LE = 2.0 LE
Figure 10.
Survival probability for two-sided α -stable probability laws with d = 0 . , L = 100 as well as β = 1 (left) and β = − (right) for different α with < α ≤ .Symbols represent simulation results based on the space-fractional diffusion equationand lines show simulations of the Langevin equation. The black lines depict theslope of the asymptotic behaviour of the survival probability following from relation(46), concretely t − /α (left) and t − /α (right). Results are obtained with time step ∆ t = 0 . and space increment ∆ x = 0 . for the solution of the space-fractionaldiffusion equation. The time step ∆ t = 10 − and averaging over N = 10 runs werechosen for the simulations of the Langevin equation. α -stable probability laws in general asymmetric form. We finally study the first-passage behaviour for asymmetric Lévy stable laws of arbitrary skewness β , againbased on the comparison between the numerical solution of the space-fractional diffusionequation and simulations of the Langevin approach for different stable index α . Theresults are shown in log-log scale in figures 10 and 11 in the range < α < . Figure10 depicts the cases β = 1 and β = − , while figure 11 shows the cases β = 0 . and β = − . . As can be seen from both figures, a positive value of the skewness leads toa slower decay (shallower slope) than for the Gaussian case, and opposite for negativevalues of β . Indeed, this behaviour is not immediately intuitive, as a positive skewnessmeans that the stable law has a longer tail on the positive axis. However, what mattersfor the short and intermediate first-passage time scales are values of the stable lawaround the most likely value, and for positive skewness this is located on the negativeaxis (compare bottom panels in figure 2). Thus, Lévy flights with a positive skewnessexperience an effective drift to the left, in our setting away from the absorbing boundary.From the Skorokhod theorem for α ∈ (0 , and β ∈ ( − , , α = 1 and β = 0 , aswell as α ∈ (1 , and β ∈ [ − , we obtained the power-law decay (see Appendix D.6) ℘ ( t ) ∼ ρ ( K α (1 + β tan ( απ/ / ) − ρ d αρ Γ(1 − ρ )Γ(1 + αρ ) t − ρ − , (46)where the positive parameter ρ is defined as ρ = 12 + 1 πα arctan( β tan( πα/ . (47) irst passage properties of asymmetric Lévy flights -4 -3 -2 -1 = 1.3 DE = 1.5 DE = 1.8 DE = 2.0 DE = 1.3 LE = 1.5 LE = 1.8 LE = 2.0 LE -4 -3 -2 -1 = 1.3 DE = 1.5 DE = 1.8 DE = 2.0 DE = 1.3 LE = 1.5 LE = 1.8 LE = 2.0 LE Figure 11.
Survival probability for asymmetric α -stable probability laws with β = 0 . (left) β = − . (right). Symbols and parameters are analogous to figure 10. The blacklines show the asymptotic t − ρ , where the exponent ρ is defined in equation (47). Following [170] (page 218) we write this in a form with a general β . This is the directgeneralisation of the classical Sparre Andersen universality for asymmetric stable jumplength distributions. It is easy to check that this result reduces to the known cases forvanishing skewness. We note that the inapplicability of the Sparre Andersen law forasymmetric jump length distributions was already pointed out by Spitzer ([171], page227). The analytical predictions from relations (46) and (47) are in excellent agreementwith the data shown in figures 10 and 11.
5. Discussion and unsolved problems
LFs are Markovian stochastic processes that are commonly used across disciplines asmodels for jump processes that exhibit the distinct propensity for very long jumps. Thescale-free nature of the underlying, Lévy stable PDF of jump lengths with its heavypower-law tail has been shown to effect a more efficient random search strategy than themore conventional Brownian search processes. We here combined numerical inversionmethods of the solution of the space-fractional diffusion equation and simulation of theLangevin equation fuelled by α -stable white noise to quantify the first-passage dynamicsof LFs with a general asymmetric jump length PDF. In particular, we demonstratedthat in all cases both approaches are in excellent agreement. As both approaches haveadvantages and disadvantages, it is very useful to have available two equally potentmethods. In addition, we verified the crossover to an exponential behaviour of the first-passage time PDF in a finite domain and the existence of a well-established power-lawdecay at intermediate times, before the random walker explores the full range of thefinite domain and thus behaves as if it were in a semi-infinite range. For symmetric α -stable laws this decay was shown to be fully consistent with the expected SparreAndersen universal law. For asymmetric cases, when the conditions of the Sparre irst passage properties of asymmetric Lévy flights α β Exact PDF Long-time asymptotic PrefactorEquation Equation2 Irrelevant d √ πK t exp (cid:16) − d K t (cid:17) (28) [165](0,2) 0 Unknown (cid:39) t − / [127, 128] (27) [125](0,1) 1 (29) [125] ∼ A ( α ) t ( α − / / (1 − α ) exp[ − B ( α ) t / (1 − α ) ] (32) [125] / K α (cid:112) /πd exp[ − ( K α t ) / d ] (36) [124, 125, 168](1,2) -1 (37) (cid:39) t − − /α [124] (41) [172] § (cid:39) t − /α (43)(0,1) (-1,1) Unknown (cid:39) t − / − ( πα ) − tan − [ β tan( πα/ [170] (46) [173, 174](1,2) Unknown Table 1.
First-passage time PDF for different stable indices α and skewnessparameters β . The fifth column refers to the equation number for the full prefactor ofthe asymptotic law in column four. Andersen theorem are no longer fulfilled, we derived the analytical behaviour from theSkorokhod theorem for specific values of the skewness. In the general case the directextension of this analytical law was shown to be fully consistent with the numericaland simulations results. The results obtained here will be of use in applications, asthese typically are involved with search processes and thus measure first-passage times.Concurrently, these results also further complete the mathematical theory of Lévy stableprocesses.The first-passage time properties of general α -stable probability laws aresummarised in table 1. For the known cases we include the references to the relevantequations of the exact PDF as well as the asymptotic prefactor. Some special casesare included, as well. Those cases with previously known results refer to the relevantreferences.It is possible to extend the studied setup to higher dimensions [175, 176, 177, 178].In this case, the scalar noise term ζ ( t ) in the Langevin equation (18) for x ( t ) is replaced § Note that the result (41) differs from that of [172] by a factor which appears due to the use of twodifferent forms of the characteristic function for the α -stable process. irst passage properties of asymmetric Lévy flights x ( t ) . Here,multivariate α -stable variables are characterised by a spectral measure defined on theunit circle [18]. For the numerical scheme of the multi-dimensional space-fractionaldiffusion equation we refer to [179, 180, 181, 182, 183, 184, 185, 186]. We note that tothe best knowledge of the authors no multi-dimensional generalisation of Skorokhod’stheorem exists. Thus, the extension of the analytical and numerical approachespresented here to higher dimensions represents a challenging problem requiring furtherstudies.Generally the formulation of non-local and/or correlated stochastic processes isnot always an easy task and, in some cases, still not fully understood. Apart fromLFs, we may allude to the debate on the formulation and solution of boundary valueproblems for fractional Brownian motion, a process fuelled with Gaussian yet long-range correlated noise [187, 188, 189]. For LFs, in addition to the results obtained hereit will be interesting to generalise the results obtained for symmetric α -stable laws inthe presence of an external drift in [113]. Similarly, it will be of interest to investigatethe PDF of first arrival times, related to the probability of hitting a small target in anotherwise unbounded environment, for the general case of asymmetric Lévy stable laws. Acknowledgments
AP acknowledges funding from the Ministry of Science, Research and Technology ofIran and University of Potsdam. This research was supported in part by PLGridInfrastructure. The computer simulations were performed at Potsdam Universityand the Academic Computer Center Cyfronet, Akademia Górniczo-Hutnicza (Kraków,Poland) under CPU grant DynStoch. AC and RM acknowledge support from DFGproject ME 1535/7-1. RM also thanks the Foundation for Polish Science for supportwithin an Alexander von Humboldt Polish Honorary Research Scholarship.
