Elect. Comm. in Probab. 15 (2010), 457–474
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
EXPLICIT SOLUTIONS TO FRACTIONAL DIFFUSION EQUATIONS
VIA GENERALIZED GAMMA CONVOLUTION
MIRKO D’OVIDIO
Department of Statistics, Probability and Applied Statistics. Sapienza University of Rome, P.le Aldo
Moro, 5 - 00185 Rome (Italy).
email: mirko.dovidio@uniroma1.it
Submitted April 18, 2010, accepted in final form September 20, 2010
AMS 2000 Subject classification: 60J65, 60J60, 26A33
Keywords: Mellin convolution formula, generalized Gamma r.v.’s, Stable subordinators, Fox functions, Bessel processes, Modified Bessel functions.
Abstract
In this paper we deal with Mellin convolution of generalized Gamma densities which leads to
integrals of modified Bessel functions of the second kind. Such convolutions allow us to explicitly
write the solutions of the time-fractional diffusion equations involving the adjoint operators of a
square Bessel process and a Bessel process.
1
Introduction and main result
In the last years, the analysis of the compositions of processes and the corresponding governing
equations has received the attention of many researchers. Many of them are interested in compositions involving subordinators, in other words, subordinated processes Y (T (t)), t > 0 (according
to [9]) where T (t), t > 0 is a random time with non-negative, independent and homogeneous
increments (see [4]). If the random time is a (symmetric or totally skewed) stable process we have
results which are strictly related to the Bochner’s subordination and the p.d.e.’s connections have
been investigated, e.g., in [6; 7; 8; 22; 23]. If the random time is an inverse stable subordinator
we shall refer to the governing equation of Y (T (t)) as a fractional equation considering that a fractional time-derivative must be taken into account. In the literature, several authors have studied
the solutions to space-time fractional equations. In the papers by Wyss [30], Schneider and Wyss
[29], the authors present solutions of the fractional diffusion equation ∂ tλ T = ∂ x2 T in terms of
Fox’s functions (see Section 2). In the works by Mainardi et al., see e.g. [17; 18] the authors have
β
shown that the solutions to space-time fractional equation x Dθα u = t D∗ u can be represented by
means of Mellin-Barnes integral representations (or Fox’s functions) and M-Wright functions (see
e.g. Kilbas et al. [13]). The fractional Cauchy problem Dαt u = L u has been thoroughly studied by
yet other authors and several representations of the solutions have been carried out, but an explicit
form of the solutions has never been obtained. Nigmatullin [25] gave a physical derivation when
L is the generator of some continuous Markov process. Zaslavsky [31] introduced the space-time
fractional kinetic equation for Hamiltonian chaos. Kochubei [14, 15] first introduced a mathemat457
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Electronic Communications in Probability
ical approach while Baeumer and Meerschaert [1] established the connections between fractional
problem and subordination by means of inverse stable subordinator when L is an infinitely divisible generator on a finite dimensional vector space. In particular, if ∂ t p = Lp is the governing
β
equation of X(t), then under certain conditions, ∂ t q = Lq + δ(x)t −β /Γ(1 − β) is the equation
governing the process X(Vt ) where Vt is the inverse or hitting time process to the β-stable subordinator, β ∈ (0, 1). Orsingher and Beghin [26, 27] found explicit representations of the solutions to
∂ tν u = λ2 ∂ x2 u only in some particlular cases: ν = (1/2)n ,1/n, n ∈ N and ν = 1/3, 2/3, 4/3. Also,
they represented the solutions to the fractional telegraph equations in terms of stable densities,
see [3; 26]. In general, the solutions to fractional equations represent the probability densities of
certain subordinated processes obtained by using a time clock (in the following we will refer to it
as L νt ) which is an inverse stable subordinator (see Section 4). For a short review on this field, see
also Nane [24] and the references therein.
