First passage time moments of asymmetric Lévy flights

Abstract

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.

Full text

Journal of Physics A: Mathematical and Theoretical
J. Phys. A: Math. Theor. 53 (2020) 275002 (43pp) https://doi.org/10.1088/1751-8121/ab9030
First passage time moments of asymmetric
Lévy flights
Amin Padash1,2, Aleksei V Chechkin2,3,Bartłomiej
Dybiec4, Marcin Magdziarz5, Babak Shokri1,6and Ralf
Metzler2,7
1Physics Department of Shahid Beheshti University, 19839-69411 Tehran, Iran
2Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm,
Germany
3Akhiezer Institute for Theoretical Physics, 61108 Kharkov, Ukraine
4Marian Smoluchowski Institute of Physics, and Mark Kac Center for Complex
Systems Research, Jagiellonian University, ul. St. Lojasiewicza 11, 30-348 Krakow,
Poland
5Faculty of Pure and Applied Mathematics and Hugo Steinhaus Centre, Wrocław
University of Science and Technology, Wyspianskiego 27, 50-370, Wrocław, Poland
6Laser and Plasma Research Institute, Shahid Beheshti University, 19839-69411
Tehran, Iran
E-mail: rmetzler@uni-potsdam.de
Received 10 February 2020, revised 18 April 2020
Accepted for publication 5 May 2020
Published 16 June 2020
Abstract
We investigate the rst-passage dynamics of symmetric and asymmetric Lévy
ights in semi-innite and bounded intervals. By solving the space-fractional
diffusion equation, we analyse the fractional-ordermoments of the rst-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 ights is presented. For the semi-innite domain, in certain
special cases analytic results are derived explicitly, and in bounded intervals a
general analytical expression for the mean rst-passage time of Lévy ights
with arbitrary skewness is presented. These results are complemented with
extensive numerical analyses.
Keywords:Lévy ight, rst passage time moments, fractional diffusion equation
(Some gures may appear in colour only in the online journal)
7Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons
Attribution 4.0 licence. Any further distribution of this work must maintain attribution
to the author(s) and the title of the work, journal citation and DOI.
1751-8121/20/275002+43$33.00 © 2020 The Author(s). Published by IOP Publishing Ltd Printed in the UK 1

J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
1. Introduction
Lévy ights (LFs) correspond to a class of Markovian random walk processes that are charac-
terised by an asymptotic power-law form for the distribution of jump lengths with a diverging
variance [1–5]. The name ‘Lévy ight’ was coined by Benoît Mandelbrot, in honour of his
formative teacher, French mathematician Paul Pierre Lévy [1,6]. The trajectories of LFs are
statistical fractals [1], characterised by local clusters interspersed with occasional long jumps.
Due to their self-similar character, LFs display ‘clusters within clusters’ on all scales. This
emerging fractality [1–3,7] makes LFs efcient search processes as they sample space more
efciently than normal Brownian motion: in one and two dimensions8Brownian motion is
recurrent and therefore oversamples the search space. LFs, in contrast, reduce oversampling
due to the occurrence of long jumps [8–18]. As search strategies LFs were argued to be
additionally advantageous as, due to their intrinsic lack of length scale they are less sensi-
tive to time-changing environments [15]. Concurrently in an external bias LFs may lose their
lead over Brownian search processes [19,20]. LFs were shown to underlie human movement
behaviour and thus lead to more efcient spreading of diseases as comparedto diffusive,Brow-
nian spreading [21–23]. LFs appear as traces of light beams in disordered media [24], and
in optical lattices the divergence of the kinetic energy of single ions under gradient cool-
ing are related to Lévy-type uctuations [25]. Finally, we mention that Lévy statistics were
originally identied in stock market price uctuations by Mandelbrot and Fama [26,27],
see also [28].
Mathematically, LFs are based on α-stable distributions (or Lévy distributions) [29,30]
which emerge as limiting distributions of sums of independent, identically distributed (i.i.d.)
random variables according to the generalised central limit theorem—that is, they have their
own, well-dened domains of attraction [2,3,29,30]. The characteristic function of an α-stable
process, which is a continuous-timecounterpart of an LF, is given as [31,32]
ˆ
α,β(k,t)=∞
−∞
α,β(x,t)eikx dx=exp −tKα|k|α[1 −iβsign(k)ω(k,α)] +iμkt,(1)
with the stability index (Lévy index) αthat is allowed to vary in the interval 0 <α2. More-
over, equation (1) includes the skewness parameter βwith −1β1, and Kα>0 is a scale
parameter. The shift parameter μcan be any real number, and the phase factor ωis dened as
ω(k,α)=⎧
⎨
⎩
tan πα
2,α=1
−2
πln |k|,α=1.(2)
Physically, the parameter μaccounts for the constant drift in the system. In this paper, we con-
sider the rst-passage time moments in the absence of a drift, μ=0. The stable index αis
responsible for the slow decay of the far asymptotics of the α-stable probability density func-
tion (PDF). Indeed, symmetric α-stable distributions in absence of a drift (β=μ=0) have the
characteristic function exp(−Kα|k|αt), whose asymptote in real space has the power-law form
Kαt|x|−1−α(‘heavy tail’ or ‘long tail’), and thus absolute moments |x|δof order δ<α
8For most search processes of animals for food or other resources these are the relevant dimensions: the case of one
dimension is relevant for animals whose food sources are found along habitat borders such as the lines of shrubbery
along streams or the boundaries of forests. Two-dimensional search within given habitats is natural for land bound ani-
mals, but even airborne or seaborne animals typically forage within a shallow depth layer compared to their horizontal
motion.
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
exist [2,3,31,33]. The scale parameter Kα(along with the stable index α) physically sets the
size of the LF-jumps. The skewness βmay be related to an effective drift or counter-gradient
effects [34,35]. LFs have been applied to explain diverse complex dynamic processes, where
scale-invariant phenomena take place or can be suspected [1,29]. According to the gener-
alised central limit theorem, each α-stable distribution with xed α<2 attracts distributions
with innite variance which decay with the same law as the attracting stable distribution. A
particular case of a stable density is the Gaussian for α=2, for which moments of all orders
exist. We note that the Gaussian law not only attracts distributions with nite variance but also
distributions decaying as |x|−3; that is, distributions, whose variance is marginally innite
[30,36]. To t real data, in particular, in nance, which feature heavy-tailed distributions on
intermediate scales, however, with nite variance, the concept of the truncated LFs has been
introduced according to which the truncation of the heavy tail at larger scales is achieved either
by an abrupt cutoff [37], an exponential cutoff [38], or by a steeper power-law decay [39–41].
The efciency of the spatial exploration and search properties of a stochastic process
is quantied by the statistics of the ‘rst-hitting’ or the ‘rst-passage’ times [42–45]. For
instance, the rst-passage of a stock price crossing a given threshold level serves as a trig-
ger to sell the stock. The event of rst-hitting would correspond to the event when exactly a
given stock price is reached. Of course, when stock prices change continuously (as is the case
for a continuous Brownian motion) both rst-passage and rst-hitting are equivalent [44]. In
contrast, for an LF with the propensity of long, non-local jumps the two denitions lead to dif-
ferent results. In general, the rst-passage will be realised earlier: it is more likely that an LF
jumps across a point in space [46] effecting so-called ‘leapovers’ [47,48]. For a foraging alba-
tross, for instance, the rst-hitting would correspond to the moment when it locates a single,
almost point-like, forage sh. The rst-passage would correspond to the event when the alba-
tross crosses the circumference of a large sh shoal. We here focus on the rst-passage time
statistic of LFs, and our main objective is the study of the moments of the rst-passage time
for asymmetric LFs in semi-innite and bounded domains. Such moments can be conveniently
used to quantify search processes. Themost commonly used moment is the mean rst-passage
time (MFPT) τ=∞
0℘(τ)τdτin terms of the rst-passage time density ℘(τ)(seebelow),
when it exists. However, other denitions such as the mean of the inverse rst-passage time,
1/τhave also been studied [19,20]. More generally, the spectrum of fractional order rst-
passage time moments τqis important to characterise the underlying stochastic process from
measurements. The characteristic times τand 1/τthus correspond to q=1andq=−1,
respectively. In what follows we study the behaviour of the spectrumof τqas function of the
LF parameters.
A set of classical results exists for the rst-passage time properties of LFs in a semi-innite
domain. In particular, [49,50] used limit theorems of i.i.d. random variables to obtain the
asymptotic behaviourof the rst-passage time distribution.Based on a continuous-timestorage
model the rst-passage time of a general class of Lévy processes was studied in [51]. By apply-
ing the laws of ladder processes the asymptotic of the rst-passage time distribution of Lévy
stable processes was investigated in [52]. After becoming clear that LFs have essential appli-
cations in different elds of science, several remarkable results were established. Thus, in [53]
it was reported that one-dimensional symmetric random walks with independent increments in
half-space have universal property.Also [54] showed that the survival probability of symmetric
LFs in a one-dimensional half-space with an absorbing boundary at the origin is independent
of the stability index αand thus displays universal behaviour. It is by now well-known that
the mentioned results are a consequence of the celebrated Sparre Andersen theorem [55,56].
Accordingly, the PDF of the rst-passage times of any symmetric and Markovian jump process
originally released at a xed point x0from an absorbing boundary in semi-innite space has
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
the universal asymptotic scaling ℘(τ)τ−3/2[43,46–48]. This law has been conrmed by
extensive numerical simulations of the rst-passage time PDF [47] and the associated survival
probability [57] of symmetric LFs within a Langevin dynamic approach (see below). Further-
more, the asymptotic of the survival probability of symmetric, discrete-time LFs was studied
in [58,59], and based on the space-fractional diffusion equation the rst-passage time PDF
and the survival probability was investigated in [60]. Starting from the Skorokhod theorem,
the Sparre Andersen theorem could be successfully reproduced analytically [48,60]. Other
analytical and numerical results that concern the rst-passage properties of asymmetric LFs
in a semi-innite domain are the following. For one-sided α-stable process (0 <α<1 with
β=1) the rst-passage time PDF and the MFPT was studied in [48]. In [47] the authors used
Langevin dynamic simulations to study the asymptotic behaviour of the rst-passage time
PDF of extremal two-sided (1 <α<2 with β=−1) α-stable laws. Moreover, by employing
the space-fractional diffusion equation the rst-passage time PDF and the survival probability
of extremal two-sided α-stable laws (1 <α<2 with β=1) and the asymptotic of the rst-
passage PDF of general, asymmetric LFs was investigated in [60]. We also mention the study
on anomalous inltration based on Lévy processes [61].
With respect to the rst-passage from a nite interval a number of classical results for sym-
metric α-stable process were reported in a series of papers in the 1950s and 1960s. To name
a few, the MFPT of one-dimensional symmetric (β=0) Cauchy (α=1) processes [62], the
MFPT of two-dimensional Brownian motion [63], and the MFPT of one-dimensional sym-
metric α-stable process with stability index 0 <α<1 were studied [64]. Moreover, for the
case 0 <α2andβ=0 the results of the rst-passage probability in one dimension [65],
the MFPT as well as the second moment of the rst-passage time PDF in Ndimensions were
reported [66]. One-sided α-stable processes with 0 <α<1andβ=1 in a nite interval were
studied with the help of arc-sine laws of renewal theory in [67] and by using the harmonic mea-
sure of a Markov process in [68]. A closed form for the MFPT by potential theory method was
obtained in [69]. For completely asymmetric LFs the rst-passage time of the two-sided exit
problem was addressed in [69–74]. The residual MFPT of LFs in a one-dimensional domain
was investigated in [75]. We also mention that necessary and sufcient conditions for the nite-
ness of the moments of the rst-passage time PDF of a general class of Lévy processes in
terms of the characteristics of the random process X(t) were shown by [76]. Additionally, har-
monic functions in a Markovian setting were dened by the mean value property concerning
the distribution of the process being stopped at the rst exit time of a domain [77]. Finally,
the authors in [78], by using the Green’s function of a Lévy stable process [79], obtained the
non-negativeharmonic functions for the stable process killed outside a nite interval, allowing
the computation of the MFPT.
We also mention that various problems of the rst-passage for symmetric and asymmet-
ric α-stable processes, as well as for two- and three-dimensional motions, were considered
by different approaches. These include Monte-Carlo simulations and the Fredholm integral
equation [80,81], Langevin dynamics simulations [82,83], fractional Laplacian operators [84,
85], eigenvalues of the fractional Laplacian [86], and the backward fractional Fokker–Plank
equation [87]. Moreover, noteworthy are simulations of radial LFs in two dimensions [7], the
effect of Lévy noise on a gene transcriptional regulatory system [88], the study of the mean
exit time and the escape probability of one- and two-dimensional stochastic dynamical sys-
tems with non-Gaussian noises [89–91]. The tail distribution of the rst-exit time of LFs from
aclosedN-ball of radius Rin a recursive mannerwas constructed in [92]. Very recently, exten-
sive simulations of the space-fractional diffusion equation and the Langevin equation were used
to investigate the rst-passage properties of asymmetric LFs in a semi-innite domain in [60].
In the same work application of the Skorokhod theorem allowed to derive a closed form for
4