Appendix A. Parametrisation of characteristic function for α -stableprocesses There are several forms for the parametrisation of α -stable laws appearing in literature,basically because of historical reasons. Each form might be useful in a particularsituation. For example, one of them is preferable for analytical calculations, whereasthe other ones can be more convenient for numerical purposes or for fitting of data.Following the exposition of the various forms of stable laws in [154, 190], we here presentfour parameterisation forms for the characteristic functions.In the main text we use the A-form of the characteristic function, ˆ (cid:96) Aα,β A ( k, t ) = exp (cid:16) − tK Aα | k | α [1 − i β A sgn( k ) ω A ( k, α )] + i µ A kt (cid:17) , (A.1) irst passage properties of asymmetric Lévy flights ω A ( k, α ) = (cid:40) tan( πα ) , α (cid:54) = 1 , − π ln | k | , α = 1 . (A.2)In this paper we exclude the case α = 1 and β (cid:54) = 0 . The B-form is helpful from ananalytical point of view, it is given by ˆ (cid:96) Bα,β B ( k, t ) = exp (cid:16) − tK Bα | k | α ω B ( k, α, β B ) + i µ B kt (cid:17) , (A.3)where (for α (cid:54) = 1 ) ω B ( k, α, β B ) = exp (cid:16) − i π β B K ( α )sgn( k ) (cid:17) (A.4)and K ( α ) = α − − α ) . The parameters have the same domains of variation asin the A-form, β A = cot (cid:16) απ (cid:17) tan (cid:16) β B K ( α ) π (cid:17) , µ A = µ B , K Aα = cos (cid:16) β B K ( α ) π (cid:17) K Bα . (A.5)The M-form is used in numerical simulations and reads ˆ (cid:96) Mα,β M ( k, t ) = exp (cid:16) − tK Mα | k | α [1 + i β M sgn( k ) ω M ( k, α, t )] + i µ M ( t ) kt (cid:17) , (A.6)where ( α (cid:54) = 1 ) ω M ( k, α, t ) = tan (cid:16) πα (cid:17) (cid:2) ( K Mα t ) /α − | k | − α − (cid:3) . (A.7)The domain of variation of the parameters in the A- and M-forms is the same andconnected by the following relations β M = β A , µ M ( t ) = µ A + ( K Aα t ) /α t β A tan (cid:16) πα (cid:17) , K Mα = K Aα . (A.8)Finally, the Z-form is represented by ˆ (cid:96) Zα,ρ ( k, t ) = exp (cid:16) − tK Zα (i k ) α exp [ − i παρ sgn( k )] (cid:17) , (A.9)where the parameters α and ρ vary within the bounds < α ≤ , − min(1 , /α ) ≤ ρ ≤ min(1 , /α ) , t > , (A.10)and the relation between the parameters in the A- and Z-forms is as follows, ρ = 12 + 1 απ arctan (cid:16) β A tan (cid:16) απ (cid:17)(cid:17) , K Zα = K Aα (cid:18) (cid:104) β A tan (cid:16) απ (cid:17)(cid:105) (cid:19) / . (A.11)In the A-, B-, and M-forms β A = 1 corresponds to β M = 1 and β B = 1 , while in thecase of the Z-form the value β A = 1 corresponds to the value ρ = 1 if α < and to thevalue ρ = 1 − /α if α > . Appendix B. Some details of the numerical scheme
With the use of equations (5) and (12) we can write equation (4) on a discrete space-timegrid as f j +1 i − f ji ∆ t = K α [ L α,β a D αx Df ( x i , t j ) + R α,β x D αb Df ( x i , t j )] . (B.1)Here we consider discretisation schemes for the three cases < α < , < α ≤ , and α = 1 separately. irst passage properties of asymmetric Lévy flights Appendix B.1. < α < By substitution of equations (12) to (13 b ) into equation (B.1) and using the relation b (cid:90) a ± ( x − y )] γ d y = 1 ± ( γ − (cid:2) ( ± ( x − b )) − γ − ( ± ( x − a )) − γ (cid:3) (B.2)where the sign + is taken for x > b > a and − for x < a < b , we obtain f j +1 i − f ji ∆ t = K α (cid:34) L α,β Γ(2 − α ) i (cid:88) k =1 f jk − f jk − ∆ x (cid:0) ( x i − x k − ) − α − ( x i − x k ) − α (cid:1) (B.3) + R α,β Γ(2 − α ) I − (cid:88) k = i f jk − f jk +1 ∆ x (cid:0) ( x k +1 − x i ) − α − ( x k − x i ) − α (cid:1)(cid:35) . (B.4)Defining λ n = n − α − ( n − − α , Ω L = L α,β K α ∆ t Γ(2 − α )(∆ x ) α , Ω R = R α,β K α ∆ t Γ(2 − α )(∆ x ) α , (B.5)we rewrite equation (B.4) as f j +1 i − f ji = Ω L i (cid:88) k =1 (cid:0) f jk − f jk − (cid:1) λ i − k +1 + Ω R I − (cid:88) k = i (cid:0) f jk − f jk +1 (cid:1) λ k − i +1 . (B.6)Changing f i → θf j +1 i + (1 − θ ) f ji , ≤ θ ≤ , on the right hand side, − θ Ω L i (cid:88) k =1 (cid:0) f j +1 k − f j +1 k − (cid:1) λ i − k +1 + f j +1 i − θ Ω R I − (cid:88) k = i (cid:0) f j +1 k − f j +1 k +1 (cid:1) λ k − i +1 = (1 − θ )Ω L i (cid:88) k =1 (cid:0) f jk − f jk − (cid:1) λ i − k +1 + f ji + (1 − θ )Ω R I − (cid:88) k = i (cid:0) f jk − f jk +1 (cid:1) λ k − i +1 . (B.7)Then the matrices A and B in equation (16) are A = A c A ,R · · · A I,R A ,L . . . . . . ...... . . . . . . A ,R A I,L · · · A ,L A c , B = B c B ,R · · · B I,R B ,L . . . . . . ...... . . . . . . B ,R B I,L · · · B ,L B c , (B.8)where A c = 1 − θ (Ω L + Ω R ) λ A i,L = θ Ω L ( λ i − λ i +1 ) , i = 1 , , . . . , IA i,R = θ Ω R ( λ i − λ i +1 ) , i = 1 , , . . . , I (B.9)and B c = 1 + (1 − θ )(Ω L + Ω R ) λ B i,L = − (1 − θ )Ω L ( λ i − λ i +1 ) , i = 1 , , . . . , IB i,R = − (1 − θ )Ω R ( λ i − λ i +1 ) , i = 1 , , . . . , I. (B.10) irst passage properties of asymmetric Lévy flights Appendix B.2. < α < Substituting equations (12), (14 a ), and (14 b ) into equation (B.1) we get f j +1 i − f ji ∆ t = K α (cid:34) L α,β Γ(3 − α ) i (cid:88) k =1 f jk +1 − f jk + f jk − (∆ x ) (cid:0) ( x i − x k − ) − α − ( x i − x k ) − α (cid:1) + R α,β Γ(3 − α ) I − (cid:88) k = i f jk +1 − f jk + f jk − (∆ x ) (cid:0) ( x k +1 − x i ) − α − ( x k − x i ) − α (cid:1)(cid:35) (B.11)with the definition λ n = n − α − ( n − − α , Ω L = L α,β K α ∆ t Γ(3 − α )(∆ x ) α , Ω R = R α,β K α ∆ t Γ(3 − α )(∆ x ) α , (B.12)and changing the notation as above, we obtain − θ Ω L i (cid:88) k =1 (cid:0) f j +1 k +1 − f j +1 k + f j +1 k − (cid:1) λ i − k +1 + f j +1 i − θ Ω R I − (cid:88) k = i (cid:0) f j +1 k +1 − f j +1 k + f j +1 k − (cid:1) λ k − i +1 = (1 − θ )Ω L i (cid:88) k =1 (cid:0) f jk +1 − f jk + f jk − (cid:1) λ i − k +1 + f ji + (1 − θ )Ω R I − (cid:88) k = i (cid:0) f jk +1 − f jk + f jk − (cid:1) λ k − i +1 . (B.13)Then the elements of the matrix A and B in equation (16) are A c = 1 − θ (Ω L + Ω R )( λ − λ ) A ,L = − θ Ω L ( λ − λ + λ ) − θ Ω R λ A ,R = − θ Ω R ( λ − λ + λ ) − θ Ω L λ A i,L = − θ Ω L ( λ i +2 − λ i +1 + λ i ) , i = 2 , , . . . , IA i,R = − θ Ω R ( λ i +2 − λ i +1 + λ i ) , i = 2 , , . . . , I (B.14)and B c = 1 + (1 − θ )(Ω L + Ω R )( λ − λ ) B ,L = (1 − θ )Ω L ( λ − λ + λ ) + (1 − θ )Ω R λ B ,R = (1 − θ )Ω R ( λ − λ + λ ) + (1 − θ )Ω L λ B i,L = (1 − θ )Ω L ( λ i +2 − λ i +1 + λ i ) , i = 2 , , . . . , IB i,R = (1 − θ )Ω R ( λ i +2 − λ i +1 + λ i ) , i = 2 , , . . . , I . (B.15) Appendix B.3. α = 1 , β = 0 By substituting equations (15) and (12) into equation (9) we obtain f j +1 i − f ji ∆ t = − K α π (cid:34) i (cid:88) k =1 f jk − f jk − ∆ x i − k ) + 1 + I − (cid:88) k = i f jk − f jk +1 ∆ x k − i ) + 1 (cid:35) . (B.16) irst passage properties of asymmetric Lévy flights λ n = 12 n + 1 , Ω L = 2 L α,β K α ∆ t ∆ x , Ω R = 2 R α,β K α ∆ t ∆ x , (B.17)changing notation as above, we obtain θ Ω L i (cid:88) k =1 (cid:0) f j +1 k − f j +1 k − (cid:1) λ i − k + f j +1 i + θ Ω R I − (cid:88) k = i (cid:0) f j +1 k − f j +1 k +1 (cid:1) λ k − i = − (1 − θ )Ω L i (cid:88) k =1 (cid:0) f jk − f jk − (cid:1) λ i − k + f ji − (1 − θ )Ω R I − (cid:88) k = i (cid:0) f jk − f jk +1 (cid:1) λ k − i . (B.18)Then the elements of the matrices A and B in equation (16) are A c = 1 + θ (Ω L + Ω R ) λ A i,L = θ Ω L ( λ i − λ i − ) , i = 1 , , . . . , IA i,R = θ Ω R ( λ i − λ i − ) , i = 1 , , . . . , I (B.19)and B c = 1 − (1 − θ )(Ω L + Ω R ) λ B i,L = − (1 − θ )Ω L ( λ i − λ i − ) , i = 1 , , . . . , IB i,R = − (1 − θ )Ω R ( λ i − λ i − ) , i = 1 , , . . . , I. (B.20) Appendix C. Error estimation of the difference scheme
We here provide some details on the error estimate of our difference scheme. For thecase < α < (equation (13)) x i (cid:90) − L f (1) ( ζ, t j )( x i − ζ ) α d ζ ≈ i (cid:88) k =1 f jk − f jk − ∆ x x k (cid:90) x k − x i − ζ ) α d ζ + O (∆ x − α ) , (C.1)for the left side derivative, and L (cid:90) x i f (1) ( ζ, t j )( ζ − x i ) α d ζ ≈ I − (cid:88) k = i f jk +1 − f jk ∆ x x k +1 (cid:90) x k ζ − x i ) α d ζ + O (∆ x − α ) , (C.2)for the right side derivative. This efficient way to approximate the Caputo derivativeof order α ( < α < ) is called L1 scheme [191, 192, 193] and its error estimate is O (∆ x − α ) (see figure 2 top left panel) [192, 194, 195]. For the case < α < thesuitable method to discretise the Caputo derivative is the L2 scheme [191, 193, 196] x i (cid:90) − L f (2) ( ζ, t j )( x i − ζ ) α − d ζ ≈ i (cid:88) k =1 f jk +1 − f jk + f jk − (∆ x ) x k (cid:90) x k − x i − ζ ) α − d ζ + O (∆ x ) , (C.3)for the left side, and L (cid:90) x i f (2) ( ζ, t j )( ζ − x i ) α − d ζ ≈ I − (cid:88) k = i f jk +1 − f jk + f jk − (∆ x ) x k +1 (cid:90) x k ζ − x i ) α − d ζ + O (∆ x ) , (C.4) irst passage properties of asymmetric Lévy flights O (∆ x ) (see figure 2 top rightpanel) [196, 197]. For the special case α = 2 the central difference scheme has truncationerror O (∆ x ) (see figure C1 top right panel). For comparison, in [194] an error estimateof order O (∆ x − α ) is presented for < α < . In [198] a computational algorithmfor approximating the Caputo derivative was developed, and the convergence order is O (∆ x ) for < α ≤ . Another difference method of order two was derived in [197] for < α ≤ .For the special case α = 1 , β = 0 , − ddx ( Hf ( x, t )) ≈ − π i (cid:88) k =1 f jk − f jk − ∆ x i − k ) + 1 − π I − (cid:88) k = i f jk − f jk +1 ∆ x k − i ) + 1 + O (∆ x ) , (C.5)the truncation error is O (∆ x ) (see figure C1 top left panel). To evaluate the truncationerror we used the relation (cid:107) e ( x ) (cid:107) = (cid:107) f ( x i , t j ) − f ji (cid:107) = (cid:118)(cid:117)(cid:117)(cid:116) I I (cid:88) i =1 ( f ( x i , t j ) − f ji ) , (C.6)where f ( x i , t j ) is the exact solution and f ji is the approximated solution of function f ( x, t ) . For (cid:107) e ( t ) (cid:107) this is similar and we use (cid:107) e ( t ) (cid:107) = (cid:118)(cid:117)(cid:117)(cid:116) J J (cid:88) j =1 ( f ( x i , t j ) − f ji ) . (C.7)The results of the error analysis for (cid:96) α,β ( x, t ) , survival probability and the first-passagetime PDF are shown in figure C1. Appendix D. A short review of the Skorokhod theorem
The Skorokhod theorem establishes a general formula for the Laplace transform ℘ ( λ ) ofthe first-passage time PDF in the semi-infinite domain for a broad class of homogeneousrandom processes with independent increments and thus has a pivotal role in the theoryof first-passage processes [132, 164]. For the process starting at x = 0 with a boundaryat x = d , ℘ ( λ ) = (cid:104) e − λt (cid:105) = ∞ (cid:90) e − λt ℘ ( t )d t = 1 − p + ( λ, d ) . (D.1)Here the auxiliary measure p + ( λ, x ) is defined via its Fourier transform as q + ( λ, k ) = ∞ (cid:90) −∞ e i kx ∂p + ( λ, x ) ∂x d x = exp ∞ (cid:90) e − λt t ∞ (cid:90) ( e i kx − f ( x, t )d x d t , (D.2) irst passage properties of asymmetric Lévy flights -2 x -10 -5 E rr o r = 0.5 = 0.7 = 1.0x x x -2 x -8 -6 -4 -2 E rr o r = 1.3 = 1.5 = 2.0x x10 -6 -4 -2 t -4 -3 -2 -1 E rr o r FPT- = 2.0SP - = 2.0FPT- = 0.5SP - = 0.5 t tt 10 -2 -1 x -4 -2 E rr o r FPT- = 1.5, = -1.0FPT- = 1.5, = 1.0xx
Figure C1.