We will present the role of the Mellin convolution formula in finding solutions of fractional diffusion equations. In particular, our result allows us to write the distribution of both stable subordinator and its inverse process whose governing equations are respectively space-fractional or
time-fractional equations. This result turns out to be useful for representing the solutions to the
following fractional diffusion equation
Dνt ũνγ,µ = Gγ,µ ũνγ,µ
(1.1)
γ,µ
ν
where ũγ,µ
ν = ũν (x, t), x > 0, t > 0, D t is the Riemann-Liouville fractional derivative, ν ∈ (0, 1]
and Gγ,µ is an operator to be defined below (see formula (3.4)). We present, for ν = 1/(2n + 1),
n ∈ N ∪ {0}, the explicit solutions to (1.1) in terms of integrals of modified Bessel functions of
the second kind (Kν ) whereas, for ν ∈ (0, 1], we obtain the solutions to (1.1) in terms of Fox’s
functions. After some preliminaries in Section 2, in Section 3 we recall the generalized Gamma
density Qγµ starting from which we define the distribution gµγ of the (generalized Gamma) process
γ,µ
γ,µ
G t and the distribution eµγ of the process E t . The latter can be seen as the reciprocal Gamma
γ,µ
γ,µ
process, indeed E t = 1/G t , or in a more striking interpretation, as the hitting time process
γ,µ
γ,µ
for which (E t < x) = (G γ,µ
> t). We shall refer to E t as the reciprocal or equivalently the
x
γ,µ
−γ,µ
γ,µ
inverse process of G t . It must be noticed that eµγ = gµ−γ because G t
= 1/G t . Furthermore,
γ,?n
we introduce the most important tool we deal with in this paper, the Mellin convolutions gµ̄
eµ̄?n
(see
eµ̄?n
1,?n
stands for eµ̄ . In Section 4 we draw some
subordinator τ̃νt and the distribution lν of the
formula (3.14)) and
(see formula (3.13)) where
useful transforms of the distribution hν of the stable
inverse process L νt . Similar calculations can be found in the paper by Schneider and Wyss [29].
The inverse (or hitting time) process is defined once again from the fact that (L νt < x) = (τ̃νx > t)
(see also [1; 4]). In Section 5 we present our main contribution. We show that the following
representations hold true:
hν (x, t) = eµ̄?n (x, ϕn+1 (t)),
x > 0, t > 0, ν = 1/(n + 1), n ∈ N
and
(n+1),?n
lν (x, t) = gµ̄
(x, ψn+1 (t)),
x > 0, t > 0, ν = 1/(n + 1), n ∈ N.
where µ̄ = (µ1 , . . . , µn ), µ j = j ν, j = 1, 2, . . . , n, ν = 1/(n + 1), n ∈ N and the time-stetching
functions are given by ϕm (s) = (s/m)m and ψm (s) = ms1/m , s ∈ (0, ∞), m ∈ N, ψ = ϕ −1 .
The discussion made so far allows us to introduce the result stated in Theorem 1. For ν = 1/(n+1),
Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution
n ∈ N ∪ {0}, the solutions to (1.1) can be written as follows
Z∞
ũνγ,µ (x, t) =
gµγ (x, s1/γ ) gµ̄
1/ν,?(1/ν−1)
(s, ψ1/ν (t)) ds,
459
x ∈ (0, ∞), t > 0
0
where, for n ∈ 2N ∪ {0}, we have
1/ν,?(1/ν−1)
gµ̄
(x, t) =
∞
 x ‹ 1−ν Z
4ν
1
π2 t 3
ν 1/2ν
Z
0
and
Q 1−ν (x, t) = K 1−ν
2
2
∞
Q 1−ν (x, s1 ) . . . Q 1−ν (sn−1 , t)ds1 . . . dsn−1
...
2
0
p
2 (x/t)1/ν ,
2
x > 0, t > 0.
γ,µ
As a direct consequence of this result we obtain ũ1 = g̃µγ , for ν = 1, where g̃µγ (x, t) = gµγ (x, t 1/γ )
and the governing equation writes
∂ γ,µ
∂ 2−γ ∂
∂ 1−γ
1
γ,µ
x
− (γµ − 1)
x
ũ1 , x > 0, t > 0.
ũ = 2
∂t 1
∂x
∂x
∂x
γ
Furthermore, for γ = 1, 2 and ν ∈ (0, 1] we obtain
Œ
‚
∂
∂2
ν 1,µ
− (µ − 2)
ũ1,µ
D t ũν = x
ν ,
∂x
∂ x2
and
Dνt ũ2,µ
ν
=
1
22
‚
∂2
∂ x2
−
∂ (2µ − 1)
∂x
x > 0, t > 0, µ > 0
(1.2)
Œ
x
ũ2,µ
ν ,
x > 0, t > 0, µ > 0.
