ARTICLE IN PRESS
Physica A 350 (2005) 245–262
www.elsevier.com/locate/physa
Advection and dispersion in time and space
B. Baeumera,1, D.A. Bensonb,2, M.M. Meerschaertc,,3
a
Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand
b
Desert Research Institute, 2215 Raggio Parkway, Reno NV 89512, USA
c
Department of Physics, University of Nevada, Reno NV 89557, USA
Received 19 July 2004; received in revised form 20 October 2004
Available online 24 November 2004
Abstract
Previous work showed how moving particles that rest along their trajectory lead to timenonlocal advection–dispersion equations. If the waiting times have infinite mean, the model
equation contains a fractional time derivative of order between 0 and 1. In this article, we
develop a new advection–dispersion equation with an additional fractional time derivative of
order between 1 and 2. Solutions to the equation are obtained by subordination. The form of
the time derivative is related to the probability distribution of particle waiting times and the
subordinator is given as the first passage time density of the waiting time process which is
computed explicitly.
r 2004 Elsevier B.V. All rights reserved.
PACS: 05.40.Fb; 02.50.Cw; 05.60.k; 47.55.Mh
Keywords: Anomalous diffusion; Continuous time random walks; First passage time; Fractional calculus;
Subdiffusion; Power laws
Corresponding author. Department of Mathematics and Statistics, Graduate Program in Hydrologic
Sciences, Unviersity of Nevada, Reno, NV 89557-0045, USA.
E-mail addresses: bbaeumer@maths.otago.ac.nz (B. Baeumer), dbenson@dri.edu (D.A. Benson),
mcubed@unr.edu (M.M. Meerschaert).
1
B. Baeumer was partially supported by the Marsden Fund administered by the Royal Society of New
Zealand.
2
D.A. Benson was supported by DOE-BES grant DE-FG03-98ER14885 and NSF grants DES-9980489
and DMS-0139943.
3
M.M. Meerschaert was partially supported by NSF grants DMS-0139927 and DMS-0417869, and the
Marsden Fund administered by the Royal Society of New Zealand.
0378-4371/$ - see front matter r 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.physa.2004.11.008
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B. Baeumer et al. / Physica A 350 (2005) 245–262
1. Introduction
Continuous time random walks (CTRW) can be used to derive governing
equations for anomalous diffusion [1–4]. The CTRW is a stochastic process model
for the movement of an individual particle [5,6]. In the long-time limit, the process
converges to a simpler form whose probability densities solve the governing
equation, leading to a useful model for anomalous diffusion. For a simple random
walk with zero-mean, finite-variance particle jumps, the limit process is a Brownian
motion AðtÞ governed by the classical diffusion equation qp=qt ¼ q2 p=qx2 where
pðx; tÞ is the probability density of the random variable AðtÞ: For symmetric infinitevariance jumps (i.e., those with probability density function tails that fall off like
jxj1a [7] with some index 0oao2), the limit process AðtÞ is an a-stable Lévy
motion, and the governing equation becomes qp=qt ¼ qa p=qjxja [8]. When waiting
times between the jumps are introduced, the limiting process is altered via
subordination [9–11]. In this study, we examine the case where the waiting times
are independent of jump size, also called an ‘‘uncoupled’’ CTRW. For infinite mean
waiting times (whose probability distribution is assumed to decay algebraically with
some index 0ogo1) the limit process is AðEðtÞÞ where EðtÞ is the inverse or first
passage time process for the g-stable subordinator. By virtue of its construction,
the process EðtÞ counts the number of particle jumps by time tX0; accounting for the
waiting time between particle jumps. In the scaling limit, EðtÞ keeps track of the
possibly nonlinear link between real time and the operational time that a particle
actually spends in motion. The governing equation becomes qg p=qtg ¼ qa p=qjxja
[2,3]. Some applications [12] seem to indicate a time derivative of order 1ogp2: In
this paper, we develop one such equation by extending the CTRW approach to
processes with finite-mean waiting times, and compute the distribution of the
relevant first passage time process.
2. The model
In the usual CTRW formalism, the long-time limit for the waiting time process is a
g-stable subordinator DðtÞ [2]. Then the inverse Lévy process EðtÞ ¼ inffx : DðxÞ4tg
counts the number of particle jumps by time tX0; reflecting the fact that the time T n
of the nth particle jump and the number N t ¼ maxfn : T n ptg of jumps by time t are
also inverse processes. When the waiting times between particle jumps have heavy
tails with 0ogo1 (i.e., infinite mean), subordination of the particle location process
AðtÞ via the inverse Lévy process EðtÞ is necessary in the long-time limit to account
for the amount of time that a particle is not participating in the motion process. The
subordination leads to a time derivative of order g in the governing equation of
motion [3]. When waiting times have heavy tails of order 1ogp2; meaning that the
probability of waiting longer than t falls off like tg ; a different model is needed [13].