Error analysis of the numerical schemes in section 2. Top left: Truncationerror for (cid:96) α,β ( x, t ) in the L1 scheme ( α = 0 . , . ) (equations (13)) and Hilbertdiscretising scheme for α = 1 , β = 0 (equation (15)). For this panel we use ∆ t = 10 − , t = 1 , L = 16 in the case α = 0 . , . and ∆ t = 10 − , t = 1 , L = 8 in the case α = 1 . Top right: Truncation error for (cid:96) α,β ( x, t ) in the L2 scheme ( α = 1 . , . )(equations (14)) and central difference discretising scheme for α = 2 . For this panelwe use ∆ t = 10 − , t = 1 , L = 8 in the case α = 1 . , . and ∆ t = 10 − , t = 1 , L = 4 in the case α = 2 . Bottom left: Truncation error for the first-passage timePDF and survival probability versus time step for Brownian motion (equation (25))and one-sided α -stable probability law with α = 0 . and β = 1 (equation (33)). Forthis panel we use ∆ x = 5 × − , t = 10 and L = 20 . Bottom right: Truncationerror for the first-passage time PDF of extremal two-sided α -stable probability lawswith stable index α = 1 . and skewness parameter β = − , (equations (34) and (39))versus space increment ∆ x . For this panel we use ∆ t = 10 − , t = 5 and L = 10 . irst passage properties of asymmetric Lévy flights f ( x, t ) is the PDF of the process, that is (cid:96) α,β ( x, t ) in our case. Theboundary condition reads p + ( λ, x ) = 0 at x = 0 . Below, for didactic purposes we firstcalculate ℘ ( t ) for Brownian motion and then proceed to symmetric ( < α ≤ and β = 0) , one-sided ( < α < and β = ± ), extremal two-sided ( < α < and β = ± ), and finally to the general case ( < α < , α (cid:54) = 1 , and − < β < ). Appendix D.1. First passage time PDF for Brownian motion
For Brownian motion the PDF reads f ( x, t ) = 1 √ πK t exp (cid:18) − x K t (cid:19) . (D.3)Substitution into equation (D.2) yields q + ( λ, k ) = exp ∞ (cid:90) e − λt t ∞ (cid:90) ( e i kx − e − x / (4 K t ) √ πK t d x d t = exp ∞ (cid:90) e − λt t (cid:110) e − tK k erfc (cid:16) − i k (cid:112) tK (cid:17) − (cid:111) = exp ln (cid:18) λλ + K k (cid:19) + ln k (cid:113) K λ − i k (cid:113) K λ , (D.4)therefore q + ( λ, k ) = λλ + K k k (cid:113) K λ − i k (cid:113) K λ / . (D.5)Now, we use the relation − i k (cid:112) K /λ = 1 + i k (cid:112) K /λ K k /λ (D.6)to get to q + ( λ, k ) = λλ + K k + i k √ K λλ + K k . (D.7)After an inverse Fourier transform according to equation (D.2) we arrive at ddx p + ( λ, x ) = 12 π ∞ (cid:90) −∞ e − i kx q + ( λ, k )d k = 12 π ∞ (cid:90) −∞ e − i kx (cid:18) λλ + K k + i k √ K λλ + K k (cid:19) d k. (D.8)For the first integral we have π ∞ (cid:90) −∞ e − i kx (cid:18) λλ + K k (cid:19) d k = λ πK ∞ (cid:90) −∞ e − i kx (cid:16) k − i (cid:113) λK (cid:17) (cid:16) k + i (cid:113) λK (cid:17) d k. (D.9) irst passage properties of asymmetric Lévy flights λ πK ∞ (cid:90) −∞ e − i kx (cid:16) k − i (cid:113) λK (cid:17) (cid:16) k + i (cid:113) λK (cid:17) d k = 12 (cid:114) λK exp (cid:32) − (cid:114) λK x (cid:33) , (D.10)For the second integral, we write π ∞ (cid:90) −∞ e − i kx (cid:18) i k √ K λλ + K k (cid:19) d k = i2 π (cid:114) λK ∞ (cid:90) −∞ k e − i kx (cid:16) k − i (cid:113) λK (cid:17) (cid:16) k + i (cid:113) λK (cid:17) d k. (D.11)Again, with the residue theorem, i2 π (cid:114) λK ∞ (cid:90) −∞ k e − i kx ( k − i (cid:113) λK )( k + i (cid:113) λK ) d k = 12 (cid:114) λK exp (cid:32) − (cid:114) λK x (cid:33) . (D.12)Therefore, by substitution of equations (D.10) and (D.12) into equation (D.8) we obtain( x > ) ddx p + ( λ, x ) = 12 π ∞ (cid:90) −∞ e − i kx ˆ q + ( λ, k )d k = (cid:114) λK exp (cid:32) − (cid:114) λK x (cid:33) . (D.13)Using the boundary condition we get p + ( λ, x ) = (cid:114) λK x (cid:90) exp (cid:32) − (cid:114) λK x (cid:33) d x = 1 − exp (cid:32) − (cid:114) λK x (cid:33) , (D.14)and thus with the help of equation (D.1), ℘ ( λ ) = exp (cid:32) − (cid:114) λK d (cid:33) , (D.15)Finally, by inverse Laplace transform we get ℘ ( t ) = d √ πK t exp (cid:18) − d K t (cid:19) . (D.16)This is the famous Lévy-Smirnov law representing a well-known result of Brownianmotion [165]. This derivation is, of course, overly complicated for the Gaussian case,but the same procedure can be applied to the general case of an asymmetric Lévy stablelaw, as we now show. Appendix D.2. First passage time PDF for symmetric α -stable processes Due to the symmetry of the PDF the function ln q + ( k, λ ) , equation (D.2), can be writtenas ln q + ( k, λ ) = A ( λ, k ) + i B ( λ, k ) , (D.17)where A ( λ, k ) = ∞ (cid:90) e − λt t ∞ (cid:90) (cos ( kx ) − (cid:96) α, ( x, t ) d x d t = 12 ln λλ + K α | k | α , (D.18) irst passage properties of asymmetric Lévy flights B ( λ, k ) = ∞ (cid:90) e − λt t ∞ (cid:90) sin ( kx ) (cid:96) α, ( x, t )d x d t. (D.19)To find B ( λ, k ) at small λ the self-similar property of the α -stable process comes inuseful, (cid:96) α, ( x, t ) = 1( K α t ) /α (cid:96) α, (cid:18) x ( K α t ) /α , (cid:19) , (D.20)where (cid:96) α, ( y,
1) = (cid:96) α, ( − y,