J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
the rst-passage time PDF of extremal two-sided α-stable laws with stability index 1 <α<2
and skewness β=±1, as well as the rst-passage time PDF asymptotic for asymmetric Lévy
stable laws with arbitrary skewness parameter β.
The rst part of this paper, based on our previous results in [60], is devoted to the study
of fractional order moments of the rst-passage time PDF of LFs in a semi-innite domain
for symmetric (0 <α<2 with β=0), one-sided (0 <α<1 with β=1), extremal two-
sided (1 <α<2 with β=±1), and a general form (α∈(0, 2] with β∈[−1, 1], excluding
α=1 with β=0) α-stable laws. Specically we obtain a closed-form solution for the frac-
tional moments of the rst-passage time PDF for one-sided and extremal two-sided α-stable
processes, and we report the conditionsfor the niteness of the fractional moments of the rst-
passage time PDF for the full class of α-stable processes. We also present comparisons with
numerical solutions of the space-fractional diffusion equation. In the second part we derive a
closed form of the MFPT of asymmetric LFs in a nite interval by solving the fractional differ-
ential equation for the moments of the rst-passage time PDF. In particular cases we present a
comparison between our analytical results with the numerical solution of the space-fractional
diffusion equation as well as simulations of the Langevin equation. Moreover, we show that the
MFPT of LFs in a nite interval is representative for the rst-passage time PDF by analysing
the associated coefcient of variation.
The structure of the paper is as follows. In section 2we introduce the space-fractional
diffusion equation in a nite interval. In section 3, the numerical schemes for the space-
fractional diffusion equation and the Langevin equation are presented. We set up the cor-
responding formalism to study the moments of the rst-passage time PDF in section 4.
Section 5then presents the analytic and numerical results of the fractional moments of the
rst-passage time PDF for symmetric, one-sided, and extremal two-sided stable distribu-
tions in semi-innite domains. We derived a closed-form solution of the MFPT for asym-
metric LFs in a nite interval in section 6and compare with the numerical solution of the
space-fractional diffusion equation and the Langevin dynamics simulations. We draw our
conclusions in section 7, and details of the mathematical derivations are presented in the
appendices.
2. Space-fractional diffusion equation in a finite domain
Fractional derivatives have been shown to be convenient when formulating the generalised
continuum diffusion equations for continuous time random walk processes with asymptotic
power-law asymptotes for both the distributions of sojourn times and jump lengths [4,5,
93–95]. We here use the space-fractional diffusion equation for innite domains and its exten-
sion to semi-innite and nite domains to describe the dynamics of LFs. From a probabilistic
point of view, the basic Caputo and Riemann–Liouville derivatives of order α∈(0, 2) can be
viewed as generators of LFs interrupted on crossing a boundary [46,48,96]. The corresponding
equation to describe LFs has the following expression for the PDF Pα,β(x,t|x0)
∂Pα,β(x,t|x0)
∂t=KαDα
xPα,β(x,t|x0)(3)
with initial condition Pα,β(x,0|x0)=δ(x−x0), where Dα
xis the space-fractional operator for
motion conned to the interval [−L,L],
Dα
xf(x)=Lα,β−LDα
xf(x)+Rα,βxDα
Lf(x).(4)
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Here −LDα
xand xDα
Lare left and right space-fractional derivatives, respectively. Let us rst
consider the case α=1and−1β1. We use the Caputo form of the fractional operators
dened by (n−1<α<n)as[97]
−LDα
xf(x)=1
Γ(n−α)x
−L
f(n)(ζ)
(x−ζ)α−n+1dζ,(5)
and
xDα
Lf(x)=(−1)n
Γ(n−α)L
x
f(n)(ζ)
(ζ−x)α−n+1dζ. (6)
Lα,βand Rα,βare the left and right weight coefcients, dened as [98,99]
Lα,β=−1+β
2cos(
απ
2),Rα,β=−1−β
2cos(
απ
2).(7)
For the case α=1andβ=0wehaveL1,0 =R1,0 =1/π, and the left and right space-fractional
operators respectively read [100]
−LD1
xf(x)=−x
−L
f(1)(ζ)
x−ζdζ,(8)
xD1
Lf(x)=L
x
f(1)(ζ)
ζ−xdζ. (9)
In the present paper, we do not consider the particular case α=1, β=0 since it cannot be
described in terms of a space-fractional operator.
We end this section by adding a remark concerning our choice of the Caputo form of
the fractional derivatives (5)and(6): it is known that there are different equivalent de-
nitions of the fractional Laplacian operator in unbounded domains [101], which in general
case loose their equivalence in bounded domains, see, e.g., [102–104]. Such ambiguity, how-
ever, does not hold in case of the rst passage problem when absorbing boundary condi-
tions are applied. In this case it is easy to verify that the Riemann–Liouville derivatives are
equivalent to the Caputo derivatives [97,100]. However, in the general case for bounded
domains the use of the Caputo derivative is preferable in applied problems for the follow-
ing reason: the Riemann–Liouville approach leads to boundary conditions, which do not
have known direct physical interpretation [97], and thus the left and right Riemann–Liouville
derivatives might be singular at the lower and upper boundaries, respectively, as discussed in
[98] in detail—a problem circumvented by dening the fractional derivative in the Caputo
sense.
3. Numerical schemes
Apart from analytical approaches to be specied below, to determine the moments of the rst-
passage time PDF of α-stable processes we will employ two numerical schemes based on the
space-fractional diffusion equation and the Langevin equation for LFs. We here detail their
specic implementation.
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
3.1. Diffusion description
Numerical methods to solve space-fractional diffusion equations are relatively sparse, and the
majority of the publications are based on the nite-difference scheme [105,106] and nite-
element methods [107–109] as well as the spectral approach [110,111]. In this paper, we use
the nite-difference scheme to solve the space-fractional diffusion equation introduced in the
preceding section. Here we only outline the essence of the method and refer to [60] for further
details. The computationally most straightforward method arises from the forward-difference
scheme in time on the left-hand side of equation (3),
∂
∂tf(xi,tj)=fj+1
i−fj
i
Δt+O(Δt), (10)
where fj
i=f(xi,tj), xi=(i−I/2)Δx,andtj=jΔt,whereΔxand Δtare step sizes in
position and time, respectively. The iand jare non-negative integers, i=0, 1, 2, ...,I,and
Δx=2L/I. Similarly, j=0, 1, 2, ...,J−1, t0=0, tJ=t,andΔt=t/J. Absorbing bound-
ary conditions for the determination of the rst-passage events imply fj
0=fj
I=0forallj.The
integrals on the right-hand side of equation (3) are discretised as follows. For 0 <α<1,
xi
−L
f(1)(ζ,tj)
(xi−ζ)αdζ=
i

k=1
fj
k−fj
k−1
Δxxk
xk−1
1
(xi−ζ)αdζ+O(Δx2−α) (11)
for the left derivative, and
L
xi
f(1)(ζ,tj)
(ζ−xi)αdζ=
I−1

k=i
fj
k+1−fj
k
Δxxk+1
xk
1
(ζ−xi)αdζ+O(Δx2−α) (12)
for the right derivative. Thisscheme is called L1 scheme and is an efcient way to approximate
the Caputo derivative of order 0 <α<1[112–114] with error estimate O(Δx2−α). For the
case 1 <α<2 the suitable method to discretise the Caputo derivative is the L2 scheme [112,
114,115], namely,
xi
−L
f(2)(ζ,tj)
(xi−ζ)α−1dζ=
i

k=1
fj
k+1−2fj
k+fj
k−1
(Δx)2xk
xk−1
1
(xi−ζ)α−1dζ+O(Δx)
(13)
for the left derivative, and
L
xi
f(2)(ζ,tj)
(ζ−xi)α−1dζ=
I−1

k=i
fj
k+1−2fj
k+fj
k−1
(Δx)2xk+1
xk
1
(ζ−xi)α−1dζ+O(Δx)
(14)
for the right derivative. We note that the truncation error of the L2 scheme is O(Δx)[115,
116]. For the special case α=1andβ=0 we approximate the derivative in space with the
backward difference scheme
xi
−L
f(1)(ζ,tj)
xi−ζdζ=
i

k=1
fj
k−fj
k−1
Δx
2
2(i−k)+1+O(Δx2) (15)
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 1. Schematic of our setup. In the interval of length 2Lthe initial condition is given
by a δ-distribution located at x0, which is chosen the distance daway from the right
boundary. At both interval boundaries we implement absorbing boundary conditions,
that is, when the particle hits the boundaries or attempts to move beyond them, it is
absorbed.
for the left derivative, and with a forward difference scheme
L
xi
f(1)(ζ,tj)
ζ−xi
dζ=
I−1