(1.3)
Equation (1.3) represents a fractional diffusion around spherical objects and thus, the solutions
we deal with obey radial diffusion equations.
2
Preliminaries
The H functions were introduced by Fox [10] in 1996 as a very general class of functions. For our
purpose, the Fox’s H functions will be introduced as the class of functions uniquely identified by
their Mellin transforms. A function f for which the following Mellin transform exists
M [ f (·)](η) =
Z
∞
x η f (x)
0
dx
x
,
ℜ{η} > 0
can be written in terms of H functions by observing that
– ™
Z∞
(a , α )
dx
i
i i=1,..,p
η m,n
m,n
x H p,q x = M p,q
(η),
(b j , β j ) j=1,..,q
x
0
where
Qm
m,n
M p,q
(η)
= Qq
j=1 Γ(b j
+ ηβ j )
j=m+1 Γ(1 −
ℜ{η} ∈ D
(2.1)
Qn
i=1 Γ(1 − ai − ηαi )
Qp
.
b j − ηβ j ) i=n+1 Γ(ai + ηαi )
(2.2)
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Electronic Communications in Probability
The inverse Mellin transform is defined as
Z θ +i∞
1
f (x) =
M [ f (·)](η)x −η dη
2πi θ −i∞
at all points x where f is continuous and for some real θ . Thus, according to a standard notation,
the Fox H function is defined as follows
™
– Z
(a , α )
1
i
i i=1,..,p
m,n
=
M m,n (η)x −η dη
H p,q x (b j , β j ) j=1,..,q
2πi P(D) p,q
where P(D) is a suitable path in the complex plane C depending on the fundamental strip (D) such
that the integral (2.1) converges. For an extensive discussion on this function see Fox [10]; Mathai
and Saxena [20]. The Mellin convolution formula
Z∞
ds
f1 ? f2 (x) =
(2.3)
f1 (x/s) f2 (s) , x > 0
s
0
turns out to be very useful later on. Formula (2.3) is a convolution in the sense that
M f1 ? f2 (·) (η) = M f1 (·) (η) × M f2 (·) (η).
(2.4)
Throughout the paper we will consider the integral
Z∞
f1 ◦ f2 (x, t) =
f1 (x, s) f2 (s, t)ds
(2.5)
0
(for some well-defined f1 , f2 ) which is not, in general, a Mellin convolution. We recall the following connections between Mellin transform and both integer and fractional order derivatives.
In particular, we consider a rapidly decreasing function f : [0, ∞) 7→ [0, ∞), if there exists a ∈ R
such that
dk
lim+ x a−k−1 k f (x) = 0, k = 0, 1, . . . , n − 1, n ∈ N, x ∈ R+
x→0
dx
then we have
n
d
Γ(η)
M
f
(·)
(η) =(−1)n
M f (·) (η − n)
(2.6)
d xn
Γ(η − n)
and, for 0 < α < 1
M
dα
d xα
f (·) (η) =
Γ(η)
Γ(η − α)
M f (·) (η − α)
(2.7)
(see Kilbas et al. [13]; Samko et al. [28] for details). The fractional derivative appearing in (2.7)
must be understood as follows
Z x
1
dn f
dα
n−α−1
f
(x)
=
(x
−
s)
(s) ds, n − 1 < α < n
(2.8)
d xα
Γ (n − α) 0
ds n
that is the Dzerbayshan-Caputo sense. We also deal with the Riemann-Liouville fractional derivative
Z x
1
dn
α
Dx f =
(x − s)n−α−1 f (s) ds, n − 1 < α < n
(2.9)
Γ (n − α) d x n 0
Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution
461
and the fact that
Dαx f =
dα
d xα
f −
f
k
dx n−1
X
dk
k=0
x k−α
x=0+
Γ(k − α + 1)
n − 1 < α < n,
,
(2.10)
see Gorenflo and Mainardi [11] and Kilbas et al. [13]. We refer to Kilbas et al. [13]; Samko et al.
[28] for a close examination of the fractional derivatives (2.8) and (2.9).