In this case, convergence of the waiting time process requires centering to the mean
waiting time w, which is not necessary when 0ogo1: Accounting for this leads
to a waiting time process W ðtÞ ¼ DðtÞ þ wt where DðtÞ is a completely positively
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skewed4 stable Lévy process with index g; so that W ðtÞ is a Lévy process with drift.
The drift ensures that W ðtÞ ! 1 with probability one as t ! 1: Since g41; the
process W ðtÞ is not strictly increasing, so it is proper to use MaxðtÞ ¼ supfW ðuÞ :
0puptg to represent the particle jump times. This substitution is explained as a
particle that makes more than one jump without intervening rest periods [13]. Then
the inverse or first passage time process HðtÞ ¼ inffx : MaxðxÞXtg counts the number
of particle jumps. The first passage time process HðtÞ serves as the subordinator in
place of (and when go1 is identical to) EðtÞ: We show this in subsequent sections.
3. First passage time density
According to the previous section we model the waiting time process as a Lévy
process with drift; i.e., W ðtÞ has characteristic function
g
E½eixW ðtÞ ¼ etðiwxþaðixÞ Þ ;
where w40 is the mean waiting time and a is a shape variable akin to the variance.
For the remainder of this paper we make the assumption w ¼ 1; which entails no loss
of generality, since any mean time w can be recovered by a simple rescaling in time.
In order to compute the density of the process H we use the fact that
PfHðTÞ4sg ¼ PfMaxðsÞoTg
for all s; TX0: Theorem 1 in Ref. [14] shows that the distribution of the maximum
process Mðs; TÞ ¼ PfMaxðsÞoTg satisfies
Z 1Z 1
u
euslT d T ðMðs; TÞÞ ds
0
0
Z 1Z 1
1
l
ix þ aðixÞg
¼ exp
ð1Þ
g dx dx
2p u
1 xðx ilÞ xðx ðix þ aðixÞ ÞÞ
for all l; u40: The integrand has two poles in the upper complex halfplane, at x ¼ il
and whenever x ¼ ix þ aðixÞg : That the second equality holds only once, follows
from the following Lemma (for a proof see Appendix A) investigating the inverse
function of azg z: This involves the following region. Let a40; 0oapp=g and
sinðyÞ
sinðaÞ
org1 o
OðaÞ:¼ reiy : aoyoa and
;
a sinðgyÞ
a sinðgaÞ
where we take sinðaÞ=a sinðgaÞ ¼ 1 when a ¼ p=g:
Lemma 1. Let a40; 1ogp2: There exists a unique holomorphic function q :
Cnð1; aðg 1ÞðagÞg=ð1gÞ ! Oðp=gÞ such that
aqðzÞg qðzÞ ¼ z :
Furthermore, there exists an analytic
F with supt40 t11=g ezt jF ðtÞjo1 for all
R 1function
g=ð1gÞ
zt
0ozoaðg 1ÞðagÞ
such that 0 e F ðtÞ dt ¼ 1=qðzÞ for z40:
4
The skew is irrelevant in the normal case g ¼ 2:
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248
Using the Lemma, we can simplify (1) to (see Appendix B)
Z 1Z 1
1 l=qðuÞ
euslT d T ðMðs; TÞÞ ds ¼
:
u þ l alg
0
0
Let Hðs; TÞ ¼ PfHðTÞpsg and recall that Mðs; TÞ ¼ PfMaxðsÞoTg: Then
Hðs; TÞ ¼ PfHðTÞpsg ¼ PfMaxðsÞXTg ¼ 1 Mðs; tÞ; and hence we can integrate
by parts to get
Z 1Z 1
Z 1Z 1
euslT d T ðMðs; TÞÞ ds ¼ l
euslT Mðs; TÞ dT ds
0
0
0
0
Z 1Z 1
¼l
euslT ð1 Hðs; TÞÞ ds dT
0
0
Z 1Z 1
¼ 1=u l
euslT Hðs; TÞ ds dT :
0
0
In other words, the Laplace–Laplace transform of the distribution function of the
first passage time process is given by
Z 1Z 1
1
1 l=qðuÞ
euslT Hðs; TÞ ds dT ¼
ul lðu þ l alg Þ
0
0
¼
1 alg1 þ u=qðuÞ
:
uðu þ l alg Þ
ð2Þ
In the following theorem, we invert this Laplace–Laplace transform to obtain an
expression for the cumulative distribution function of the first passage time process
(corresponding in our model to the ‘‘number of jumps by time t’’).
~
Theorem 2. Let 1ogp2; let m be the function with Laplace transform mðuÞ
¼ 1=qðuÞ
given by Lemma 1 and gg be the maximally skewed, standard g-stable density; i.e., the
g
Fourier transform of gg is g^ g ðkÞ ¼ eðikÞ : Then the cumulative distribution function of the
first passage time of a Lévy-stable waiting time process with drift is given by
!