k=i
fj
k−fj
k+1
Δx
2
2(k−i)+1+O(Δx2) (16)
for the right derivative. We note that here the truncation error is the order O(Δx2). By
substitution of equations (10)–(16)into(3) we obtain
Afj+1=Bfj, (17)
where the coefcients Aand Bhave matrix form of dimension (I+1) ×(I+1) and
j=0, 1, 2, ...,J−1. In the numerical scheme for the setup used in our numerical simula-
tions (see section 4and gure 1below) the initial condition f(x,0)=δ(x−x0)atx0=L−d
is approximated as
f(xi,0)=(Δx)−1,i=(2L−d)/Δx
0, otherwise .(18)
In the next step, the time evolution of the PDF is obtained by applying the absorbing boundary
conditions fj
0=fj
I=0forallj.
3.2. Langevin dynamics
The fractional diffusion equation (3) can be related to the LF Langevin equation [57,117,118]
d
dtx(t)=K1/α
αζ(t), (19)
where ζ(t) is Lévy noise characterised by the same αand βparameters as the space-fractional
operator (3) and with unit scale parameter. The Langevinequation (19) provides a microscopic
(trajectory-wise) representation of the space-fractional diffusion equation (3). Therefore, from
an ensemble of trajectories generated from equation (19), it is possible to estimate the time-
dependent PDF whose evolution is described by equation (3). In numerical simulations, LFs
can be described by the discretised form of Langevin equation
x(t+Δt)=x(t)+K1/α
α(Δt)1/αζt, (20)
where ζtstands for the sequence of i.i.d. α-stable random variables with unit scale parameter
[31,119] and identical index of stability αand skewness βas in equation (19). Relation (20)
is exactly the Euler–Maruyama approximation [120–122] to a general α-stable Lévy process.
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
From the trajectories x(t), see equations (19)and(20), it is also possible to estimate the
rst-passage time τas
τ=min{t:|x(t)|L}.(21)
From the ensemble of rst-passage times, it is then possible to obtain the survival probability
S(t), which is the complementary cumulative density of rst-passage times. More precisely,
the initial condition is S(0) =1, and at every recorded rst-passage event at time τi,S(t)is
decreased by the amount 1/Nwhere Nis the overall number of recorded rst-passage events.
4. First passage time properties of α-stable processes
For an α-stable random process, the survival probability and the rst-passage time are observ-
able statistical quantities characterising the stochastic motion in bounded domains with absorb-
ing boundary conditions. In the following, we investigate the properties of the rst-passage
time moments in a semi-innite and nite interval for symmetric and asymmetric α-stable laws
underlying the space-fractional diffusion equation. In addition a comparison with the Langevin
approach and with analytical expressions for the MFPT of LFs in a nite interval is presented.
To this end, we use the setup shown in gure 1, in which the absorbing boundaries are located
at −Land L, and the centre point of the initial δ-distribution is located the distance daway
from the right boundary.
The survival probability that up until time ta random walker remains ‘alive’ within the
interval [−L,L]isdenedas[43,45]
S(t|x0)=L
−L
Pα,β(x,t|x0)dx, (22)
Recall that Pα,β(x,t|x0) is the PDF of an LF conned to the interval [−L,L] which starts at x0.
The associated rst-passage time PDF reads
℘(t|x0)=−dS(t|x0)
dt.(23)
The rst-passage time PDF satises in particular the normalisation
∞
0
℘(t|x0)dt=1, (24)
and the positive integer moments of this random variable are dened as
τm(x0)=∞
0
tm℘(t|x0)dt=∞
0
mtm−1S(t|x0)dt,m=1, 2, .... (25)
Employing the Laplace transform,
f(t)÷L{f(t); s}=∞
0
e−st f(t)dt, (26)
we obtain
τm(x0)=(−1)m∂m
∂sm℘(s|x0)s=0
.(27)
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Conversely, following the procedure suggested in [84], by substitution of equation (22)into
equation (25)weget
τm(x0)=∞
0
mtm−1L
−L
Pα,β(x,t|x0)dxdt.(28)
Applying the backward space-fractional Kolmogorov operator Dα
x0in a nite domain9(see
details in appendix A),
Dα
x0f(x0)=Rα,β−LDα
x0f(x0)+Lα,βx0Dα
Lf(x0), (29)
to both sides of equation (28),
Dα
x0τm(x0)=∞
0
mtm−1L
−L
Dα
x0Pα,β(x,t|x0)dxdt, (30)
and using the corresponding backward Kolmogorov equation
∂Pα,β(x,t|x0)
∂t=KαDα
x0Pα,β(x,t|x0), (31)
we get
Dα
x0τm(x0)=m
Kα∞
0
tm−1∂
∂tL
−L
Pα,β(x,t|x0)dxdt.(32)
In the limit m=1,
Dα
x0τ(x0)=1
KαL
−L
Pα,β(x,∞|x0)dx−L
−L
Pα,β(x,0|x0)dx.(33)
Then, by including the initial condition of the density function Pα,β(x,0|x0)=δ(x−x0),
where x0∈[−L,L], we get the functional relation
Dα
x0τ(x0)=−1
Kα
(34)
for the MFPT. This result is similar to equation (41) in [84], except that instead of a symmetric
Riesz–Feller operator we here employ a more general formof the fractional derivative operator
Dα
x0which is called backward space-fractional Kolmogorov operator in a nite domain. We
note that in comparison with the forward space-fractionalderivative dened by equation (4)in
equation (29) the left and right weight coefcients are exchanged.
For the case m=2, we have
Dα
x0τ2(x0)=2
Kα∞
0
t∂
∂tL
−L
Pα,β(x,t|x0)dxdt.(35)
Changing the order of integration,
Dα
x0τ2(x0)=2
KαL
−L∞
0
t∂
∂tPα,β(x,t|x0)dtdx, (36)
9More precisely, Dα
x0is the generator of LFs killed upon leaving the domain.
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
integrating by parts in the inner integral,
Dα
x0τ2(x0)=−2
KαL
−L∞
0
Pα,β(x,t|x0)dtdx, (37)
and, once again, changing the order of integration, we nd
Dα
x0τ2(x0)=−2
Kα∞
0L
−L
Pα,β(x,t|x0)dxdt.(38)
Calling on equation (28) with m=1, we obtain the functional relation
Dα
x0τ2(x0)=−2
Kατ(x0) (39)
for the second moment of the rst-passage time PDF.
More generally, by using this recursion relation one can write
Dα
x0τm(x0)=−m
Kατm−1(x0), m=1, 2, .... (40)
By applying Dα
x0on both sides,
(Dα
x0)2τm(x0)=−m
Kα
Dα
x0τm−1(x0), (41)
and with equation (40)wehave
(Dα
x0)2τm(x0)=m(m−1)
Kα2τm−2(x0).(42)
By repeating this procedure, we derive
(Dα
x0)mτm(x0)=(−1)mΓ(1 +m)
Kαm.(43)
This equation is the generalisation of the result obtained in [84] for symmetric LFs (see
equation (44) there).
5. First passage time properties of LFs in a semi-infinite domain
In this section, we investigate the rst-passage time properties of LFs in a semi-innite domain.
The motion starts at x0, and the boundary is located at x=L,insuchawaythatinoursetup
L−x0=d. In order to reproduce numerically the results for semi-innite domain with the
scenario shown in gure 1,weemployLas well as x0, as large as possible in order to allow a
constant d(L=1012 in our simulations).
5.1. Symmetric LFs in a semi-infinite domain
For a semi-innite domain with an absorbing boundary condition, as said above it is well known
that the rst-passage time density for any symmetric jump length distribution in a Markovian
setting has the universal Sparre Andersen scaling ℘(t)t−3/2[43,55,56]. In the theory of
a general class of Lévy processes, that is, homogeneous random processes with independent
increments, there exists a theorem, that provides an analytical expression for the PDF of rst-
passage times in a semi-innite interval, often referred to as the Skorokhodtheorem [32,123].
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Based on this theorem the asymptotic expression for the rst-passage time PDF of symmetric
α-stablelawsis[48]
℘(t)∼dα/2
α√πKαΓ(α/2)t−3/2, (44)
which species an exact expression for the prefactor in the Sparre Andersen scaling law. The
existence of this long-timetail leads to the divergence of the MFPT τin equation (25).This
means that the LF will eventually cross the boundary dwith unit probability, but the expected
time that this takes is innite. For Brownian motion (α=2), the PDF for the rst-passage time
has the well known Lévy–Smirnov form [42]
℘(t)=d
√4πK2t3exp −d2
4K2t, (45)
which is exact for all times [42,43] and whose asymptote coincides with result (44)forthe
appropriate limit α=2.
For the moments of Brownian motion (α=2) we have
τq=∞
0
tqd
√4πK2t3exp −d2
4K2tdt, (46)
where by change of variables u=d2/4K2tand using the integral form of the Gamma function,
Γ(z)=∞
0
ζz−1e−ζdζ,Re(z)>0, (47)
we get (see p 84 in [43])
τq=Γ(1
2−q)
22q√π
d2q
Kq
2
=Γ(1 −2q)
Γ(1 −q)
d2q
Kq
2
,−∞ <q<1/2.(48)
In the last step we used the duplication rule 22zΓ(z)Γ(z+1/2) =2√πΓ(2z).
To nd a closed form of the rst-passage time PDF of LFs based on general symmetric
α-stable probability laws (0 <α<2) remains an unsolved problem. We show the short time
behaviour for symmetric LFs in gure 2, bottom left panel. As can be seen, only for the case of
Brownian motion (α=2) the PDF has value zero at t=0, while for LFs with α<2therst-
passage time PDF exhibitsa non-zero value at t=0, thus demonstrating that LFs can instantly
cross the boundary with their rst jump away from their initial position x0. The magnitude of
℘(t→0) can be estimated from the survival probability, as shown by equations (3) and (A.5)
in [124] for symmetric LFs and here by equation (71) in section 5.2.5 below for asymmetric
LFs with α∈(0, 2] and β∈(−1, 1] (excluding α=1 with β=0). Of course, in the case of
symmetric LFs (β=0) equation (71) coincides with equation (3) in [124]. The values of the
rst-passage time PDF at t=0 obtained by numerical solution of the space-fractional diffusion
equation are in perfect agreement with those obtained from equation (71). Fractional moments
of the rst-passage time PDF for symmetric α-stable laws in a semi-innite domain for differ-
ent ranges of the stability index αare shown in the top left panel of gure2. As can be seen the
fractional moments are nite only for −1<q<1/2, as expected from the Sparre Andersen
universal scaling with exponent 3/2.
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 2. Top left: fractional order moments of the rst-passage time PDF for symmet-
ric (0 <α2, β=0) α-stable laws in a semi-innite domain with d=0.5. Here and
in the following we set Kα=1. Results are shown for the case when Lis sufciently
large (here we used L=1012). Dashed lines represent the numerical solution of the
space-fractional diffusion equation and the solid line shows the analytical result (48)for
Brownian motion. Top right: fractional order moments of the rst-passage time PDF for
one-sided (0 <α<1, β=1) α-stable laws in a semi-innite domain. Symbols show
the numerical solution of the space-fractional diffusion equation with d=0.5, Δx=
0.01, and Δt=0.001, and lines represent the analytic result (56). Bottom left: rst-
passage time PDF of symmetric α-stable laws with 0 <α2andβ=0. Lines cor-
respond to the numerical solution of the space-fractional diffusion equation and the
solid line shows result (45). Bottom right: rst-passage time PDF of one-sided (0 <
α<1 with β=1) α-stable laws obtained by numerical solution of the space-fractional
diffusion equation.
5.2. Asymmetric LFs in a semi-infinite domain
5.2.1. One-sided α-stable processes with 0<α<1and β=1.By applying the Skorokhod
theorem, it can be shown that the rst-passage time PDF of one-sided α-stable laws has the
exact form [48]
℘(t)=ξ
dαMαξt
dα(49)
with
ξ=Kα
cos(απ(ρ−1/2)),ρ=1
2+1
απ arctan(βtan(απ/2)), (50)
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
which connects to our case here via
ξ=Kα
cos(απ/2) ,ρ=1.(51)
Here Mα(z) is the Wright M-function [97,125] (also sometimes called Mainardi function) with
the integral representation [125] (p 241)
Mα(z)=1
2πi
Ha
eσ−zσαdσ
σ1−α,z∈C,0<α<1, (52)
where the contour of integration Ha (the Hankel path) is the loop starting and ending at −∞
and encircling the disk |σ||z|1/α counterclockwise, i.e., |arg(σ)|πon Ha. Here and below
for the asymptotic behaviour of the rst-passage time PDFs we refer the reader to our recent
paper [60]. The asymptotics of the M-function at short and long times is presented in appendix
Eof[60]. The long-time asymptotics of the PDF (49) is given by equations (31) and (32) of
[60], while the short-time asymptotics of (49) is given by equation (33) of [60] (or equivalently,
equation (71) below with ρ=1). By denition (25) of the moments of the rst-passage time
PDF and the rst-passage time PDF (49) of one-sided stable laws, we nd
τq=∞
0
tqξ
dαMαξt
dαdt
=ξ
dα∞
0
tq1
2πi
Ha
σα−1eσ−ξt(σ/d)αdσdt
=ξ
dα
1
2πi
Ha
σα−1eσ∞
0
tqe−ξt(σ/d)αdtdσ. (53)
By change of variables u=ξt(σ/d)αin the inner integral and with the help of equation (47)
we get
τq=dqαΓ(1 +q)
ξq
1
2πi
Ha
σ−qα−1eσdσ. (54)
Using Hankel’s contour integral
1
Γ(z)=1
2πi
Ha
ζ−zeζdζ,z∈C, (55)
we then obtain the fractional order moments of the rst-passage time PDF for one-sided α-
stable laws with 0 <α<1andβ=1,
τq=Γ(1 +q)
Γ(1 +qα)
dqα
ξq,q>−1.(56)
The MFPT (q=1) for one-sided α-stable process was derived in [48,69]. Also, from
equation (27) and the Laplace transform of the rst-passage time PDF, which has the form
of the Mittag-Lefer function [48], it is possible to nd all moments explicitly. In the right
panel of gure 2we show the results for the fractional order moments of one-sided α-stable
laws obtained by numerically solving the space-fractional diffusion equation, along with the
analytical results of equation (56).
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
5.2.2. One-sided α-stable processes with 0<α<1and β=−1.One-sided α-stable laws
with the stability index 0 <α<1 and skewness parameter β=−1 satisfy the non-positivity
of their increments. Therefore, the random walker never crosses the right boundary d.Inthe
semi-innite domain therefore the survival probability remains unity (S(t)=1) and the rst-
passage time PDF ℘(t)=0. Therefore, the fractional moments read
τq=⎧
⎪
⎪
⎨
⎪
⎪
⎩
0, q<0
1, q=0
∞,q>0
.(57)
Due to normalisation of the rst-passage time PDF, τq=1whenq=0.
5.2.3. Extremal two-sided α-stable processes with 1<α<2and β=−1.Stable laws with
stability index 1 <α<2 and skewness β=1orβ=−1 are called extremal two-sided
skewed α-stable laws [128]. Let us rst consider the case 1 <α<2, β=−1. By applying
the Skorokhod theorem it can be shown that the rst-passage time PDF of extremal two-sided
α-stable laws with 1 <α<2andβ=−1 has the following exact form [60]
℘(t)=t−1−1/αd
αξ1/α M1/α d
(ξt)1/α , (58)
in terms of the Wright M-function M1/α. The long-time asymptotic of the PDF (58)isgiven
by equation (41) of [60] or, equivalently, equation (68) below with ρ=1/α. Respectively, the
short-time asymptotic of equation (58) is given by equation (39) of [60], or by equation (71)
below with ρ=1/α.
For the considered case of extremal two-sided α-stable laws with 1 <α<2andβ=−1
by recalling the integral representation (52)oftheM-function, the rst-passage time PDF
moments become
τq=d
αξ1/α ∞
0
tq−1−1/αM1/α d
(ξt)1/α dt
=d
αξ1/α ∞
0
tq−1−1/α 1
2πi
Ha
eσ−d(σ/ξt)1/α dσ
σ1−1/α dt
=d
αξ1/α
1
2πi
Ha
σ1/α−1eσ∞
0
tq−1−1/αe−d(σ/ξt)1/α dtdσ. (59)
Changing variables, u=d(σ/ξt)1/α in the inner integral and with the help of equation (47),
we nd
τq=dqαΓ(1 −qα)
ξq
1
2πi
Ha
σq−1eσdσ=Γ(1 −qα)
Γ(1 −q)
dqα
ξq,−∞ <q<1/α,
(60)
where in the last equality we used equation (55) to get the desired result. In the limit α=2we
recover the fractional moments of the rst-passage time PDF (48) for a Gaussian process. The
left panel of gure 3shows the results of equation (60) along with numerical solutions of the
space-fractional diffusion equation. As can be seen the fractional order moments −∞ <q<
1/α are nite, as they should.
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 3. Left: fractional order moments of the rst-passage time PDF for extremal
two-sided α-stable laws in a semi-innite domain with stability index 1 <α2and
skewness β=−1. Symbols represent the numerical solution of the space-fractional dif-
fusion equation and lines correspond to the analytic result (60). Right: same as in the left
panel but with skewness β=1. Lines show results of equation (67). In both panels, we
used d=0.5andL=1012.
5.2.4. Extremal two-sided α-stable processes with 1<α<2and β=1.Applying the Sko-
rokhod theorem it can be shown that the rst-passage time PDF of the extremal two-sided
α-stable law with stability index 1 <α<2 and skewness β=1 has the following series
representation [60] (see equation (D.73))
℘(t)=t−2+1/αdα−1
αξ1−1/α
∞