3
Mellin convolution of generalized Gamma densities
In this section we introduce and study the Mellin convolution of generalized gamma densities. In
the literature, it is well-known that generalized Gamma r.v. possess density law given by
Qγµ (z) = γ
z γµ−1
exp {−z γ } ,
Γ µ
z > 0, γ > 0, µ > 0.
Our discussion here concerns the function
x‹
1
xγ
x γµ−1
gµγ (x, t) = sign(γ) Qγµ
exp − γ ,
= |γ| γµ
t
t
t Γ(µ)
t
Let us introduce the convolution
Z∞
gµγ11 ? gµγ22 (x, t) =
0
gµγ11 (x, s)gµγ22 (s, t)ds = sign(γ1 γ2 )
x > 0, t > 0, γ 6= 0, µ > 0.
1
Z
t
∞
0
Qγµ11 (x/s)Qγµ22 (s/t)
for which we have (see formula (2.4))
i
i
h
i
h
h
M gµγ11 ? gµγ22 (·, t) (η) = M gµγ11 (·, t 1/2 ) (η) × M gµγ22 (·, t 1/2 ) (η)
ds
s
(3.1)
(3.2)
(3.3)
as a straightforward calculation shows. We now introduce the generalized Gamma process (GGP in
γ,µ
short). Roughly speaking, the function (3.1) can be viewed as the distribution of a GGP {G t , t >
γ,µ
0} in the sense that ∀t the distribution of the r.v. G t is the generalized Gamma distribution (3.1).
Thus, we make some abuse of language by considering a process without its covariance structure.
n
In the literature there are several non-equivalent definitions of the distribution on R+
of Gamma
distributions, see e.g. Kotz et al. [16] for a comprehensive discussion. In Section 5 (Corollary 1)
we will show that the distribution (3.1) satisfies the p.d.e.
∂
gµγ =
∂t
where
Gγ,µ f =
1
γ2
∂
∂x
d(t γ )
x 2−γ
dt
∂
∂x
Gγ,µ gµγ ,
− (γµ − 1)
x > 0, t > 0
∂
∂x
x 1−γ
f,
x > 0, t > 0
(3.4)
and γ 6= 0, f ∈ D(Gγ,µ ). For γ = 1, equation (3.1) becomes the distribution of a 2µ-dimensional
(2µ)
squared Bessel process {BESSQ t/2 , t > 0} and, for γ = 2 we obtain the distribution of a 2µ(2µ)
dimensional Bessel process {BES t/2 , t > 0}, both starting from zero. Some interesting distributions can be realized through Mellin convolution of distribution gµγ . Indeed, after some algebra we
arrive at
γ
x γµ1 −1 t γµ2
gµγ1 ? gµ−γ
(x,
t)
=
, x > 0, t > 0, γ > 0
(3.5)
2
B(µ1 , µ2 ) (t γ + x γ )µ1 +µ2
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Electronic Communications in Probability
and
x γµ2 −1 t γµ1
γ
gµ−γ
? gµγ2 (x, t) =
1
B(µ1 , µ2 )
(t γ
+ x γ )µ1 +µ2
x > 0, t > 0, γ > 0
,
(3.6)
where B(·, ·) is the Beta function (see e.g. Gradshteyn and Ryzhik [12, formula 8.384]). Moreover,
in light of the Mellin convolution formula (2.4), the following holds true
h
i
h
i
−γ
γ
M gµγ1 ? gµ−γ
(·,
t)
(η)
=
M
g
?
g
(·,
t)
(η).
µ
µ
2
2
1
A further distribution arising from convolution can be presented. In particular, for γ 6= 0, we have
r !
µ1 +µ2
2|γ| (x γ /t γ ) 2
xγ
γ
γ
Kµ2 −µ1 2
, x > 0, t > 0
(3.7)
gµ1 ? gµ2 (x, t) =
x Γ(µ1 )Γ(µ2 )
tγ
which proves to be very useful further on. The function Kν appearing in (3.7) is the modified Bessel
function of imaginary argument (see e.g [12, formula 8.432]). For the sake of completeness we
have writen the following Mellin transforms:
η−1
h
i
Γ γ +µ
M gµγ (·, t) (η) =
t η−1 , t > 0, ℜ{η} > 1 − γµ, γ 6= 0,
Γ µ
and
M
h
η
i
Γ µ− γ
x η−1 ,
gµγ (x, ·) (η) =
Γ µ
x > 0, ℜ{η} > γµ, γ 6= 0.