1) = 1 π ∞ (cid:90) cos ( ky ) e −| k | α d k. (D.21)When λ tends to zero, from equation (D.19) we get B ( λ, k ) = ∞ (cid:90) e − λt t ∞ (cid:90) sin ( kx ) 1( K α t ) /α (cid:96) α, (cid:0) x/ ( K α t ) /α , (cid:1) d x d t ≈ ∞ (cid:90) t ∞ (cid:90) sin ( ky ( K α t ) /α ) (cid:96) α, ( y, y d t = α ∞ (cid:90) (cid:96) α, ( y, ∞ (cid:90) sin ( kys ) s d s d y = απ k ) . (D.22)By substitution of equations (D.18) and (D.22) into equation (D.17) we get q + ( λ, k ) ≈ √ λ √ K α | k | α/ e isgn( k ) απ/ , λ → . (D.23)Then the inverse Fourier transform of the above equation renders ddx p + ( λ, x ) = 12 π ∞ (cid:90) −∞ e − i kx ˆ q + ( λ, k )d k = √ λ √ K α Γ( α/ x − α/ , (D.24)and, after applying the boundary condition, p + ( λ, x ) = 2 √ λα √ K α Γ( α/ x α/ . (D.25)Recalling equation (D.1), ℘ ( λ ) ≈ − d α/ α √ K α Γ( α/ λ / . (D.26)Now, with the help of the Tauberian theorem [165] (Chapter XIII, section 5) we findthat the small- λ asymptotic of the Laplace transform ψ ( λ ) ≈ − b λ µ , b = b Γ(1 − µ ) /µ, λ → (D.27)corresponds to the long-time asymptotic of the PDF ([199], chapter 3) ψ ( t ) ∼ b t − − µ , < µ < , b > . (D.28) irst passage properties of asymmetric Lévy flights α -stable process has the form [125] ℘ ( t ) ≈ d α/ α √ πK α Γ( α/ t − / . (D.29) Appendix D.3. First passage time PDF for one-sided α -stable processes, < α < and β = 1 Due to the monotonic growth of the process in this case there exists a simple relationbetween the cumulative probabilities of the first-passage time and the α -stable processitself, see, e.g., [168]. However, for a didactic purpose in this Appendix we obtain theresult by the use of Skorokhod’s method. Since for one-sided α -stable processes with < α < and β = 1 the PDF (cid:96) α, ( x, t ) vanishes for x < , we get ∞ (cid:90) (cid:0) e i kx − (cid:1) (cid:96) α, ( x, t )d x = ∞ (cid:90) −∞ (cid:0) e i kx − (cid:1) (cid:96) α, ( x, t )d x = exp [ − tK α | k | α (1 − isgn( k ) tan( απ/ − . (D.30)After plugging this into equation (D.2), q + ( λ, k ) = exp ∞ (cid:90) e − λt t (exp [ − tK α | k | α (1 − isgn( k ) tan( απ/ −
1) d t . (D.31)Then q + ( λ, k ) = exp (cid:26) ln (cid:18) λλ + ζ (cid:19)(cid:27) = λλ + ζ , (D.32)where ζ = K α | k | α (cid:16) − isgn( k ) tan πα (cid:17) = K α cos ( απ/
2) ( − i k ) α . (D.33)Therefore, ∂p + ( λ, x ) /∂x follows from result (D.32) by inverse Fourier transform, ddx p + ( λ, x ) = 12 π ∞ (cid:90) −∞ e − i kx q + ( λ, k )d k = 12 π ∞ (cid:90) −∞ e − i kx (cid:32) λ ( − i k ) α K α cos ( απ/ + λ (cid:33) d k. (D.34)Defining s = − i k , we have dd x p + ( λ, x ) = 12 π i i ∞ (cid:90) − i ∞ e sx λs α K α / [cos( απ/ λ d s. (D.35)Recalling relation (E.5) we obtain dd x p + ( λ, x ) = − dd x E α (cid:18) − λ cos ( απ/ K α x α (cid:19) , (D.36)where E α ( z ) = (cid:80) ∞ n =0 z n / Γ(1 + αn ) is the Mittag-Leffler function, see [138, 167] andAppendix E. With the boundary condition p + ( λ, x ≤
0) = 0 we get p + ( λ, x ) = 1 − E α (cid:18) − λ cos ( απ/ K α x α (cid:19) . (D.37) irst passage properties of asymmetric Lévy flights ℘ ( λ ) = 1 − p + ( λ, d ) wehave ℘ ( λ ) = E α ( − λ cos( απ/ d α /K α ) . (D.38)The Laplace inversion is then immediately accomplished in terms of the Wright function(see equation (E.12)) for < α < [125], ℘ ( t ) = K α cos( απ/ d α M α (cid:18) K α t cos( απ/ d α (cid:19) . (D.39)By using relation (E.14) we make sure that ℘ ( t ) is normalised, ∞ (cid:90) ℘ ( t )d t = ∞ (cid:90) K α cos( απ/ d α M α (cid:18) K α t cos( απ/ d α (cid:19) d t = 1 . (D.40)This can be also shown by taking the integral form (E.10) of the M -function. Bychanging the order of integration we get ∞ (cid:90) ℘ ( t )d t = ∞ (cid:90) K α cos( απ/ d α π i (cid:90) Ha exp (cid:18) σ − K α t cos( απ/ d α σ α (cid:19) d σσ − α d t = 12 π i (cid:90) Ha σ α − e σ K α cos( απ/ d α ∞ (cid:90) exp (cid:18) − K α σ α cos( απ/ d α t (cid:19) d t d σ = 12 π i (cid:90) Ha σ − e σ d σ = 1 , (D.41)where Ha denotes the Hankel path, and in the last step we made use of equation (E.3). Appendix D.4. First passage time PDF for extremal two-sided α -stable process, < α < and β = − To apply the Skorokhod theorem we need to calculate the following integral ∞ (cid:90) (cid:0) e i kx − (cid:1) (cid:96) α, − ( x, t )d x. (D.42)To this end we use the Laplace transform of α -stable processes with < α ≤ and β B = − , which is derived in [154] (page 169, equation (2.10.9)) in dimensionless B-formwith K Bα = 1 and t = 1 , α E /α ( − s ) = ∞ (cid:90) e − sx (cid:96) Bα, − ( x, x. (D.43)In dimensional variables this equation reads α E /α (cid:16) − s (cid:0) K Bα t (cid:1) /α (cid:17) = ∞ (cid:90) e − sx (cid:96) Bα, − ( x, t )d x. (D.44) irst passage properties of asymmetric Lévy flights ∞ (cid:90) e − sx (cid:96) Aα, − ( x, t )d x = 1 α E /α (cid:32) − s (cid:18) K Aα t | cos ( απ/ | (cid:19) /α (cid:33) . (D.45)Now we go back to equation (D.42) which can be written as (we again omit the index A in what follows) ∞ (cid:90) e − sx (cid:96) α, − ( x, t ) | s = − i k d x − ∞ (cid:90) e − sx (cid:96) α, − ( x, t ) | s =0 d x. (D.46)Using equation (D.45) we find ∞ (cid:90) (cid:0) e i kx − (cid:1) (cid:96) α, − ( x, t )d x = 1 α (cid:34) E /α (cid:32) i k (cid:18) K α t | cos ( απ/ | (cid:19) /α (cid:33) − (cid:35) , (D.47)and after plugging this expression into equation (D.2), q + ( λ, k ) = exp α ∞ (cid:90) e − λt t (cid:34) E /α (cid:32) i k (cid:18) K α t | cos ( απ/ | (cid:19) /α (cid:33) − (cid:35) d t . (D.48)To calculate expression (D.48) we first find ∂∂λ ln q + ( λ, k ) = − α ∞ (cid:90) e − λt (cid:34) E /α (cid:32) i k (cid:18) K α t | cos ( απ/ | (cid:19) /α (cid:33) − (cid:35) d t = 1 αλ − λ /α λ /α − i k (cid:16) K α | cos( απ/ | (cid:17) /α , (D.49)where we employ the Laplace transform (E.4) of the Mittag-Leffler function. By takingthe indefinite integral over λ we obtain q + ( λ, k ) = λ /α λ /α − i k (cid:16) K α | cos( απ/ | (cid:17) /α , (D.50)and then from equation (D.50), by inverse Fourier transform, ddx p + ( λ, x ) = 12 π ∞ (cid:90) −∞ e − i kx q + ( λ, k )d k = (cid:18) | cos ( απ/ | K α (cid:19) /α λ /α exp (cid:34) − (cid:18) | cos ( απ/ | K α (cid:19) /α xλ /α (cid:35) . (D.51)With the boundary condition p + ( λ, x = 0) = 0 we get p + ( λ, x ) = 1 − exp (cid:34) − (cid:18) | cos ( απ/ | K α (cid:19) /α xλ /α (cid:35) . (D.52) irst passage properties of asymmetric Lévy flights ℘ ( λ ) we obtain ℘ ( λ ) = 1 − p + ( λ, d ) = exp (cid:34) − d (cid:18) | cos ( απ/ | K α (cid:19) /α λ /α (cid:35) , (D.53)which is of a stretched exponential form. Recalling now the Laplace transform pair(E.13) for the M -function, we finally arrive at the first-passage time PDF for theextremal α -stable process with < α ≤ and β = − , ℘ ( t ) = t − − /α dα (cid:16) K α | cos ( απ/ | (cid:17) /α M /α (cid:32) d (cid:18) K α t | cos ( απ/ | (cid:19) − /α (cid:33) . (D.54)Let us show the normalisation of this function. By using the integral form (E.10) of the M -function and changing the order of integration we have ∞ (cid:90) ℘ ( t )d t = dα (cid:16) K α | cos( απ/ | (cid:17) /α ∞ (cid:90) t − − /α M /α (cid:32) d (cid:18) K α t | cos ( απ/ | (cid:19) − /α (cid:33) d t = ∞ (cid:90) t − α (cid:20) K α td α | cos ( απ/ | (cid:21) − /α × πi (cid:90) Ha exp (cid:32) σ − (cid:20) K α td α | cos ( απ/ | (cid:21) − /α σ /α (cid:33) d σσ − /α d t = 12 πi (cid:90) Ha e σ σ /α − ∞ (cid:90) t − α (cid:20) K α td α | cos ( απ/ | (cid:21) − /α × exp (cid:32) − (cid:20) K α d α | cos ( απ/ | σ (cid:21) − /α t − /α (cid:33) d t d σ. (D.55)By change of variable u = [ K α /d α | cos ( απ/ | σ ] − /α t − /α , performing the inner integraland using the Hankel formula (E.3) for the Gamma function we obtain the necessarynormalisation condition. Now, if we employ the series expansion (E.9) for the M -function, from equation (D.54) we arrive at a series which corresponds to that in equation(2.25) of [172]. Note that in our case the additional factor K α / | cos( απ/ | appears dueto a different starting form for the characteristic function of the α -stable process. Appendix D.5. First passage time PDF for extremal two-sided α -stable processes, < α < , β = 1 Similar to above, at first we obtain the Laplace transform of the α -stable PDF with < α ≤ and β = 1 . We write ∞ (cid:90) f ( u )d u = ∞ (cid:90) −∞ f ( u )d u − (cid:90) −∞ f ( u )d u, (D.56) irst passage properties of asymmetric Lévy flights (cid:96) α,β ( x, t ) = (cid:96) α, − β ( − x, t ) to get ∞ (cid:90) e i kx (cid:96) α, ( x, t )d x = ∞ (cid:90) −∞ e i kx (cid:96) α, ( x, t )d x − (cid:90) −∞ e i kx (cid:96) α, − ( − x, t )d x. (D.57)The second integral on the right side can be written as (cid:90) −∞ e i kx (cid:96) α, − ( − x, t )d x = ∞ (cid:90) e − i kx (cid:96) α, − ( x, t )d x. (D.58)To take the first integral on the right hand side of equation (D.57) we use thecharacteristic function in the A-form. To take the second integral (D.58) we employthe Laplace transform of the PDF with < α < and β = − given by equation (D.45)with s = i k . Thus, relation (D.57) in the A-form has the shape ∞ (cid:90) e i kx (cid:96) Aα, ( x, t )d x = exp (cid:18) ( − i k ) α K Aα t | cos ( απ/ | (cid:19) − α E /α (cid:32)(cid:20) ( − i k ) α K Aα t | cos ( απ/ | (cid:21) /α (cid:33) . (D.59)With the help of equation (D.59) we write (again the index A is omitted in what follows) ∞ (cid:90) (cid:0) e i kx − (cid:1) (cid:96) α, ( x, t )d x = ∞ (cid:90) e − sx (cid:96) α, ( x, t ) | s = − i k d x − ∞ (cid:90) e − sx (cid:96) α, ( x, t ) | s =0 d x = (cid:34) exp (cid:18) ( − i k ) α K α t | cos ( απ/ | (cid:19) − α E /α (cid:32)(cid:20) ( − i k ) α K α t | cos ( απ/ | (cid:21) /α (cid:33) − α (cid:35) . (D.60)By substituting this expression into equation (D.2) we get ln q + ( λ, k ) = ∞ (cid:90) e − λt t (cid:20) exp (cid:18) ( − i k ) α K α t | cos ( απ/ | (cid:19) − α E /α (cid:32)(cid:20) ( − i k ) α K α t | cos ( απ/ | (cid:21) /α (cid:33) − α (cid:35) d t. (D.61)The derivative with respect to λ reads ∂∂λ ln q + ( λ, k ) = − ∞ (cid:90) e − λt (cid:20) exp (cid:18) ( − i k ) α K α t | cos ( απ/ | (cid:19) − α E /α (cid:32)(cid:20) ( − i k ) α K α t | cos ( απ/ | (cid:21) /α (cid:33) − α (cid:35) d t = − λ − ( − i k ) α K α | cos ( απ/ | + 1 α λ /α − λ /α − (cid:16) ( − i k ) α K α | cos ( απ/ | (cid:17) /α + (cid:18) − α (cid:19) λ , (D.62)where for the second term we employ the Laplace transform (E.4) of the Mittag-Lefflerfunction. By taking the indefinite integral over λ we obtain q + ( λ, k ) = λ − /α (cid:18) λ /α − (cid:16) ( − i k ) α K α | cos ( απ/ | (cid:17) /α (cid:19) λ − ( − i k ) α K α | cos ( απ/ | . (D.63) irst passage properties of asymmetric Lévy flights dp + ( λ, x ) /dx follows from equation (D.63) by inverse Fourier transform, ddx p + ( λ, x ) = 12 π ∞ (cid:90) −∞ e − i kx k (cid:16) K α | cos ( απ/ | λ (cid:17) /α − ( − i k ) α K α | cos ( απ/ | λ d k. (D.64)By defining s = − i k we have dd x p + ( λ, x ) = 12 π i i ∞ (cid:90) − i ∞ e sx − s (cid:16) K α | cos ( απ/ | λ (cid:17) /α − s α K α | cos ( απ/ | λ d s = 12 π i i ∞ (cid:90) − i ∞ e sx − s α K α | cos ( απ/ | λ d s − π i i ∞ (cid:90) − i ∞ e sx s (cid:16) K α | cos ( απ/ | λ (cid:17) /α − s α K α | cos ( απ/ | λ d s. (D.65)Using the properties of the Mittag-Leffler function, equations (E.5) and (E.6) we canwrite dd x p + ( λ, x ) = − ddx E α (cid:18) | cos ( απ/ | λx α K α (cid:19) + (cid:18) K α | cos ( απ/ | λ (cid:19) /α d d x E α (cid:18) | cos ( απ/ | λx α K α (cid:19) , (D.66)and with the boundary condition p + ( λ, x = 0) = 0 we get p + ( λ, x ) = 1 − E α (cid:18) | cos ( απ/ | λx α K α (cid:19) + (cid:18) K α | cos ( απ/ | λ (cid:19) /α dd x E α (cid:18) | cos ( απ/ | λx α K α (cid:19) . (D.67)Thus, for the Laplace transform ℘ ( λ ) = 1 − p + ( λ, d ) we obtain ℘ ( λ ) = E α (cid:18) λ | cos ( απ/ | K α d α (cid:19) − (cid:18) λ | cos ( απ/ | K α (cid:19) − /α dd d E α (cid:18) λ | cos ( απ/ | K α d α (cid:19) . (D.68)By applying the inverse Laplace transform with respect to λ and using the seriesrepresentation (E.1) of the Mittag-Leffler function for the first term on the right side ofequation (D.68) we have π i (cid:90) Ha e λt E α (cid:18) λ | cos ( απ/ | K α d α (cid:19) d λ = 12 π i (cid:90) Ha e λt ∞ (cid:88) n =0 (cid:16) λ | cos ( απ/ | K α d α (cid:17) n Γ( αn + 1) d λ = ∞ (cid:88) n =0 (cid:16) | cos ( απ/ | K α d α (cid:17) n Γ( αn + 1) 12 πi (cid:90) Ha e λt λ n d λ = ∞ (cid:88) n =0 (cid:16) | cos ( απ/ | K α d α (cid:17) n Γ( αn + 1)Γ( − n ) = 0 , (D.69) irst passage properties of asymmetric Lévy flights / Γ( − n ) = 0 for n = 0 , , , . . . . For the second term on the right side ofequation (D.68) we calculate the derivative with respect to d of the series representation(E.1) of the Mittag-Leffler function and get − (cid:18) λ | cos ( απ/ | K α (cid:19) − /α dd d E α (cid:18) λ | cos ( απ/ | K α d α (cid:19) = − (cid:18) λ | cos ( απ/ | K α (cid:19) − /α ∞ (cid:88) n =1 (cid:16) λ | cos ( απ/ | K α d α (cid:17) n Γ( αn ) . (D.70)After inverse Laplace transform and using the integral form (E.3) of the Gammafunction, we obtain − π i (cid:90) Ha e λt (cid:18) λ | cos ( απ/ | d α K α (cid:19) − /α ∞ (cid:88) n =1 (cid:16) λ | cos ( απ/ | d α K α (cid:17) n Γ( αn ) d λ = − ∞ (cid:88) n =1 (cid:16) | cos ( απ/ | d α K α (cid:17) n − /α Γ( αn ) 12 π i (cid:90) Ha e λt λ n − /α d λ = − ∞ (cid:88) n =1 (cid:16) | cos ( απ/ | d α K α (cid:17) n − /α t − n − /α Γ( αn )Γ(1 /α − n ) . (D.71)Rewriting this expression and using the relation − Γ( αn )Γ(1 /α − n ) = α Γ( αn − /α − n ) yields ℘ ( t ) = t − /α d α − α ( K α / | cos ( απ/ | ) − /α ∞ (cid:88) n =1 ( | cos ( απ/ | d α /K α t ) n − Γ( αn − /α − n ) . (D.72)To obtain a closed-form solution by help of equation (30) we rewrite equation (D.72)as ℘ ( t ) = t − /α d α − αξ − /α ∞ (cid:88) n =1 ( d α /ξt ) n − Γ( αn − /α − n )= t − /α d α − αξ − /α ∞ (cid:88) n =0 ( d α /ξt ) n Γ( αn + α − − n + 1 /α ) . (D.73)Now, the generalised four-parametric Mittag-Leffler function has the series representa-tion (page 129, equation (6.1.1) [200]) E α ,β ; α ,β ( z ) = ∞ (cid:88) k =0 z k Γ( α k + β )Γ(Γ( α k + β ) , z ∈ C , (D.74)for α , α ∈ R ( α + α (cid:54) = 0) and β , β ∈ C . It is an entire function, and if α + α > it has the Mellin-Barnes integral form (page 132, equation (6.1.14) of [200]) E α ,β ; α ,β ( z ) = 12 πi (cid:90) L Γ( s )Γ(1 − s )Γ( β − α s )Γ( β − α s ) ( − z ) − s d s, (D.75) irst passage properties of asymmetric Lévy flights L = L −∞ is a contour running in a horizontal strip, from −∞ + iφ to −∞ + iφ ,with −∞ < φ < < φ < ∞ . This contour separates the poles of the Gammafunctions Γ( s ) and Γ(1 − s ) . The function E α ,β ; α ,β ( z ) with α + α > converges forall z (cid:54) = 0 . For real values of the parameters α , α ∈ R and complex values of β , β ∈ C the four-parametric Mittag-Leffler function E α ,β ; α ,β can be represented in termsof the generalised Wright function and the Fox H -function. If α > , α < and thecontour of integration in expression (D.75) is chosen as L = L −∞ for α + α > , byidentification with the corresponding Mellin-Barnes integral definiont of the H -functionone can obtain the following representation of E α ,β ; α ,β in terms of the H -function(page 135, equation (6.1.28) [200]) E α ,β ; α ,β ( z ) = H , , (cid:34) − z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (0 , , ( β , − α )(0 , , (1 − β , α ) (cid:35) . (D.76)From equations (D.72) and(D.74) by setting α = α , β = α − , α = − , β = 1 /α ,and z = d α /ξt , we finally obtain ℘ ( t ) = t − /α d α − αξ − /α H , , (cid:34) − d α ξt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (0 , , (1 /α, , , (2 − α, α ) (cid:35) . (D.77) Appendix D.6. Asymptotic of the first-passage time PDF of α -stable processes with α ∈ (1 , , β ∈ [ − , , or α ∈ (0 , , β ∈ ( − , , or α = 1 , β = 0 We write equation (D.2) as ln q + ( λ, k ) = ∞ (cid:90) λ ∞ (cid:90) e − ut ∞ (cid:90) ( e i kx − f ( x, t )d x d t d u, (D.78)and split this expression into two terms, ln q + ( λ, k ) = ∞ (cid:90) λ ∞ (cid:90) e − ut ∞ (cid:90) e i kx f ( x, t )d x d t d u − ∞ (cid:90) λ ∞ (cid:90) e − ut ∞ (cid:90) f ( x, t ) d x d t d u. (D.79)We now employ theorem 4 from [201], which says that the Laplace transform withrespect to x of an α -stable law in the Z-form of the characteristic function has theform (cid:107) (cid:96) Zα,ρ ( s, t ) = ∞ (cid:90) e − sx (cid:96) Zα,ρ ( x, t )d x = sin ( πρ ) π ∞ (cid:90) exp( − tK Zα ( sx ) α ) x + 2 x cos ( πρ ) + 1 d x. (D.80) (cid:107) Except for α ∈ (0 , , β = 1 , − irst passage properties of asymmetric Lévy flights s → − i k ,while for the second term s → . Then we get ln q + ( λ, k ) = ∞ (cid:90) λ ∞ (cid:90) e − ut sin ( πρ ) π ∞ (cid:90) exp( − tK Zα ( − i kx ) α ) x + 2 x cos ( πρ ) + 1 d x d t d u − ∞ (cid:90) λ ∞ (cid:90) e − ut sin ( πρ ) π ∞ (cid:90) x + 2 x cos ( πρ ) + 1 d x d t d u. (D.81)In this expression we change the order of integration and first take the integrals over t , ln q + ( λ, k ) = sin ( πρ ) π ∞ (cid:90) λ ∞ (cid:90) x + 2 x cos ( πρ ) + 1 ∞ (cid:90) (cid:16) e − tu − tK Zα ( − i kx ) α − e − tu (cid:17) d t d x d u = sin ( πρ ) π ∞ (cid:90) λ ∞ (cid:90) ( u + K Zα ( − i kx ) α ) − − ( u ) − x + 2 x cos ( πρ ) + 1 d