n=0
(dα/ξt)n
Γ(αn+α−1)Γ(−n+1/α).(61)
Now, with the help of Euler’s reection formula Γ(1 −z)Γ(z)sin(πz)=πand the relation
sinπ(z−n)=(−1)nsin(πz) we rewrite this expression in the form
℘(t)=sin(π/α)t−2+1/αdα−1
παξ1−1/α
∞

n=0
Γ(n+1−1/α)(−dα/ξt)n
Γ(αn+α−1) .(62)
To obtain the long-time asymptotics of the PDF we take n=0inequation(62) and arrive at
the power-law decay given by equation (43) of [60] or, equivalently, equation (68) below with
ρ=1−1/α.
To calculate the moments of the rst-passage time we use the relation between the Wright
generalised hypergeometric function and the H-function [126] (see equations (1.123) and
(1.140)). We arrive at
℘(t)=sin(π/α)t−2+1/αdα−1
παξ1−1/α H1,2
2,2 dα
ξt
(0, 1), (1/α,1)
(0, 1), (2 −α,α).(63)
Further, with the help of the inversion property of the H-function [126] (property 1.3, equation
(1.58)), we have
℘(t)=sin(π/α)t−2+1/αdα−1
παξ1−1/α H2,1
2,2 ξt
dα
(1, 1), (α−1, α)
(1, 1), (1 −1/α,1).(64)
At short times the H-function representation of the rst-passage PDF leads to equation (44)
of [60] or, equivalently, equation (71) below with ρ=1−1/α. Substitution of equation (64)
16

J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 4. Left: rst-passage time PDF at short times for the extremal α-stable processes
in the semi-innite domain with stability index α=1.5. Right: long-time behaviour of
the same PDF on log–log. The black lines show the asymptotic behaviour of the PDFs.
In both panels d=0.5, symbols represent the numerical solution of the space-fractional
diffusion equation, and the dotted lines show the analytical results namely, equation (58)
for β=−1 and equation (61)forβ=1.
into (25) yields
τq=∞
0
sin(π/α)tq−2+1/αdα−1
παξ1−1/α H2,1
2,2 ξt
dα
(1, 1), (α−1, α)
(1, 1), (1 −1/α,1)dt.(65)
Recalling the Mellin transform of the H-function [126] (p 47, equation (2.8)), we nd
τq=sin(π/α)Γ(1 −1/α −q)Γ(q+1/α)Γ(q)
παΓ(qα)
dqα
ξq.(66)
Using Euler’s reection formula Γ(1 −z)Γ(z)sin(πz)=π, we nally get
τq=sin(π/α)
sin(π(q+1/α))
Γ(1 +q)
Γ(1 +qα)
dqα
ξq,−1<q<1−1/α, (67)
where ξisgivenbyequation(50). The same result with a different method was given in dimen-
sionless form in [127] (see proposition 4). For α=2, we again consistently recover result
(48). In the right panel of gure 3we plot the numerical result for the space-fractional diffu-
sion equation and the analytic result corresponding to equation(67). As expected, moments of
order −1<q<1−1/α are nite.
For completeness in gure 4we also provide a comparison of the rst-passage time PDFs
for the extremal two-sided α-stable processes in the semi-innite domain with β=−1and
β=1. One can see (left panel) that in the limit t→0 the rst-passage time PDF tends to zero
for β=−1 and attains a nite value for β=1. Respectively, in the long-time limit (right
panel) the PDFs decay differently, faster for β=−1 (like t−1−1/α)andslowerforβ=1
(like t−2+1/α).
5.2.5. General asymmetric form of α-stable processes. In this section we present the
rst-passage properties of α-stable processes in general form. By applying the Skorokhod
theorem for α∈(0, 1) with β∈(−1, 1), for α=1 with β=0, as well as for α∈(1, 2]
17