(3.8)
Formula (3.8) suggests that
i
i
h
i
h
h
M gµγ11 ? gµγ22 (x, ·) (η) = M gµγ11 (x 1/2 , ·) (η) × M gµγ22 (x 1/2 , ·) (η).
γ,µ
For the one-dimensional GGP we are able to define the inverse generalized Gamma process {E t ,
t > 0} (IGGP in short) by means of the following relation
γ,µ
P r{E t
< x} = P r{G γ,µ
x > t}.
The density law eµγ = eµγ (x, t) of the IGGP can be carried out by observing that
γ,µ
eµγ (x, t) = P r{E t
∈ d x}/d x =
∞
Z
t
∂
∂x
gµγ (s, x) ds,
x > 0, t > 0
(3.9)
and, making use of the Mellin transform, we obtain
Z∞
h
i
∂ γ
γ
M eµ (·, t) (η) =
M
g (s, ·) (η) ds, ℜ{η} < 1
∂x µ
t
Z∞
h
i
= by (2.6) = −(η − 1)
M gµγ (s, ·) (η − 1) ds
t
= by (3.8) = −(η − 1)
∞
Z
t
Γ µ−
η−1
γ
Γ µ
sη−2 ds =
Γ µ−
η−1
γ
Γ µ
t η−1
Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution
The derivative under the integral sign in (3.9) is allowed from the fact that Ξ1 (s) =
L 1 (R+ ) as a function of s. From (2.2) and the fact that
™
™
– – (a , α )
i
i i=1,..,p
m,n
c (ai , cαi )i=1,..,p
m,n
= c H p,q x H p,q x (b j , cβ j ) j=1,..,q
(b j , β j ) j=1,..,q
for all c > 0 (see Mathai and Saxena [20]), we have that

 γ 1,0 t γ (µ, 0)
γ
 , x > 0, t > 0, γ > 0.
eµ (x, t) = H1,1  γ x
x (µ, 1)
∂
∂x
463
gµγ (s, x) ∈
(3.10)
(3.11)
h
i
By observing that M eµγ (·, t) (1) = 1, we immediately verify that (3.11) integrates to unity. The
density law gµγ can be expressed in terms of H functions as well, therefore we have

 γ
γ
x
(µ,
0)
1,0
 , x > 0, t > 0, γ > 0.
gµγ (x, t) = H1,1  γ (3.12)
x
t (µ, 1)
In view of (3.11) and (3.12) we can argue that
γ,µ law
Et
and
eµγ (x, t)
=
gµ−γ (x, t),
−γ,µ law
= Gt
γ,µ
= 1/G t ,
t > 0, γ > 0, µ > 0
γ > 0, x > 0, t > 0.
1,1/2
, t > 0} can be written as
p
1,1/2
Et
= inf{s; B(s) = 2t}
Remark 1. We notice that the inverse process {E t
where B is a standard Brownian motion. Thus, E 1,1/2
p can be interpreted as the first-passage time
of a standard Brownian motion through the level 2t.
In what follows we will consider the Mellin convolution eµ̄?n (x, t) = eµ1 ?. . .?eµn (x, t) (see formulae
(2.4) and (3.2)) where µ̄ = (µ1 , . . . , µn ), µ j > 0, j = 1, 2, . . . , n and, for the sake of simplicity,
eµ (x, t) = eµ1 (x, t). For the density law eµ̄?n (x, t), x > 0, t > 0 we have
€
Š
n
n
h
i
h
i
Y
Y
Γ µj + 1 − η
?n
1/n
η−1
€ Š
M eµ̄ (·, t) (η) =
M eµ j (·, t ) (η) = t
(3.13)
Γ µj
j=1
j=1
γ,?n
with ℜ{η} < 1. Furthermore, for the Mellin convolution gµ̄ (x, t) = gµγ1 ?, . . . , ?gµγn (x, t) we have
η−1
n
n Γ
h
i
h
i
+ µj
Y
Y
γ
γ,?n
€ Š
(3.14)
M gµ̄ (·, t) (η) =
M gµγ j (·, t 1/n ) (η) = t η−1
Γ µj
j=1
j=1
with ℜ{η} > 1 − min j {µ j }.