Z s
Z 1
mðs uÞ
tu
gg ðuÞ du þ
gg
Hðs; tÞ ¼
du :
(3)
1=g
ðauÞ1=g
ðtsÞ=ðasÞ1=g
0 ðauÞ
Remark 3. For tb0;
Z 1
Hðs; tÞ ðtsÞ=ðasÞ1=g
gg ðuÞ du ¼ PðW ðsÞXtÞ
(4)
in view of the fact that W ðsÞ is identically distributed with ðasÞ1=g W g þ s where W g is
the stable random variable with density gg : If W ðsÞ were an increasing process, the
left-hand and right-hand expressions in (4) would be equal. Hence the second term in
(3) accounts for the fact that W ðsÞ is not monotone.
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4. The differential equations
In this section, we show that the density of the uncoupled CTRW limit process
solves a fractional partial differential equation with an extra time derivative term of
order 1ogp2 on the right-hand side whose coefficient is negative. This is in contrast
to Mainardi and colleagues’ [15–18] investigation of the transition via fractional
derivatives from the second order hyperbolic equation to the first order parabolic
case. Our equations stay parabolic in nature (given an elliptic operator in space).
We begin with a related result concerning the first passage time density. We say
that a mild solution to a fractional partial differential equation is a function whose
Laplace transform solves the equivalent algebraic equation in Laplace–Laplace
space. The following theorem employs the Caputo derivative ðd=dtÞg ; which can be
defined for 1ogp2 by requiring that ðd=dtÞg F ðtÞ has Laplace transform lg F~ ðlÞ lg1 F ð0Þ lg2 F 0 ð0Þ where F~ ðlÞ is the Laplace transform of F ðtÞ; see for example
Refs. [19,20].
Theorem 4. There exists a unique distribution f such that the density
uðt; sÞ ¼ dHðs; tÞ=ds of the first passage time distribution Hðs; tÞ in (3) is the unique
mild solution to
g
d
d
d
a
(5)
uðt; sÞ þ uðt; sÞ ¼ uðt; sÞ þ f ðsÞdðtÞ
dt
dt
ds
with conditions: uð0; sÞ ¼ dðsÞ; uðt; 0Þ ¼ ut ð0; sÞ ¼ 0 8s; t40; and s7!uðt; sÞ is a probability density for all t40: For any other distribution f Eq. (5) has no solution.
We now extend this result to incorporate more general spatial derivative
operators. When the CTRW particle jumps are in the generalized domain of
attraction of an operator stable limit with probability distribution n; the long-time
limiting particle location process AðtÞ is operator stable with distribution nt [21]. For
a simple random walk, the density pðx; tÞ of AðtÞ defines
R a strongly continuous
semigroup ðTðtÞÞtX0 on L1 ðRd Þ via the formula TðtÞf ðxÞ ¼ f ðx yÞpðy; tÞ dy: If L is
the generator of this semigroup, then Cðx; tÞ ¼ TðtÞf ðxÞ also solves the abstract
Cauchy problem dC=dt ¼ LC; Cð0Þ ¼ f : For an uncoupled CTRW with infinite
mean waiting times, the limiting particle location is given by the subordinated
process AðEðtÞÞ: The density hðx; tÞ of this process solves a fractional Cauchy
problem ðd=dtÞg h ¼ Lh where 0ogo1 [22]. The next theorem extends this result to
1ogp2: Theorem 4.1 in Ref. [13] shows that the long-time CTRW limit process in
this caseR is AðHðtÞÞ; and Corollary 4.2 in Ref. [13] shows that this limit process has
1
density 0 pðx; sÞd s ðHðt; sÞÞ: The next theorem shows that this density is the Green’s
function solution to a fractional partial differential equation (6) with time derivative
of order 1ogp2: This result extends Theorem 3.1 in Ref. [22] showing that these
CTRW limits provide stochastic solutions of fractional Cauchy problems with time
derivative of order 1ogp2:
Theorem 5. Let L be the generator of an operator stable semigroup ðTðtÞÞtX0 on
^
L1 ðRd Þ such that for f 2 L1 ðRd Þ; Lf ¼ F1 ðLðkÞf
ðkÞÞ: Let a40; 1ogp2: Then there
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250
exists a unique distribution g such that
g
d
d
CðtÞ þ CðtÞ ¼ LCðtÞ þ dðtÞg
a
dt
dt
(6)
with conditions
C 0 ð0Þ ¼ 0;
Cð0Þ ¼ f ;
CðtÞ 2 L1 ðRd Þ
for all tX0 has a mild solution and this unique solution is given by
Z 1
Cðt; xÞ ¼
TðsÞf ðxÞd s ðHðt; sÞÞ :
0
For any other distribution g Eq. (6) has no solution.
5. Examples
In this section we give two examples showing how the more differentiated scaling
in time influences the resulting model.
Example 6. Consider a random walk composed of zero-mean, finite-variance
Gaussian jumps with intervening waiting times that are in the domain of attraction
of a Gaussian with unit mean and variance. In this case, the first passage time density
of the Gaussian waiting times (with drift) is well known; see, for example, Ref. [23].