J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 5. Left: fractional order moments of the rst-passage time PDF (top) and the rst-
passage time PDF (bottom) of α-stable laws in the semi-innite domain with skewness
β=−0.5. Right: the same but for skewness β=0.5. For all panels we used d=0.5
and L=1012. The lines represent numerical solutions of the space-fractional diffusion
equation and vertical lines in the top panels represent the limit q=ρ.
with β∈[−1, 1], it was shown that the rst-passage time PDF has the following power-law
decay [60]
℘(t)∼ρ(Kα(1 +β2tan2(απ/2))1/2)−ρdαρ
Γ(1 −ρ)Γ(1 +αρ)t−ρ−1=1
αΓ(1 −ρ)Γ(αρ)
dαρ
ξρt−ρ−1, (68)
where ξand ρare dened in equation (50). It is obvious that the corresponding integral (25)
is nite for moments q<ρ, otherwise the integral diverges. To estimate the behaviour of the
rst-passage time PDF at short times, we employ the asymptotic expression of LFs for large x.
For the purpose of this derivation, we follow the method introduced in [124] and assume that
the starting position is at x0=0 while the boundary is located at x=d, which is identical to
our setting in a semi-innite domain. Therefore, the survival probability at short times reads
S(t|0) =d
−∞
Pα,β(x,t|0)dx=1−∞
d
Pα,β(x,t|0)dx, (69)
where the α-stable law with the stability index α∈(0, 2] (α=1) and skewness β∈(−1, 1]
in the limit x→∞is given by [128]
Pα,β(x,t|0) ∼π−1(1 +β2tan2(απ/2))1/2sin(απρ)Γ(1 +α)Kαt
x1+α
=π−1sin(απρ)Γ(1 +α)ξt
x1+α.(70)
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
By substitution into equation (69) and recalling equation (23) we arrive at
℘(t→0) =π−1sin(απρ)Γ(α)ξ
dα.(71)
It is easy to check with the use of equation (50) that the rst-passage time PDF is only zero
for Brownian motion (α=2andρ=1/2) at short times. Otherwise, the boundary is crossed
immediately with a nite probability on the rst jump. To support our conclusion regarding the
existence of fractional order moments of the rst-passage time PDF for general asymmetric
form of the α-stable law, we plot the fractional order moments and the rst-passage time PDF
for two sets of the skewness, β=0.5and−0.5, and different values of the stability index α.
The results are shown in gure 5, and it can be seen moments with −1<q<ρare nite. The
lower bound (−1<q), arising due to the nite jump in the rst-passage time PDF at t→0,
can be seen in the bottom panels of gure 5. Similar to the symmetric case with β=0shown
in gure 2the values of the rst-passage time PDF at t→0 obtained by numerical solution of
the space-fractional diffusion equation are in perfect agreement with the behaviour provided
by equation (71). We also note that in [76] (theorem 2) presented a sufcient condition for
the niteness of the moments of the rst-passage time of the general Lévy process which is in
agreement with our results for LFs in general asymmetric form.
6. First passage time properties of LFs in a bounded domain
In this section we consider an LF in the interval [−L,L] with initial point x0and absorb-
ing boundary conditions at both interval borders (gure 1). Eventually, the LF is absorbed,
and our basic goal is to characterise the time dependence of this trapping phenomenon. From
equation (34) and with the space-fractional operators (5)and(6)wend
KαRα,β
Γ(n−α)x0
−L
τ(n)(ζ)
(x0−ζ)α−n+1dζ+Lα,β(−1)n
Γ(n−α)L
x0
τ(n)(ζ)
(ζ−x0)α−n+1dζ=−1.(72)
Applying the boundary condition τ(±L)=0 and the fact that 0Dα
L±x0(L±x0)α−n+1=const
[100] (p 626, equation (30.81)) leads us to a solution of equation (72) in the following form
τ(x0)=Cα,β(L−x0)μ(L+x0)ν, (73)
where Cα,βis a normalisation factor. First we consider the case 0 <α<1(n=1). After sub-
stitution of equation (73)into(72) and some calculations (see details in appendix B) we obtain
μ=αρ,ν=α−αρ (74)
and
Cα,β=cos(απ(ρ−1/2))
Γ(1 +α)Kα
=1
Γ(1 +α)ξ.(75)
For the case 1 <α2(n=2) a similar procedure leads to the same result, and formulas
(74)and(75) are valid for all α∈(0, 2] with β∈[−1, 1] (excluding the case α=1, β=0).
Finally, the MFPT for LFs in a bounded domain [−L,L] reads
τ=(L−x0)αρ(L+x0)α−αρ
Γ(1 +α)ξ, (76)
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
where ρand ξare given in expression (50). We note that in [78] from theGreen’s function of a
Lévy stable process [79] the MFPT of LFs in the interval (−1, 1) in the dimensionless Z-form
of the characteristic function (KZ
α=1) is given (see remark 5 in [78]). To see the equivalence
between equation (76) and the result in [78] we note that the following relation between the
parameters in the A-andZ-forms is established and reads (see equation (A.11) in [60])
ρ=1
2+1
απ arctan βAtan απ
2,KZ
α=KA
α
cos(απ(ρ−1/2)).(77)
Here,weusethestandardA-from parameterisation for the characteristic function.
6.1. Symmetric α-stable processes
For symmetric α-stable processes in a bounded domain, the MFPT for stability index 0 <α
2and|x0|<Lin N-dimension is given by [66]
τ=K(α,N)(L2−x2
0)α/2, (78)
where
K(α,N)=Γ
N
22αΓ1+α
2ΓN+α
2−1
.(79)
In one dimension by using the duplication rule 22zΓ(z)Γ(z+1/2) =2√πΓ(2z) this equation
reads [81,84]
τ=(L2−x2
0)α/2
Γ(1 +α).(80)
For the setup in gure 1,x0=L−dand by dening l=d/L, in dimensional variables the
MFPT yields in the form
τ=(d(2L−d))α/2
Γ(1 +α)Kα
=Lα(l(2 −l))α/2
Γ(1 +α)Kα
.(81)
This result is consistently recovered from the general formula (76) by setting ρ=1/2(or,
equivalently, β=0).
The second moment of the rst-passage time PDF for symmetric α-stable process with
stability index 0 <α2and|x0|<Lin Ndimensions was derived in [66],
τ2=αLαK(α,N)2L2
x2
0s−x2
0α/2−1F−α
2;N
2;N+α
2;sL−2ds, (82)
where Fis the Gauss hypergeometric function dened in equation (B.5). Analogous to the
MFPT we set N=1, x0=L−d, and in order to make time dimensional, equation (82)hasto
be divided by K2
α. Equation (82) is reduced to a simple form for Brownian motion only [43],
τ2=L4
12K2
α
(l2−2l)(l2−2l−4), (83)
where l=d/L. The behaviour of arbitrary-order moments is similar and reads τm∝Lmα/Km
α
(see gure 9), for the case when we start the process at the centre of the interval [−L,L].
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 6. MFPT versus distance dof the initial point of the random process from the
right side boundary for symmetric α-stable processes (β=0) and different sets of the
stability index α.Left:L=0.7. Right: L=2.5. Dashed lines show the analytic solu-
tion (81) and symbols represent numerical solutions of the space-fractional diffusion
equation.
In gure 6we study the MFPT for symmetric α-stable processes with varying initial posi-
tion. We employ two different interval lengths and plot the MFPT versus dfor different sets of
the stability index α. As can be seen, for interval length of L=0.7, regardless of the starting
point of the random walker the MFPT is always longer for smaller α. In contrast, for inter-
val length L=2.5, when the starting point of the random walker is close to the centre of the
interval, for larger αthe MFPT is longer. When the starting point gets closer to the bound-
aries, the behaviour is opposite. These observations are in line with the fact that LFs have a
propensity for long but rarer jumps, a phenomenon becoming increasingly pronounced when
the value of αdecreases. Conversely, LFs have short relocation events with a higher frequency
for values αclose to 2. Therefore, for small intervals (left panel of gure 6)itiseasiertocross
the boundaries when short relocation events happen with a high frequency, corresponding to
Lévy motion with αcloser to 2. In the opposite case, LFs with low-frequency large jumps
(α→0) can escape more efciently from large intervals (right panel of gure 6), except for
initial positions close to the boundaries. We also note that in both panels of gure 6, when the
stability index αgets closer to 0, the MFPT becomes atter away from the boundaries. This
result implies that with different starting points the random walker crosses the interval by a
single jump—concurrently, the MFPT has a small variation.
6.2. Asymmetric α-stable processes
6.2.1. One-sided α-stable processes with 0<α<1and β=1.This type of jump length dis-
tribution is dened on the positive axis. Therefore the situation for this process in semi-innite
and bounded domains is similar and moments for the rst-passage time PDF turn out to be
exactly the same as in equation (56) obtained above. Another method to nd the moments of
the rst-passage time PDF is to employ relation (43), addressed originally in [84] for symmet-
ric α-stable laws. The space-fractional operator for one-sided α-stable laws (0 <α<1and
β=1) reads
Dα
x0=−1
cos(απ/2) x0Dα
L.(84)
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
We apply the space-fractional integration operator D−mα
x0on both sides of equation (43) and get
(see appendix Cfor details)
τm(x0)=x0D−mα
L
cosm(απ/2)Γ(1 +m)
Kαm, (85)
where the sequential rule was used, namely, (Dα
x0)m=Dmα
x0[97] (p 86, equation (2.169)). The
space-fractional integration operator x0D−mα
Lused here is dened as [97] (p 51, equation (2.40))
x0D−mα
Lf(x)=1
Γ(mα)L
x0
f(ζ)
(ζ−x0)1−mαdζ. (86)
By substitution of equation (86) with f(ζ)=1into equation (85) we arrive at
τm(x0)=cosm(απ/2)Γ(1 +m)
Γ(mα)KαmL
x0
(ζ−x0)mα−1dζ=Γ(1 +m)
Γ(1 +mα)
dmα
ξm.
(87)
This result is the same as equation (56) with parameter ξdened in (50)andd=L−x0.
Thesameresultform=1isalsoshownin[48,69]. Moreover from equation (76)byset-
ting ρ=1orβ=1(0<α<1) we arrive at above expression with m=1. The left panels of
gure 7show the MFPT of one-sided LFs (0 <α<1andβ=1) for different values of the
stability index αfor two interval lengths (top: L=0.7, bottom: L=2.5). For interval length
L=0.7, smaller αvalues lead to longer MFPTs for different initial positions, except for the
situations when the LF starts really close to the left boundary. This observation is due to the
lower frequency of long-range jumps compared to high-frequency shorter-range jumps for
larger αvalues, similar to the above. For interval length L=2.5, when the initial position
of the random walker is located a distance d<2 away from the right boundary, for smaller
αit takes longer to cross the right boundary. For larger dvalues the smaller αvalues over-
take the LFs with the intermediate stable index α=0.5. Note, however, that the MFPT for
α=0.9 remains shorter than for LFs with the smaller stable index. For increasing interval
length low-frequency long jumps will eventually win out unless the particle is released close
to an absorbing boundary,compare also the discussion in [19,20]. Thus, the crossing of curves
with different αvalues in the left panel of gure 7has a simple physical meaning: it reects
the growing role of long jumps with smaller αwhen the distance dto the right boundary
(respectively, the interval length L) increases.
6.2.2. One-sided α-stable processes, 0<α<1,β=−1.For one-sided α-stable processes
with 0 <α<1andβ=−1 the space-fractional operator reads
Dα
x0=−1
cos(απ/2) −LDα
x0, (88)
and following a similar procedure as for the case 0 <α<1 with β=1, we obtain
τm(x0)=cosm(απ/2)Γ(1 +m)
Γ(mα)Kαmx0
−L
(ζ−x0)mα−1dζ=(2L−d)mα
ξm
Γ(1 +m)
Γ(1 +mα).(89)
By setting ρ=0orβ=−1(0<α<1) for m=1 we recover the same result as in
equation (76). The behaviourof the MFPT for this section is similar to the left panelsof gure 7,
apart from substituting dfor 2L−d.
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 7. MFPT versus distance dof the initial position of the LF from the right bound-
ary. Top panels: interval length L=0.7. Bottom panels: interval length L=2.5. Left
panels: extremal one-sided α-stable processes with β=1 and different sets of the sta-
bility index α. Dashed lines represent the analytic result (87) while symbols represent
the numerical solution of the space-fractional diffusion equation. Right panels: MFPT
for extremal two-sided α-stable processes with skewness β=1. Dashed lines show the
analytic result (91) and symbols represent the numerical solution of the space-fractional
diffusion equation.
6.2.3. Extremal two-sided α-stable processes with 1<α<2and β=−1, 1.For extremal
two-sided α-stable processes with stability index 1 <α<2, when the initial position is the
distance daway from the right boundary and for skewness β=−1(orρ=1/α)in(76)we
obtain the MFPT
τ=d(2L−d)α−1
Γ(1 +α)ξ, (90)
where ξdeed in equation (50). For the case β=1, by setting ρ=1−1/α in equation (76)
the following result yields,
τ=dα−1(2L−d)
Γ(1 +α)ξ.(91)
In contrast to the completely one-sided cases above, in results (90)and(91) two factors appear
that include the distances dand 2L−d. As a direct consequence, we recognise the completely
different functional behaviourin the right panels of gure 7. Namely, the MFPT decays to zero
at both interval boundaries.
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
For completely asymmetric LFs the rst-passage of the two-sided exit problem was
addressed in [69–74]. A different expression (instead of dα−1in equation (91)itisdα)forthe
MFPT of completely asymmetric LFs with 1 <α<2andβ=1 in dimensionless form was
derived with the help of the Green’s function method in [69] (see equation (1.8)). In [72] the dis-
tribution of the rst-exit time from a nite interval for extremal two-sided α-stable probability
laws with 1 <α<2andβ=−1 was reported in the Laplace domain.
In the right panels of gure 7we show the MFPT for extremal α-stable processes with
skewness β=1 for two different interval lengths as function of the initial distance dfrom 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
τ|α=2−τ|α=0, (92)
with ξdened in equation (50). By solving for d,wend
d=2cos(απ(1/2−1/α))
Γ(1 +α)1/(2−α)
.(93)
For α=1.1andα=1.5 the MFPT is equal with the Brownian case for d=0.261 and
d=1.132, respectively. The right side panels of gure 7indeed demonstrate that as long as
the distance dof the initial position of the LF is within the range 0 <d<0.261 from the right
boundary for α=1.1 and in the range 0 <d<1.132 for α=1.5, Brownian motion has a
shorter MFPT, otherwise the LF is faster. In general, if dis less than the term on the right-
hand side of equation (93) for arbitrary α∈(1, 2), Brownian motion is faster on average. In
the opposite case, long-range relocation events and left direction effective drift of LFs with
positive skewness parameter lead to shorter MFPTs.
6.2.4. General asymmetric α-stable processes. We nally show the result for the rst-
passage time moments of asymmetric α-stable processes with arbitrary skewness β.Thecor-
responding result for the MFPT with α∈(0, 2] and β∈[−1, 1] (excluding the case α=1and
β=0) has the following expression
τ=(L−x0)αρ(L+x0)α−αρ
Γ(1 +α)ξ.(94)
Setting d=L−x0and 2L−d=L+x0we nd
τ=dαρ(2L−d)α−αρ
Γ(1 +α)ξ(95)
with ρand ξdened in equation (50). In gure 8, analogous to gure 7, we show the MFPT
for α-stable processes with skewness β=0.5 and two different interval lengths (L=0.7and
L=2.5). The left panels of gure 8show the MFPT versus the distance dfrom the right bound-
ary for α-stable processes with 0 <α<1 and skewness β=0.5, for the two different lengths.
As can be seen for the smaller interval, increasing αfrom 0.1 to 0.9, regardless of the initial
position the MFPT decreases. This result can be explained as follows. An α-stable process
with stability index 0 <α<1 and skewness β=0.5, has a longer tail on the positive axis
and a shorter tail on the negative axis. Moreover, with increasing αfrom 0.1 to 0.9, the pro-
cess experiences a larger effective drift to the right boundary. Concurrently, when αdecreases
(increases), larger (shorter) jumps are possible with lower (higher) frequency. Therefore, with
increasing αthe possibility of shorter jumps with higher frequency and a larger effective drift
24