Lemma 1. The functions gµγ and eµγ are commutative under ?-convolution.
Proof. Consider the Mellin convolution (3.13). Let eµ j be the distribution of the process X σ j , then
formula (3.13) means that
¦
©η−1
E {X σ1 (X σ2 (. . . X σn (t) . . .))}η−1 = E X σ1 (t 1/n )X σ2 (t 1/n ) · · · X σn (t 1/n )
for all possible permutations of {σ j }, j = 1, 2, . . . , n. The same result can be shown for eµγ j . Suppose
now that the process X σ j possesses distribution gµγ j , from (3.14) we obtain the claimed result.
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Electronic Communications in Probability
4
Stable subordinators
(ν)
The ν-stable subordinators {τ̃ t , t > 0}, ν ∈ (0, 1), are defined as non-decreasing, (totally)
positively skewed, Lévy processes with Laplace transform
(ν)
E exp{−λτ̃ t } = exp {−tλν } ,
t > 0,
λ>0
(4.1)
and characteristic function
(ν)
E exp{iξτ̃ t } = exp{−tΨν (ξ)},
where
Ψν (ξ) =
Z
∞
(1 − e−iξu )
0
ξ∈R
ν
(4.2)
du
Γ (1 − ν) uν+1
(see Bertoin [4]; Zolotarev [32]). After some algebra we get

 πν ‹‹
πν ξ
ν
ν
Ψν (ξ) =σ|ξ| 1 − i sgn(ξ) tan
= |ξ| exp −i
.
2
2 |ξ|
(ν)
For the density law of the ν-stable subordinator {τ̃ t , t > 0}, say hν = hν (x, t), x > 0, t > 0 we
have the t-Mellin transforms
”
—
πην ξ
M ĥν (ξ, ·) (η) = |ξ|−ην exp i
Γ η
(4.3)
2 |ξ|
and
”
—
M h̃ν (λ, ·) (η) = λ−ην Γ η
(4.4)
where
ĥν (ξ,
t) = F hν (·, t) (ξ) is the Fourier transform appearing in (4.2) and h̃ν (λ, t) =
L hν (·, t) (λ) is the Laplace transform (4.1). By inverting (4.3) we obtain the Mellin transform with respect to t of the density hν which reads
Z
”
—
1
M hν (x, ·) (η) =
e−iξx M ĥ(ξ, ·) (η)dξ
(4.5)
2π R
)
πην
( i πην
e 2
e−i 2
Γ η Γ 1 − ην
+
=
2π
(i x)1−ην
(−i x)1−ην
§
§ π
ª
§ π
ªª
Γ η Γ 1 − ην
=
exp
−i
+
iπην
+
exp
i
−
iπην
1−ην
2
2
2π
x
Γ η Γ 1 − ην
Γ η
x ην−1 , x > 0, ν ∈ (0, 1)
=
sin πην =
π x 1−ην
Γ ην
where ℜ{ην} ∈ (0, 1). Formula (4.5) can be also obtained by inverting (4.4). We are also able to
evaluate the Mellin transform with respect to x of the density law hν . From (4.3) and the fact that
Z∞
Γ(η)
νπ ξ
ν
ν
x η−1 e−iξx d x =
,
where
(±iξ)
=
|ξ|
exp
±i
, ν ∈ (0, 1)
(4.6)
(iξ)η
2 |ξ|
0
we obtain
Γ η
M hν (·, t) (η) =
2π
Z
|ξ|
R
−η
exp −i
πη ξ
2 |ξ|
− tΨν (ξ) dξ
Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution
=
=
Γ(η)
¨
e
2π
−i
πη
2
∞
Z
ξ
−η −tΦν (ξ)
e
dξ + e
i
πη
2
0
Γ(η)
Γ
1−η
Z
∞
ξ
−η −tΦν (−ξ)
e
465
«
dξ
0
t
η−1
ν
¦
e iπ(1−η) + e−iπ(1−η)
2πν
ν
η−1
1−η
t ν
,
=Γ
ν
νΓ 1−η
©
ℜ{η} ∈ (0, 1), t > 0.
(4.7)
The inversion of Fourier and Laplace transforms by making use of Mellin transform has been also
treated by Schneider and Wyss [29].