The governing equation becomes
a
d2
d
d2
Cðt; xÞ þ Cðt; xÞ ¼ D 2 Cðt; xÞ þ dðtÞg
2
dt
dt
dx
with Cð0; xÞ ¼ f ðxÞ and qC=qt ¼ 0 when t ¼ 0: The solution is given by
Z 1
t 1
ðt sÞ2
1
x2
pffiffiffiffiffiffiffiffiffiffi exp pffiffiffiffiffiffiffiffiffiffiffi exp Cðt; xÞ ¼
ds%f ;
s 4pas
4as
4Ds
4pDs
0
with ‘%’ being the convolution operator in x.
Example 7. To generalize the previous example, let the waiting times be regularly
varying with unit mean and infinite variance, so the limit of waiting times DðtÞ
converges to a Lévy motion with index 1ogo2: Furthermore, let the jump sizes be
symmetric and heavy tailed with index 1oap2: Then the particle position density
Cðx; tÞ; assuming an initial particle distribution Cð0; xÞ ¼ f ðxÞ and qC=qt ¼ 0 when
t ¼ 0; follows
g
d
d
d
qa
Cðt; xÞ þ
a
Cðt; xÞ þ Cðt; xÞ ¼ v
Cðt; xÞ þ dðtÞg :
(7)
dt
dt
dx
qjxja
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251
10
t = 0.1; a = 0.1
t = 1; a = 0.1
t = 2; a = 0.1
t = 0.1; a = 1
t = 1; a = 1
t = 2; a = 1
1
0.1
0.01
0.001
0.0001
-10
-5
0
5
10
Fig. 1. Solutions to Example 6 with pulse initial condition. Solid lines are with a ¼ 0:1 and dashed lines
are for a ¼ 1; shown for various t and with diffusion parameter D set to 1.
The mild solution is given by
Z 1
1
x vs
Cðt; xÞ ¼
g
%f d s ðHðt; sÞÞ :
s1=a a s1=a
0
These two examples show that the particle motion has classical advective
(drift) motion with additional dispersion in both space and time. In the first
example, the effect of the second time derivative vanishes for large time. Note,
however, that when the coefficient on the second time derivative a is larger,
the density is more peaked at the origin and has more weight in the tails (Fig. 1).
The main effect of the parameter g; when less than two, is to cause particles
to spend heavy-tailed amounts of time in an immobile state that decays very
slowly (Fig. 2). The density of the first passage time, which provides a map between
the number of jumps (a particle’s operational time) and the clock time, shows
fewer jumps at any time for lower values of g: For smaller values of g; the time
dispersion is noticeable for prolonged time periods (Fig. 2), hence the effect
of the g-order derivative is important for longer periods. The solution to
Example 7 is obtained by subordinating the shifted Lévy motion against these
densities.
6. Conclusions
The study of classical random walks and, more recently, CTRW, has focused on
the spatial dispersion of the particles. Temporal dispersion in the limit process is
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252
10
γ = 1.1
γ = 1.5
Density h(s,t)
γ = 1.9
t=1
t = 10
t = 100
t = 1,000
t = 10,000
1
0.1
0.01
0.001
0.0001
0.00001
0.1
1
10
100 1,000 10,000
0.1
1
10
100 1,000 10,000
0.1
1
10 100 1,000 10,000
Number of jumps (s) at time t
Fig. 2. First passage time densities d s ðHðs; tÞÞ corresponding to g-stable waiting time processes with
a ¼ 0:1 and unit drift. In the model, the first passage time tracks the random number of jumps at time t.
commonly assumed to be restricted to the infinite-mean waiting time CTRW
[6,9,1,4]. However, by viewing the time process in a similar mathematical light as the
space process, one sees that the time ‘‘drift’’ follows a different scaling procedure
than the time dispersion. The time drift measures the linear portion of the map
between clock time and the number of jumps, resulting in the familiar first time
derivative in the governing equation of motion. If deviations in the waiting times
have a power law distribution with finite mean but infinite variance, then the effects
of the deviations do not disappear in the limit, and the governing equation has an
additional time operator of order 1ogo2: If deviations in the waiting times have
finite mean and variance, this procedure leads to an additional second-order time
derivative. The resulting model gives a sharper description of space-time diffusive
processes.
Appendix A. Proof of Lemma 1
First we show uniqueness. Let 0oaop=g;
(
)
sinðyÞ 1=ðg1Þ iy
G ¼
e : 0pypa
a sinðgyÞ
and
(
Gr ¼
sinðaÞ
a sinðgaÞ
1=ðg1Þ
)
iy
e : apypa
:
Then q OðaÞ ¼ G:¼G þ Gr Gþ is clearly a simply closed path around OðaÞ (see for
example Ref. [24, Theorem 10.40]). Let
pðzÞ ¼ azg z :
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Then
!
sinðyÞ 1=ðg1Þ iy
sinðyÞ g=ðg1Þ igy
sinðyÞ 1=ðg1Þ iy
¼a
p
e
e
e
a sinðgyÞ
a sinðgyÞ
a sinðgyÞ
sinðyÞ g=ðg1Þ
sinðyÞ 1=ðg1Þ
¼a
cosðgyÞ cosðyÞ
a sinðgyÞ
a sinðgyÞ
!