J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 8. Left: MFPT of a general asymmetric α-stable process versus initial distance
dfrom the right boundary for skewness β=0.5andα∈(0, 1). Right: the same for
α∈(1, 2). Top: interval length L=0.7. Bottom: interval length L=2.5. Symbols are
numerical solutions of the space-fractional diffusion equation, the dashed lines represent
equation (95).
towards the right side absorbing boundary arises and leads to shorter MFPTs. The decay of the
MFPT around d=1.4, when the initial position is close to the left boundary, shows us the effect
of small jumps of the negative short tail of the underlying α-stable law. The behaviour of the
MFPT in the larger interval is more complicated. For initial positions with distance d<1 from
the right boundary increasing αleads to decreasing MFPTs. This is due to the dominance of
an effectivedrift to the right and a higher frequency of long jumps when αchanges from 0.1 to
0.9. Conversely, when d>1 we observe two scenarios. First, for 0.1<α<0.6, with increas-
ing αMFPT increases. We can explain this result as follows. By increasing αin the range
(0.1, 0.6) the long relocation events dominate the effective drift and higher frequency events
with shorter jump length. Second, for 0.6<α<0.9, with increasing αthe MFPT decreases.
This is now due to the dominance of the effective drift and higher frequency of shorter jump
events against long-range jumps in the range 0.6<α<0.9.
α-stable processes with 1 <α<2andβ=0.5, have a heavier tail on the positive axis and
a resulting effective drift to the left. Based on the above properties, the behaviour of MFPT
is quite rich, as can be seen in the right panels of gure 8. For instance, for interval length
L=0.7, when α∈(1.4, 2), regardless of the initial position, Brownian motion always has a
shorter MFPT, whereas for α∈(1, 1.4) it does depends on the initial position. For the interval
length L=2.5, when the initial position is located in 2.5<d<5, smaller αalways has a
shorter MFPT. Otherwise, the superiority of LFs over the Brownian particle depends on its
initial position.
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 9. Top: MFPT versus interval length Lwhen the initial point is in the centre of
the interval (d=L) for different values of the skewness βin log–log scale. Symbols
show numerical solutions of the space-fractional diffusion equation and dashed lines
represent equation (95). Bottom: higher order moments of the rst-passage time PDF
versus interval length Lfor β=1 and two values of the stability index αin log–log
scale. Symbols show the numerical solutions of the space-fractional diffusion equation
and dashed lines are equations (56)and(91).
When the initial point of the random process is kept at the centre of the interval (x0=0), we
show results for the MFPT and higher moments of the rst-passage time PDF for different sta-
bility αas function of the interval length for symmetric and asymmetric α-stable processes in
gure 9. As can be seen, the moments of the rst-passage time PDF scale like τm∼Lmα/Km
α
independent of the skewness β.
6.3. Further properties of the MFPT
In this section, we study the MFPT versus the index of stability α. In gure 10 we x the
initial position of the random process to the centre of the interval (d=Lin gure 1) and plot
the MFPT versus the stability index αfor different skewness β, for three different interval
lengths L. As can be seen, there is a perfect agreement between the results based on the space-
fractional diffusionequation and the Langevin dynamic approach with the analytic result (95).
To elucidate the behaviour of the MFPT in gure 10 we remind the reader of some properties
of α-stable laws. First, α-stable laws with smaller αhave a heavier tail and the associated
frequency of long-range relocation events is smaller compared to laws with larger α,forwhich
short jumps with higher frequency are dominant. Second, symmetric α-stable probability laws
have the same tail on both sides. Third, α-stable laws with 0 <α<1 and skewness β>0have
an effective drift to the right and a longer tail on the positive axis. Moreover, when α→1−
26

J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 10. MFPT of an asymmetric α-stable process for L=0.5 (top left), L=1.0(top
right), and L=3 (bottom) versus αwhen the initial position is in the centre of the inter-
val (d=Lin gure 1). Symbols show results of Langevin equation simulations, dashed
lines are based on the numerical solution of the space-fractional diffusion equation, and
the dotted lines show the analytic solution (95).
with β>0, the effective drift to the right direction increases. Conversely, α-stable laws with
1<α<2 and skewness β>0 have an effective drift to the left and a longer tail on the positive
axis (see the bottom panel of gure 3 in [60]). When α→1+with β>0, the effective drift to
the left increases.
For a small interval length (L=0.5, top left panel of gure 10), short relocation events with
higher frequency (larger α) of symmetric LFs cross the boundaries quite quickly (full black
circles), whereas in large intervals (L=3, bottom panel of gure 10), long-range relocation
events of symmetric LFs lead to shorter MFPTs (full black circles). For intermediate inter-
val length (L=1, top right panel in gure 10), by increasing αfrom 0 to ≈0.46 the MFPT
increases, but for α∈(0.46, 2] this behaviour reverts. This observation is due to the tipping
balance between long jumps with low frequency and short jumps with high frequency for α
less and larger than 0.46, respectively.
Conversely, as can be seen from all panels in gure 10, on converging to the limit α→1
from both sides with skewness β=0, the MFPT tends to zero, which is in agreement with
the analytical result (95). To explain this phenomenon we follow [31,33] and rst rewrite the
characteristic function (1)and(2)oftheLFsas
α,β(k,t)=exp Kαt−|k|α+ikω(k,α,β)+iμkt, (96)
27