x d u. (D.82)In the next step we again change the order of integration, ln q + ( λ, k ) = − sin ( πρ ) π ∞ (cid:90) K Zα ( − i kx ) α x + 2 x cos ( πρ ) + 1 (cid:90) ∞ λ u ( u + K Zα ( − i kx ) α ) d u d x = sin ( πρ ) π ∞ (cid:90) ln λ − ln ( λ + K Zα ( − i kx ) α ) x + 2 x cos ( πρ ) + 1 d x (D.83)Now, we split the above equation into two terms, ln q + ( λ, k ) = sin ( πρ ) π ∞ (cid:90) ln λx + 2 x cos ( πρ ) + 1 d x − sin ( πρ ) π ∞ (cid:90) ln ( λ + K Zα ( − i kx ) α ) x + 2 x cos ( πρ ) + 1 d x. (D.84)By defining the first term in the right hand side of equation (D.84) as ln r ( λ ) = sin ( πρ ) π ∞ (cid:90) ln λx + 2 x cos ( πρ ) + 1 d x (D.85)and using ∞ (cid:90) x + 2 x cos ( πρ ) + 1 d x = πρ sin ( πρ ) , (D.86)we get ln r ( λ ) = ρ ln λ. (D.87) irst passage properties of asymmetric Lévy flights λ → and then useequation (D.86) and the integral ∞ (cid:90) ln xx + 2 x cos ( πρ ) + 1 d x = 0 . (D.88)The result is sin ( πρ ) π ∞ (cid:90) ln ( K Zα ( − i kx ) α ) x + 2 x cos ( πρ ) + 1 d x = ρ ln ( K Zα ( − i k ) α ) . (D.89)Combining equations (D.87) and (D.89) we obtain the asymptotic of ln q + ( λ, k ) at small λ in the form ln q + ( λ, k ) ≈ ρ ln λ − ρ ln ( K Zα ( − i k ) α ) , (D.90)and thus q + ( λ, k ) ≈ λ ρ ( K Zα ( − i k ) α ) ρ , λ → . (D.91)Going back to equation (D.2) by inverse Fourier transform we find ( x > ) ddx p + ( λ, x ) ≈ π ∞ (cid:90) −∞ e − i kx (cid:18) λ ρ ( K Zα ( − i k ) α ) ρ (cid:19) d k = λ ρ x αρ − ( K Zα ) ρ Γ( αρ ) . (D.92)With the boundary condition p + ( λ, x = 0) = 0 we arrive at p + ( λ, x ) = λ ρ x αρ ( K Zα ) ρ Γ(1 + αρ ) . (D.93)Following equation (D.1), ℘ ( λ ) ∼ − d αρ ( K Zα ) ρ Γ(1 + αρ ) λ ρ . (D.94)Finally, with the help of the Tauberian theorem [165], see equation (D.28), the long timeasymptotic for the cases α ∈ (1 , , β ∈ [ − , , or α ∈ (0 , , β ∈ ( − , , and α = 1 , β = 0 has the form ℘ ( t ) ∼ ρ ( K Zα ) − ρ d αρ Γ(1 − ρ )Γ(1 + αρ ) t − ρ − . (D.95)In this expression the exponent of t was obtained in [170], Proposition VIII.1.2, p. 219,while the prefactor was derived by another method in [173], Corollary, p. 564, and [174],Theorem 3b, p. 285.To represent the above equation in the A-form of the characteristic function weneed to use relation (A.11). Thus, we arrive at the desired result (46). irst passage properties of asymmetric Lévy flights Appendix E. Some properties of the Mittag-Leffler and the Wrightfunctions
The (one-parameter) Mittag-Leffler function is defined by the following seriesrepresentation, which is convergent in the whole complex plane [138, 167] E α ( z ) = ∞ (cid:88) n =0 z n Γ( αn + 1) , α > , z ∈ C . (E.1)Its integral representation is E α ( z ) = 12 π i (cid:90) Ha ζ α − e ζ ζ α − z d ζ, α > , z ∈ C , (E.2)where the Hankel integration path is a loop which starts and ends at −∞ and followsthe circular disk | ζ | ≤ | z | /α in the positive sense, − π ≤ arg ζ ≤ π on Ha. Theequivalence between the series and integral representations can be proven by using theHankel formula for the Gamma function z ) = 12 π i (cid:90) Ha e ζ ζ − z d ζ, z ∈ C . (E.3)The Mittag-Leffler function is completely monotonous on the negative real axis ( z < )if < α ≤ . The Mittag-Leffler function is connected to the Laplace integral throughthe identity [167] E α ( − λx α ) ÷ L { E α ( − λx α ); s } = ∞ (cid:90) e − sx E α ( − λx α )d x = s α − s α + λ , (E.4)for Re( s ) > | λ | /α . From here we easily get two useful formula, see also equations (E.52),(E.54), and (E.55)) in [167], − λ ddx E α ( − λx α ) ÷ s α + λ , (E.5)and − λ d dx E α ( − λx α ) ÷ ss α + λ , (E.6)where α > and Re( s ) > | λ | /α .The Wright W function has the series representation [167] (convergent in the wholecomplex plane) W α,β ( z ) = ∞ (cid:88) n =0 ( z ) n n !Γ( αn + β ) , α > − , β ∈ C . (E.7)The integral representation of this function is W α,β ( z ) = 12 π i (cid:90) Ha e σ + zσ − α d σσ β , α > − . (E.8)For α = 0 we get W ,β ( z ) = e z / Γ( β ) . irst passage properties of asymmetric Lévy flights M function has the series representation [167] M α ( z ) = ∞ (cid:88) n =0 ( − z ) n n !Γ(1 − α − αn ) = 1 π ∞ (cid:88) n =1 ( − z ) n − ( n − αn ) sin ( απn ) , (E.9)where < α < . We note that M α (0) = 1 / Γ(1 − α ) . The radius of convergence of thepower series is infinite for < α < . The integral representation of the M -function is M α ( z ) = 12 π i (cid:90) Ha e σ − zσ α d σσ − α , z ∈ C , < α < . (E.10)Since the M -function is entire in z the exchange between the series and the integral in thecalculations is legitimate. For the special case α = 1 / the M -function can be expressedin terms of the known functions M / ( z ) = exp( − z / / √ π . Another importantproperty of the M -function is the asymptotic representation M α ( x ) as x → + ∞ . By asaddle-point approximation it is shown in [202] that M α ( x/α ) ∼ x ( α − / / (1 − α ) (cid:112) π (1 − α ) exp (cid:18) − − αα x / (1 − α ) (cid:19) . (E.11)Recalling the integral representation for large argument of the Mittag-Leffler function(E.2), for the Laplace transform of the M α ( r ) one can write M α ( r ) ÷ E α ( − s ) , < α < . (E.12)The relevant Laplace transform pair related to the M α ( r − α ) function is [167] λαr α +1 M α ( λr − α ) ÷ e − λs α , < α < , λ > . (E.13)The M-function is non-negative, integrable, and normalised on the positive semi-axis[167] ∞ (cid:90) M α ( r )d r = 1 , < α < , (E.14)and also ∞ (cid:90) αr − − α M α ( r − α )d r = 1 , < α < . (E.15) References [1] van Kampen NG 1981
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