sinðyÞ g=ðg1Þ
sinðyÞ 1=ðg1Þ
sinðgyÞ sinðyÞ
i a
a sinðgyÞ
a sinðgyÞ
sinðyÞ 1=ðg1Þ sinðyÞ
cosðgyÞ cosðyÞ
¼
a sinðgyÞ
sinðgyÞ
1=ðg1Þ
sinðyÞ
sinððg 1ÞyÞ
¼
:
a sinðgyÞ
sinðgyÞ
A quick calculation shows that for 0oyop=g; y7! sinðyÞ= sinðgyÞ is an increasing
function which implies that y7!ðsinðyÞ=a sinðgyÞÞ1=ðg1Þ sinððg 1ÞyÞ= sinð½g=ðg 1Þðg 1ÞyÞ is also increasing. Since
sinðyÞ 1=ðg1Þ sinððg 1ÞyÞ
¼ aðg 1ÞðagÞg=ð1gÞ
lim
y!0 a sinðgyÞ
sinðgyÞ
we have that the image of G under p is a path on the negative real axis,
" #
sinðaÞ 1=ðg1Þ sinððg 1ÞaÞ
; aðg 1ÞðagÞg=ð1gÞ :
pðG Þ ¼ a sinðgaÞ
sinðgaÞ
Investigating pðGr Þ we see that for z 2 Gr ;
sinðaÞ g=ðg1Þ igy
sinðaÞ 1=ðg1Þ iy
pðzÞ ¼ a
e e
a sinðgaÞ
a sinðgaÞ
sinðaÞ 1=ðg1Þ sinðaÞ
ðcosðgyÞ þ i sinðgyÞÞ cosðyÞ i sinðyÞ
¼
a sinðgaÞ
sinðgaÞ
sinðaÞ 1=ðg1Þ sinðaÞ cosðgyÞ
sinðaÞ sinðgyÞ
¼
cosðyÞ þ i
sinðyÞ
:
a sinðgaÞ
sinðgaÞ
sinðgaÞ
Using again that y7! sinðyÞ= sinðgyÞ is increasing for y40; the imaginary part is
positive iff y is positive. Furthermore,
sinðaÞ 1=ðg1Þ sinðaÞ
jpðzÞjX
1
a sinðgaÞ
sinðgaÞ
for all z 2 Gr ; and this lower bound tends to infinity as a ! p=g: Hence we obtain for
a large enough that pðGr Þ is a closed contour going once counterclockwise around the
origin, which implies that pðGÞ is a closed contour going once counterclockwise
around the origin.
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254
Fix w 2 Cnð1; aðg 1ÞðagÞg=ð1gÞ ; and then choose aop=g (a depends on w)
such that pðGÞ goes around w. Using the counting formula for zeros and poles, we see
that there is exactly one z 2 OðaÞ (see for example Ref. [25, Theorem 13.2.2]) such
that pðzÞ ¼ w: Furthermore (see for example Ref. [26, p. 153]),
Z
1
p0 ðzÞ
dz
qðwÞ:¼p1 ðwÞ ¼
z
2pi G pðzÞ w
is a holomorphic function on Cnð1; aðg 1ÞðagÞg=ð1gÞ :
Next, we show that 1=q is the Laplace transform of an analytic function f on
ð0; 1Þ with t11=g eot f ðtÞ bounded for o4 aðg 1ÞðagÞg=ð1gÞ :
Since aqðzÞg qðzÞ ¼ z we have that for jzj large enough, ða þ 1ÞjqðzÞjg 4jaqðzÞg qðzÞj ¼ jzj: Furthermore, qðzÞ 2 Oðp=gÞ and thus qðzÞ is bounded away from zero.
Hence there exists M40 such that
MjqðzÞjg Xjzj
(A.1)
for all zeð1; aðg 1ÞðagÞ
rðzÞ:¼
g=ð1gÞ
: Let
d 1
1
¼ qðzÞ2
dz qðzÞ
agqðzÞg1 1
using the fact that dq=dz ¼ 1=ðdp=dzÞ: The function z7!1=agqðzÞg1 1 has a single
pole at z ¼ aðg 1ÞðagÞg=ð1gÞ and is bounded off a neighborhood of that pole.