J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 11. MFPT versus αwith d=0.5 for two interval lengths. Top: L=1. Bottom:
L=3. Dotted lines show the result (95) and symbols are the numerical solution of the
space-fractional diffusion equation.
where
ω(k,α,β)=|k|α−1βtan(πα/2), α=1
−(2/π)βln |k|,α=1.(97)
The function ω(k,α,β) is not continuous at α=1andβ=0. However, setting
μ1=μ+βKαtan(πα/2), α=1
μ,α=1(98)
yields the expression
α,β=exp Kαt−|k|α+ikω1(k,α,β)+iμ1kt, (99)
where
ω1(k,α,β)=β|k|α−1−1tan(πα/2), α=1
−(2/π)βln |k|,α=1(100)
is a function that is continuous in α. Thus for β=0, as the Lévy index αapproaches unity,
the absolute value of the effective drift βKαtan(πα/2) tends to innity. For β>0, as seen in
gure 10, the effectivedrift is directed to the right as αapproaches unity from below, α→1−,
and, respectively, to the left as α→1+.
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 12. Second moment of the rst-passage time PDF for interval length L=0.5
(top left), L=1.0 (top right) and L=3 (bottom) versus α. The initial position is in
the centre of the interval (d=L). Symbols show the numerical solution of the space-
fractional diffusion equation and dotted lines show the analytic solution (82)forthe
symmetric case (β=0) and (87) with m=2 (one-sided 0 <α<1, β=1).
We now change the scenario and set the initial position at a distance d=0.5 away from the
right boundary. Figure 11 analyses the MFPT versus αand different skewness βfor two dif-
ferent interval lengths (L=1andL=3). As can be seen, there is a perfect agreement between
the results based on the numerical solution of the space-fractional diffusion equation and the
analytic solution (95). In comparison with the symmetric initial position of the random process
in gure 10, for positive values of the skewness parameter and when α∈(0, 1), since the initial
point is closer to the right boundary and the effective drift is in direction of the positive axis,
the MFPT decreases. For α∈(1, 2) and positive skewness, the effective drift is towards the
left, and the MFPT increases rapidly. The opposite behaviour is observed when the skewness
is negative (gure 11, right panels): for α∈(0, 1) and α∈(1, 2) with β<0, the effective drift
is to the left and right directions, respectively.
In gure 12, analogous to gure 10, we show the results for the second momentof the rst-
passage time PDF versus the stability index αfor different sets of the skewness parameter β
when the initial position is in the centre of the interval (d=L).
Finally, in gure 13 we show the coefcient of variation
f=τ2−τ2
τ2.(101)
When f>1 the underlying distribution is broad and we need to study higher order moments
to get the complete information of the rst-passage time PDF. When f<1, the distribution
29