Choose oo0 such that o4 aðg 1ÞðagÞg=ð1gÞ : Then for Sa :¼freid :
r40; aodoag we have that
z o
1
o1
sup jðz oÞrðzÞj ¼ sup 2
g1
1
z2oþSdþp=2
z2oþSdþp=2 qðzÞ agqðzÞ
for all 0odop=2: Let Cþ ¼ fz 2 C : RðzÞ40g: By the analytic representation
theorem for Laplace transforms [27, Theorem 2.6.1] there exists a holomorphic
function f : Cþ ! C such that supz2Cþ jeoz f ðzÞjo1 and
Z 1
rðzÞ ¼
ezt f ðtÞ dt :
0
Furthermore,
sup jzz
RðzÞ40
1=g
2=g
z
zðg1Þ=g rðzÞj ¼ o1 :
qðzÞ2 agqðzÞg1 1
Using the complex representation theorem ([27, Theorem 2.5.1] with b ¼ 1=g and
qðzÞ ¼ z1=g rðzÞ) there exists g 2 C½0; 1Þ with supt40 t1=g jgðtÞjo1 such that
Z 1
z1=g rðzÞ ¼ z1=g
ezt gðtÞ dt :
0
By the uniqueness of the Laplace transform f ¼ g and hence
sup jt1=g eot f ðtÞjo1 :
tX0
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255
Clearly, F ðtÞ ¼ f ðtÞ=t 2 L1 ð0; 1Þ and hence, using Fubini,
Z 1
Z 1Z 1
Z 1
zt
st
e F ðtÞ dt ¼ e f ðtÞ dt ds ¼ rðsÞ ds ¼ 1=qðzÞ
z
0
0
z
for z40: Thus the Laplace transform of F ðtÞ is 1=q and the growth conditions follow
with z ¼ o: &
Appendix B. Simplifying the Laplace–Laplace transform of the maximum process
Clearly, z40 implies qðzÞ40: Thus the second pole of the integrand of (1),
l
ix þ aðixÞg
;
xðx ilÞ xðx ðix þ aðixÞg ÞÞ
is at x ¼ iqðxÞ: For Gn ¼ fneiy : 0pyppg there exists M40 such that
Z 1
l
ix þ aðixÞg
lim
g dx
n!1 2p G xðx ilÞ xðx ðix þ aðixÞ ÞÞ
n
Z
1
M
p lim
dx ¼ lim M=2n ¼ 0 :
n!1 2p G n2
n!1
n
Thus, we can apply the residue theorem and obtain
Z
1 1
l
ix þ aðixÞg
dx
2p 1 xðx ilÞ xðx ðix þ aðixÞg ÞÞ
Z
1
l
1 þ aðixÞg1
¼ lim
g dx
n!1 2pi ½n;n[G ðx ilÞ xðx ðix þ aðixÞ ÞÞ
n
1 þ alg1
l
1 þ aðqðxÞÞg1
g þ
xðx ðl þ al ÞÞ iqðxÞ il xði þ igaqðxÞg1 Þ
l þ alg
l
1 þ aðqðxÞÞg1
¼
þ
:
xðx þ l alg Þ xðqðxÞ lÞ ð1 gaqðxÞg1 Þ
¼l
Since q is an inverse function we can compute its derivative
d
1
qðxÞ ¼
dx
agqðxÞg1 1
as in the proof of Lemma 1. Thus for u4alg l;
Z 1
l þ alg
l
1 þ aðqðxÞÞg1
dx
þ
g
xðqðxÞ lÞ ð1 gaqðxÞg1 Þ
xðx þ l al Þ
u
Z 1
1 þ axg1
¼ ½lnð1 þ ðl alg Þ=xÞ1
dx
l
u
g
qðuÞ ðax xÞðx lÞ
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B. Baeumer et al. / Physica A 350 (2005) 245–262
¼ lnð1 þ ðl alg Þ=uÞ ½lnð1 l=xÞ1
qðuÞ
1 l=qðuÞ
u ul=qðuÞ
¼ ln
:
¼ ln
1 þ ðl alg Þ=u
u þ l alg
Using L’Hopital’s rule, we see that for alg l40; limu!alg l lnððu ul=qðuÞÞ=ðu þ
l alg ÞÞo1; since the expression inside the logarithm tends to a finite constant as
u ! alg l: Then it is not hard to show that the above equality holds for all u40
(integrate from u to alg l e and alg l þ e to infinity and then let e ! 0). Thus,
using (1),
Z 1Z 1
1 l=qðuÞ
euslT d T ðMðs; TÞÞ ds ¼
:
u þ l alg
0
0
Appendix C. Proof of Theorem 2
Clearly, the inverse in u of (2) is given by5
Z s
g1
~
Hðs; lÞ ¼ ð1 al Þ
expðrðl alg ÞÞ dr
0
Z s
þ
mðs rÞ expðrðl alg ÞÞ dr
0
Z s
1 expðsðl alg ÞÞ
þ
mðs rÞ expðrðl alg ÞÞ dr :
¼
l
0
Inverting with respect to l is a bit more delicate. Using the complex inversion
formula (see for example Ref. [28, Theorem 7.6]) we obtain
Z t
Z
1
elt 1 expðsðl alg ÞÞ
Hðs; rÞ dr ¼
2pi cþiR l
l
0
Z s
þ
mðs rÞ expðrðl alg ÞÞ dr dl
0
Z
1
elt 1 expðsðl alg ÞÞ
¼
dl
2pi cþiR l
l
Z
Z s
1
elt
expðrðl alg ÞÞ dl dr
mðs rÞ
þ
2pi
l
0
cþiR
the second equality holding due to Fubini since
Z s
Z
jelt j
j expðrðl alg ÞÞj dl dr
jmðs rÞj
0
cþiR jlj
5
As the Laplace transform in l has to stay bounded as l ! 1; we see that convolution with the
function m has to have the following effect for all s40:
m% expðsðl alg ÞÞ expðsðl alg ÞÞ=l :
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B. Baeumer et al. / Physica A 350 (2005) 245–262
Z
257
Z
j expðrðl alg ÞÞj
dl dr
jlj
0
cþiR
Z s
Z
j expðrðl alg ÞÞj
dl dr
þ ect
jmðs rÞj
jlj
s=2
cþiR
Z s=2
Z
j expðrðl alg ÞÞj
dr dl
pect sup jmðrÞj
jlj
s=2oros
cþiR 0
Z
j expðrðl alg ÞÞj
þ ect kmk1 sup
dl
jlj
s=2oros cþiR
¼ ect
s=2
jmðs rÞj
o1 :
Making a change of variables we see that on the right-hand side we have an
expression akin to the formula for the inversion of the Fourier transform.
Z
1
elt 1 expðsðl alg ÞÞ
dl
2pi cþiR l
l
Z
Z s
1
elt
expðrðl alg ÞÞ dl dr
mðs rÞ
þ
2pi cþiR l
0
Z
ect 1 iut 1 expðsðiu þ c aðiu þ cÞg ÞÞ
e
¼
du
2p 1
ðiu þ cÞ2
Z
Z s
ect 1 iut expðrðiu þ c aðiu þ cÞg ÞÞ
du dr :
ðC:1Þ
mðs rÞ
e
þ
iu þ c
2p 1
0
Now the Fourier transform of a shifted g-stable distribution is
!
Z 1
g
1
xr
ikx
e
g
dx ¼ eikrþraðikÞ :
1=g g
1=g
ðarÞ
ðarÞ
1
Since ecx gg ðxÞ is bounded for all x 2 R for some cX0 (e.g., Ref. [29, Theorem 4.7.1])
we obtain that
!
Z 1
cx
g
e
x
r
eiux
g
dx ¼ erðiuþcþaðiuþcÞ Þ :
1=g g
1=g
ðarÞ
ðarÞ
1
Hence, the expressions in (C.1) are indeed inverse Fourier transforms and
Z
ect 1 iut 1 expðsðiu þ c aðiu þ cÞg ÞÞ
e
du
2p 1
ðiu þ cÞ2
Z
Z s
ect 1 iut expðrðiu þ c aðiu þ cÞg ÞÞ
du dr
mðs rÞ
e
þ
iu þ c
2p 1
0
!
Z t Z w
1
xs
¼t
g
dx dw
1=g g
ðasÞ1=g
1 1 ðasÞ
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258
Z
Z
s
t
1
1
ðarÞ1=g
mðs rÞ
þ
0
gg
xr
!
dx dr :
ðarÞ1=g
Therefore,
Z
t
1
!
Z
s
mðs rÞ
tr
!
gg
dx þ
dr
1=g
ðasÞ1=g
ðarÞ1=g
0 ðarÞ
!
!
Z s
Z 1
1
xs
mðs rÞ
tr
gg
gg
dx þ
dr
¼
1=g
ðasÞ1=g
ðasÞ1=g
ðarÞ1=g
t
0 ðarÞ
!
Z s
Z 1
mðs uÞ
tu
gg ðuÞ du þ
gg
du :
&
¼
1=g
ðauÞ1=g
ðtsÞ=ðasÞ1=g
0 ðauÞ
Hðs; tÞ ¼ 1 1
ðasÞ1=g
gg
xs
ðC:2Þ
Appendix D. Proof of Theorem 4
Assume there exists a solution to (5). Using the Caputo derivative and the fact that
~~ rÞ lg1 : Then
uð0; sÞ ¼ dðsÞ; the Laplace–Laplace transform of ðd=dtÞg uðt; sÞ is lg uðl;
it follows easily that
g1
~
~~ rÞ ¼ 1 al þ fg ðrÞ :
uðl;
r þ l al
~ rÞjp1: Hence, the Laplace–Laplace
Since s7!uðt; sÞ is a probability density, juðt;
transform for each r is analytic in the right halfplane in l: Thus 1 alg1 þ f~ðrÞ ¼ 0
if r þ l alg ¼ 0 or equivalently, l ¼ qðrÞ; q given by Lemma 1. Hence
g
l þ al
¼ r=qðrÞ :
f~ðrÞ ¼
l
Since the range of l7!alg l contains the right halfplane, f~ is uniquely determined,
and so is f.