J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Figure 13. Coefcient of variation fversus αfor the rst-passage time PDF with initial
distance L=d.
is narrow and higher order moments are not needed. For the one-sided α-stable process (0 <
α<1andβ=1) recalling equation (87), the coefcient of variation reads
f=
Γ(1+2)
Γ(1+2α)d2α
ξ2−Γ(1+1)
Γ(1+α)dα
ξ2
Γ(1+1)
Γ(1+α)dα
ξ2=2Γ(1 +α)2
Γ(1 +2α)−1, (102)
which is always less than one, comparealso gure 13. Thus, the MFPT is a fairly good measure
for the rst-passage process.
7. Discussion and unsolved problems
LFs are relevant proxy processes to study the efciency and spatial exploration behaviour
of random search processes, from animals (‘movement ecology’) and humans to robots and
computer algorithms. Apart from the MFPT such processes can be studied in terms of the
mean inverse rst-passage time 1/τas well as fractional order moments. Here we quantied
the rst-passage dynamics of symmetric and asymmetric LFs in both semi-innite and nite
domains and obtained the moments of the associated rst-passage time PDF. These moments
were analyses as functions of the process parameters, the stable index αand skewness β,as
well as the system parameters, the initial distance dand the interval length L(if not innite). As
seen in the results the behaviour for different parameters can be quite rich and requires careful
interpretation. Table 1summarises the main features.
We here studied the one-dimensional case, for which the effect of LF versus Brownian
search is expected to be most signicant. A one-dimensional scenario is relevant for the vertical
search of seaborne predators [12,13] as well as random search along, for instance, natural
boundaries such as eld-forest boundaries or the shrubberygrowing along streams. Other direct
applications include search in computer algorithms [129,130] or the effective one-dimensional
search on linear polymer chains where LFs are effected by jumps to different chain segments
at points where the polymer loops back onto itself [14,131]. In a next step it will be of interest
to extend these results to two dimensions, which is the relevant situation for a large number of
search and movement processes. Another important direction of future research is to study the
inuence of interdependence on the rst-passage properties for processes with innite variance.
Indeed, when the specic stochastic process is considered in a bounded domain the analysis of
correlations in this process is important [132,133]. Fractional LFs with long-range dependence
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Tab le 1. First-passage time PDF moments for different αand skewness β.
αβSemi-innite domain Bounded domain
2 Irrelevant (48), −∞ <q<1/2[43]τ→(81)[43,66,81]
τ2→(83)[43,66]
(0, 2) 0 Unknown, −1<q<1/2τ→(81)[66,81,84]
τ2→(82)[66]
(0,1) 1 (56), (87), −1<q<∞,(q=m=1[48,69])
(1, 2) −1(60), −∞ <q<1/α τ→(90)
1(67), −1<q<1−1/α τ→(91)
(0, 1) (−1, 1) Unknown, −1<q<ρ[76]τ→(95)
(1, 2)
have been detected in beat-to-beat heart rate uctuations [134], in solar are time series [135],
and they have been shown to be a model qualitatively mimicking self-organized criticality
signatures in data [136]. Apparently, correlations or spectral power analysis, strictly speaking,
cannot be used for LFs, and alternative measures of dependence are necessary, see, e.g., the
review [137].
In many situations for diffusive processes cognisance of the MFPT is insufcient to fully
characterise the rst-passage statistic. This statement was quantied in terms of the uni-
formity index statistic in [138,139]. Instead, it is important to know the entire PDF of
rst-passage times, even in nite domains [140–143]. Such notions are indeed relevant for
biological processes, for instance, in scenarios underlying gene regulation, for which the
detailed study reveals a clear dependence on the initial distance, which thus goes beyond
the MFPT [144–146]. While we here saw that the coefcient of variation of the rst-passage
statistic is below unity, it will have to be seen, for instance, how this changes to situations of
rst-arrival to a partially reactive site. Another feature to be included are many-particle effects,
for instance, ocking behaviour provoking different hunting strategies [147–149].
Acknowledgments
AP acknowledges funding from the Ministry of Science, Research and Technology of Iran
and Potsdam University in Germany. Computer simulations were performed at the Shahid
Beheshti University (Tehran, Iran) and Potsdam University (Potsdam, Germany). This research
was supported in part by PL-Grid Infrastructure. ACh and RM acknowledge support from
the DFG project 1535/7-1. RM also acknowledges support from the Foundation for Polish
Science (Fundacja na rzecz Nauki Polskiej) within an Alexander von Humboldt Polish Hon-
orary Research Scholarship. MM acknowledges support from NCN-DFG Beethoven Grant
No.2016/23/G/ST1/04083.
Appendix A. Generator and backward Kolmogorov equation for an LF killed
upon leaving the domain
Let τ=min{t:|x(t)|L}be the rst-passage time of an LF x(t). Let us dene the corre-
sponding killed process on [−L,L]as
¯x(t)=x(t)ift<τ
∂if tτ.(A.1)
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Here, ∂is the so-called ‘cemetery state’. It is a domain outside of the interval [−L,L]. Note
that the process ¯x(t) describes the dynamics of the LF conned to the interval [−L,L]. When
the LF leaves the domain, ¯x(t) moves to the cemetery state and stays there forever.
The key property here is that ¯x(t) is also a Markov process [150]. Therefore one can dene
its generator Dα
xin a usual way. This generator is equal to the generator of LFs conned to the
interval [−L,L][150]. It has the form
Dα
xf(x)=Rα,β−LDα
xf(x)+Lα,βxDα
Lf(x), (A.2)
for appropriately smooth function f(x). Here −LDα
xand xDα
Lare the fractional derivatives
dened in (5)and(6), respectively. Moreover, Lα,βand Rα,βare the constants dened in
equation (7). Here we employ an important property, namely, that under absorbing bound-
ary conditions the adjoint operator of the left derivative (5) is equal to the right derivative (6)
and vice versa [151].
Consequently, it follows from the general theory of Markov processes [152] that the PDF
Pα,β(x,t|x0) of the killed process starting at x0satises the backward Kolmogorov equation
∂Pα,β(x,t|x0)
∂t=KαDα
x0Pα,β(x,t|x0), (A.3)
where Dα
x0isgivenby(A.2) with xreplaced by x0. Finally, knowing the generator of ¯x(t)and
the corresponding backward Kolmogorov equation one can apply the usual method of nding
the mean rst-passage time of the LF described in detail in section 4.
Appendix B. Derivation of MFPT for general α-stable process in a finite
interval
Here we compute the MFPT of LFs with stability index α∈(0, 2] and skewness β∈[−1, 1]
(excluding α=1 with β=0). To determine the parameters μand ν, by substitution of
equation (73)into(72)weget
KαCα,βRα,β
Γ(n−α)x0
−L
((L−ζ)μ(L+ζ)ν)(n)
(x0−ζ)α−n+1dζ
+(−1)nKαCα,βLα,β
Γ(n−α)L
x0
((L−ζ)μ(L+ζ)ν)(n)
(ζ−x0)α−n+1dζ=−1.(B.1)
Let us rst consider the case n=1(0<α<1). By taking the rst derivative
KαCα,βRα,β
Γ(1 −α)x0
−L
ν(L−ζ)μ(L+ζ)ν−1−μ(L−ζ)μ−1(L+ζ)ν
(x0−ζ)αdζ
−KαCα,βLα,β
Γ(1 −α)L
x0
ν(L−ζ)μ(L+ζ)ν−1−μ(L−ζ)μ−1(L+ζ)ν
(ζ−x0)αdζ=−1, (B.2)
then, by change of variables y=(x0−ζ)/(x0+L)andy=(ζ−x0)/(L−x0)intherstand
second integral on the left-hand side, respectively, we have
(L+x0)ν−α(L−x0)μ1
0
ν(1 +L+x0
L−x0y)μ(1 −y)ν−1−μL+x0
L−x0(1 +L+x0
L−x0y)μ−1(1 −y)ν
yαdy,
(B.3)
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
and
(L+x0)ν(L−x0)μ−α1
0
νL−x0
L+x0(1 −y)μ(1 +L−x0
L+x0y)ν−1−μ(1 −y)μ−1(1 +L−x0
L+x0y)ν
yαdy.
(B.4)
Then, dening z=(L+x0)/(L−x0) and using the integral representation of the Gauss
hypergeometric function [153] (see equation (9.1.6)),
F(a;b;c;x)=Γ(c)
Γ(b)Γ(c−b)1
0
tb−1(1 −t)c−b−1
(1 −xt)adt,(B.5)
where Re(c)>Re(b)>0and|arg(1 −z)|<π, we obtain
KαCα,βRα,β
Γ(1 −α)(L+x0)ν−α(L−x0)μΓ(1 −α)Γ(1 +ν)
Γ(1 +ν−α)F(−μ;1−α;1+ν−α;−z)
−μzΓ(1 −α)Γ(1 +ν)
Γ(2 +ν−α)F(1 −μ;1−α;2+ν−α;−z)(B.6)
−KαCα,βLα,β
Γ(1 −α)(L+x0)ν(L−x0)μ−α
×ν
z
Γ(1 −α)Γ(1 +μ)
Γ(2 +μ−α)F(1 −ν;1−α;2+μ−α;−z−1)
−Γ(1 −α)Γ(1 +μ)
Γ(1 +μ−α)F(−ν;1−α;1+μ−α;−z−1)=−1.(B.6)
Moreover, by applying the relation [153] (see equation (9.5.9))
F(a;b;c;x)=(−x)−aΓ(c)Γ(b−a)
Γ(c−a)Γ(b)F(a;1+a−c;1+a−b;x−1)
+(−x)−bΓ(c)Γ(a−b)
Γ(c−b)Γ(a)F(b;1+b−c;1+b−a;x−1), (B.7)
where |arg(−x)|<π,|arg(1 −x)|<π,anda–b=0, ±1, ±2, ..., one can write
Γ(1 −α)Γ(1 +ν)
Γ(1 +ν−α)F(−μ;1−α;1+ν−α;−z)
=zμΓ(1 +μ−α)Γ(1 +ν)
Γ(1 +μ+ν−α)F(−μ;α−μ−ν;α−μ;−z−1)
+zα−1νΓ(1 −α)Γ(α−μ−1)
Γ(−μ)F(1 −α;1−ν;2+μ−α;−z−1),
(B.8)
and
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
μzΓ(1 −α)Γ(1 +ν)
Γ(2 +ν−α)F(1 −μ;1−α;2+ν−α;−z)
=μzμΓ(1 +ν)Γ(μ−α)
Γ(1 +μ+ν−α)F(1 −μ;α−μ−ν;1+α−μ;−z−1)
+μzαΓ(1 −α)Γ(α−μ)
Γ(1 −μ)F(1 −α;−ν;1+μ−α;−z−1).(B.9)
By substitution into equation (B.6)
KαCα,βRα,β
Γ(1 −α)(L+x0)ν−α(L−x0)μνzμB(ν,1+μ−α)F(−μ;α−μ−ν;α−μ;−z−1)
+νzα−1B(1 −α,α−μ−1)F(1 −α;1−ν;2+μ−α;−z−1)−μzμB(1 +ν,μ−α)
×F(1 −μ;α−μ−ν;1+α−μ;−z−1)−μzαB(1 −α,α−μ)
×F(1 −α;−ν;1+μ−α;−z−1)−KαCα,βLα,β
Γ(1 −α)(L+x0)ν(L−x0)μ−α
ν
zB(1 −α,1+μ)F(1 −ν;1−α;2+μ−α;−z−1)
−μB(1 −α,μ)F(−ν;1−α;1+μ−α;−z−1)=−1.(B.10)
Here, B(a,b)=Γ(a)Γ(b)/Γ(a+b) is the Beta function and with the help of the symmetry
property of the Gauss hypergeometric function, F(a;b;c;x)=F(b;a;c;x)[153] (see equation
(9.2.1)), we have
F(1 −α;1−ν;2+μ−α;−z−1)=F(1 −ν;1−α;2+μ−α;−z−1)
F(1 −α;−ν;1+μ−α;−z−1)=F(−ν;1−α;1+μ−α;−z−1).
(B.11)
By substitution into equation (B.10), we get
KαCα,βRα,β
Γ(1 −α)(L+x0)μ+ν−ανB(ν,1+μ−α)F(−μ;α−μ−ν;α−μ;−z−1)
−μB(1 +ν,μ−α)F(1 −μ;α−μ−ν;1+α−μ;−z−1)
+KαCα,β
Γ(1 −α)(L+x0)ν−1(L−x0)1+μ−ανRα,βB(1 −α,α−μ−1) −νLα,βB(1 −α,1+μ)
F(1 −ν;1−α;2+μ−α;−z−1)−KαCα,β
Γ(1 −α)(L+x0)ν(L−x0)μ−α
μRα,βB(1 −α,α−μ)−μLα,βB(1 −α,μ)×F(1 −α;−ν;1+μ−α;−z−1)=−1.
(B.12)
Then, by rearranging we obtain
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
KαCα,βLα,β
Γ(1 −α)(L+x0)μ+ν−ανB(ν,1+μ−α)F(−μ;α−μ−ν;α−μ;−z−1)
−μB(1+ν,μ−α)F(1−μ;α−μ−ν;1+α−μ;−z−1)+KαCα,β
Γ(1 −α)(L+x0)ν(L−x0)μ−α
×νRα,βB(1 −α,α−μ−1) −Lα,βB(1 −α,1+μ)z−1F(1 −α;1−ν;2
+μ−α;−z−1)−μRα,βB(1 −α,α−μ)−Lα,βB(1 −α,μ)
×F(1 −α;−ν;1+μ−α;−z−1)=−1.(B.13)
The left-hand side mustbe independent of zsince μand νdo not depend on z. This requirement
leads to the relations below. For the rst term on the left-hand side, with the help of F(a,b=
0, c,x)=1[153] (see section (9.8)), we have
F(−μ;α−μ−ν;α−μ;−z−1)=1
F(1 −μ;α−μ−ν;1+α−μ;−z−1)=1, (B.14)
where
b=α−μ−ν=0→α=μ+ν. (B.15)
For the second term, we nd
Rα,βB(1 −α,α−μ−1) −Lα,βB(1 −α,1+μ)=0
Rα,βB(1 −α,α−μ)−Lα,βB(1 −α,μ)=0.(B.16)
By denition of the Beta function,
Rα,β
Γ(1 −α)Γ(α−μ−1)
Γ(−μ)=Lα,β
Γ(1 −α)Γ(1 +μ)
Γ(2 +μ−α)
Rα,β
Γ(1 −α)Γ(α−μ)
Γ(1 −μ)=Lα,β
Γ(1 −α)Γ(μ)
Γ(1 +μ−α).
(B.17)
Using Euler’s reection formula Γ(1 −z)Γ(z)sin(πz)=π, we obtain
Rα,β
sin(π(α−μ−1)) =Lα,β
−sin(πμ)
Rα,β
sin(π(α−μ)) =Lα,β
sin(πμ), (B.18)
which are identical. With the help of the weight coefcients (see equation (7))
Lα,β=−sin(παρ)
sin(πα)cos(πα(ρ−1/2))
Rα,β=−sin(πα(1 −ρ))
sin(πα)cos(πα(ρ−1/2)), (B.19)
where ρis dened in equation (50). Substitution into equation (B.18), we nd
sin(πα(1 −ρ))
sin(π(α−μ)) =sin(παρ)
sin(πμ).(B.20)
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
Therefore, the parameters μand νhave the following form (see equation (B.15))
μ=αρ,ν=α−αρ. (B.21)
To determine the normalisation factor, by substituting equation (74) into equation (B.13)we
obtain
KαCα,βRα,β
Γ(1 −α)[νB(ν,1+μ−α)−μB(1 +ν,μ−α)]=−1.(B.22)
Using the Beta function,
KαCα,βRα,β
Γ(1 −α)[−αΓ(1 +ν)Γ(μ−α)]=−1, (B.23)
we get
Cα,β=Γ(1 −α)
αKαRα,βΓ(1 +α−αρ)Γ(αρ −α)=1
Γ(1 +α)Kα
sin(πα(ρ−1))
Rα,βsin(πα),
(B.24)
where the last equality follows from Euler’s reection formula. Finally by substitution of Rα,β
(equation (B.19)), we get the desire result (75).
Appendix C. Fractional integration of a fractional derivative
Here we show the composition rule for the right Riemann–Liouville fractional integral and
the right fractional derivative in the Caputo form of the operator. The right Riemann–Liouville
fractional integral is given by (p∈Re >0) [97]
xD−p
Lf(x)=1
Γ(p)L
x
f(ζ)
(ζ−x)1−pdζ,(C.1)
and with the right Caputo form of the fractional derivative as (n−1<q<n)
xDq
Lf(x)=(−1)n
Γ(n−q)L
x
f(n)(ζ)
(ζ−x)q−n+1dζ,(C.2)
we write
xD−p
LxDq
Lf(x)=1
Γ(p)L
x
ζDq
Lf(ζ)
(ζ−x)1−pdζ. (C.3)
Then, with the help of equation (C.2)wend
xD−p
LxDq
Lf(x)=(−1)n
Γ(p)Γ(n−q)L
x
1
(ζ−x)1−pL
ζ
f(n)(y)
(y−ζ)q−n+1dydζ.
(C.4)
Now, we change the integration order,
L
xL
ζ
f(x,ζ,y)dydζ=L
xy
x
f(x,ζ,y)dζdy,(C.5)
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J. Phys. A: Math. Theor. 53 (2020) 275002 A Padash et al
and get
xD−p
LxDq
Lf(x)=(−1)n
Γ(p)Γ(n−q)L
x
f(n)(y)y
x
1
(ζ−x)1−p(y−ζ)q−n+1dζdy.
(C.6)
After change of variable, ζ=x+z(y−x) in the inner integral, we arrive at
xD−p
LxDq
Lf(x)=(−1)n
Γ(p)Γ(n−q)L
x
f(n)(y)
(y−x)1−n1
0
1
z1−p(1 −z)q−n+1dzdy.
(C.7)
Then, with the help of
1
0
1
z1−p(1 −z)q−n+1dz=Γ(p)Γ(n−q)
Γ(n),(C.8)
we nd
xD−p
LxDq
Lf(x)=(−1)n
Γ(n)L
x
f(n)(y)
(y−x)1−ndy.(C.9)
For our case in section 6.2.1 with p=q=mαand f(x0)=τm(x0), when n=1(0<α<1,
m=1) this becomes
x0D−mα
Lx0Dmα
Lτm(x0)=τm(x0)−τm(L), (C.10)
and when n=2(0<α<1, m=2) after integration by part we get
x0D−mα
Lx0Dmα
Lτm(x0)=(L−x0)∂τm(y)
∂yy=L
+τm(x0)−τm(L).(C.11)
With a similar procedure for n3 it can be deduce that in order to get result (85), all deriva-
tives of the order n−1<mαof τm(y)aty=Lshould be zero. The fact that τm(y)vanishes
at y=Lis intuitively clear, when the initial point of the random walker is located right at the
absorbing boundary x0=L, it will be removed immediately. We also note that by differentiat-
ing the result (87) it is easy to check that the assumption that all derivatives of τm(y)vanish
at y=Lis reasonable.
ORCID iDs
Amin Padash https://orcid.org/0000-0002-3289-6556
Bartłomiej Dybiec https://orcid.org/0000-0002-6540-3906
Babak Shokri https://orcid.org/0000-0002-8242-5111
Ralf Metzler https://orcid.org/0000-0002-6013-7020
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