The first passage time distribution has Laplace–Laplace transform
Z 1Z 1
1 alg1 þ r=qðrÞ
erslT Hðs; TÞ ds dT ¼
rðr þ l alg Þ
0
0
by (2). Since Hð0; TÞ ¼ 0 for all T40 we have that
Z 1Z 1
1 alg1 þ r=qðrÞ
erslT d s ðHðs; TÞÞ dT ¼
r þ l alg
0
0
(D.1)
and hence the density of the first passage time distribution has the same
Laplace–Laplace transform as the solution to the differential equation and it is
therefore a mild solution. &
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259
Appendix E. Proof of Theorem 5
Taking the Fourier transform of (6) we obtain
g
d
^ kÞ ¼ LðkÞ
^ kÞ þ dðtÞgðkÞ
^ kÞ þ d Cðt;
^ Cðt;
^
a
:
Cðt;
dt
dt
Taking the Laplace transform in t yields
~^
~^
~^
^ Cðl;
^
alg Cðl;
kÞ f^ðkÞ ¼ LðkÞ
kÞ þ alg1 f^ðkÞ þ lCðl;
kÞ þ gðkÞ
:
Thus
^
f^ðkÞ alg1 f^ðkÞ þ gðkÞ
~^
:
Cðl;
kÞ ¼
g
^
al þ l LðkÞ
~
Since CðlÞ
2 L1 ðRd Þ we know that its Fourier transform has to be bounded for all
RðlÞ40: This implies that the numerator of the above equation has to be zero
^
whenever the denominator is equal to zero, or whenever qðLðkÞÞ
¼ l: Hence
g1 ^
^ f^ðkÞ=qðLðkÞÞ
^
^
^
Þf ðkÞ ¼ LðkÞ
gðkÞ
¼ ð1 þ aqðLðkÞÞ
is uniquely determined. Thus,
^
^
1 alg1 LðkÞ=qð
LðkÞÞ
~^
f^ðkÞ :
Cðl;
kÞ ¼
g
^
al þ l LðkÞ
R1
To see that CðtÞ ¼ 0 TðsÞf d s ðHðt; sÞÞ has the same Fourier–Laplace transform,
take the Fourier transform and observe that
Z 1
^
^ kÞ ¼
esLðkÞ d s ðHðt; sÞÞf^ðkÞ :
Cðt;
0
But this is the Laplace transform in s of the first passage time density evaluated at
^
LðkÞ
times f^ðkÞ; i.e., taking the Laplace transform in t, using (D.1) we see that
Z 1Z 1
^
~^
eltsðLðkÞÞ d s ðHðt; sÞÞ dt f^ðkÞ
Cðl;
kÞ ¼
0
¼
0
g1
^
^
LðkÞ=qð
LðkÞÞ
f^ðkÞ
^
al þ l LðkÞ
1 al
g
and therefore Cðt; xÞ is indeed the mild solution of (6).
Finally we prove that g is a distribution. The proof depends on the fact that for
some positive real constant C we have
^
jLðkÞjpC
maxfkkk; kkk2 g
for all k 2 Rd :
To see this, use the Lévy Representation [7] to write
Z 1
ikx
^
LðkÞ
¼ ika kAk þ
eikx 1 fðdxÞ ;
2
1 þ kxk2
xa0
(E.1)
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260
where a 2 Rd ; A is a nonnegative definite matrix, and f is a s-finite Borel measure on
Rd nf0g such that
Z
minf1; kxk2 gfðdxÞo1 :
(E.2)
xa0
The integral term I in the Lévy Representation satisfies jIjpjI 1 j þ jI 2 j with
Z
ikx
I1 ¼
eikx 1 fðdxÞ
1 þ kxk2
0okxko1
¼ I 11 þ I 12 ;
where
Z
ikx
jI 11 j ¼ ðe 1 ikxÞfðdxÞ
0okxko1
Z
1
p
kkk2 kxk2 fðdxÞ
0okxko1 2
pC 1 kkk2
and
Z
jI 12 j ¼ ik
x
fðdxÞ
2
1 þ kxk
0okxko1
Z
kxk3
pkkk
fðdxÞ
2
0okxko1 1 þ kxk
x
pC 2 kkk ;
while
eikx 1 Z
jI 2 j ¼
kxkX1
ikx
fðdxÞ
1 þ kxk2
pD þ D þ Dkkk ;
where D ¼ ffx : kxkX1go1 using the fact that jeikx j ¼ 1 and kxk=ð1 þ kxk2 Þp1
for kxkX1: Then (E.1) holds.
^ f^ðkÞ=qðLðkÞÞ
^
^
Since gðkÞ
¼ LðkÞ
and jqðzÞjXM 0 jzj1=g for almost all z 2 Rd by
(A.1) we have
LðkÞ
^
1 ^
jLðkÞj11=g
p
^
qðLðkÞÞ
M0
and note that 1 1=g40: Using (E.1) we obtain
11=g
^
^
; kkk22=g g
jgðkÞjpM
1 jf ðkÞj maxfkkk
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261
and then it follows easily that for some D40 we have
Z
^ dko1 :
ð1 þ kkk2 ÞD gðkÞ
Now Example 7.12 (b) on p. 191 of Ref. [30] shows that g is a tempered
distribution. &
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