We investigate the relationship between stable subordinators and their inverse processes. For a
(ν)
(ν)
ν-stable subordinator {τ̃ t , t > 0} and an inverse process {L t , t > 0} (ISP in short) such that
(ν)
P r{L t
< x} = P r{τ̃(ν)
x > t}
we have the following relationship between density laws
Z∞
∂
(ν)
hν (s, x)ds,
lν (x, t) = P r{L t ∈ d x}/d x =
∂x
t
x > 0, t > 0.
(4.8)
We observe that ∂∂x hν (s, x) exists and there exists ζ(s) ∈ L 1 (R+ ) such that Ξ2 (s) = ∂∂x hν (s, x) =
const · Dsν hν (s, x) ≤ ζ(s). The function hν is the distribution of a totally skewed stable process,
n
thus hν (x), x ∈ R+
belongs to the space of functions in D((−4)ν/2 ), see Samko et al. [28]. Thus,
the integral in (4.8) converges. The density law (4.8) can be written in terms of Fox functions by
observing that
Z∞
∂
M lν (·, t) (η) =
M
hν (s, ·) (η) ds
∂x
t
Z∞
= by (2.6) = −(η − 1)
M hν (s, ·) (η − 1) ds
t
= by (4.5) = −
∞
Z
t
=
Γ η
Γ η
Γ ην − ν
t ν(η−1) ,
Γ ην − ν + 1
Thus, by direct inspection of (2.2), we recognize that
 
1 1,0 x (1 − ν, ν)
,
lν (x, t) = ν H1,1  ν t
t (0, 1)
sην−ν−1 ds
ℜ{η} < 1/ν, t > 0.
x > 0, t > 0, ν(0, 1).
(4.9)
(4.10)
Density (4.10) integrates to unity, indeed M lν (·, t) (1) = 1. The t-Laplace transform
L [lν (x, ·)](λ) = λν−1 exp {−xλν } ,
λ > 0, ν ∈ (0, 1)
comes directly from the fact that
Z∞
Z∞
Γ η
−λt
x η−1 L lν (x, ·) (λ) d x.
e M lν (·, t) (η) d t = ην−ν+1 =
λ
0
0
(4.11)
466
Electronic Communications in Probability
From (4.11) we retrieve the well-known fact that L lν (·, t) (λ) = Eν (−λt ν ) (see also Bondesson
et al. [5]) where Eβ is the Mittag-Leffler function which can be also written as
Z∞
¦
©
x ν−1 sin πν
1
ν
exp −λ1/ν t x
Eν (−λt ) =
d x, t > 0, λ > 0.
(4.12)
π 0
1 + 2x ν cos πν + x 2ν
ν
The distribution lν satisfies the fractional equation ∂∂t ν lν = − ∂∂x lν , x > 0, t > 0 subject to
lν (x, 0) = δ(x) where the fractional derivative must be understood in the Dzerbayshan-Caputo
sense (formula (2.8)). The governing equation of lν can be also presented by considering the
Riemann-Liouville derivative (2.9) and the relation (2.10) (see e.g. Baeumer and Meerschaert
[1]; Meerschaert and Scheffler [21]; Baeumer et al. [2]). It is well-known that the ratio involv(ν)
(ν)
ing two independent stable subordinator { 1 τ̃ t , t > 0} and { 2 τ̃ t , t > 0} has a distribution, ∀t,
given by
(ν)
(ν)
r(w) = P r{ 1 τ̃ t / 2 τ̃ t
∈ dw}/dw =
w ν−1 sin πν
1
π 1 + 2w ν cos πν + w 2ν
,
w > 0, t > 0.
(ν)
(4.13)
(ν)
Here we study the ratio of two independent inverse stable processes { 1 L t , t > 0} and { 2 L t , t >
0} by evaluating its Mellin transform as follows
n
oη−1
1 sin νπ − ηνπ
(ν)
(ν)
E 1 Lt / 2 Lt
=M lν (·, t) (η) × M lν (·, t) (2 − η) =
(4.14)
ν
sin ηπ
with ℜ{η} ∈ (0, 1). By inverting (4.14) we obtain
k(x) =
1
sin νπ
νπ 1 + 2x cos νπ +
x2
=
1
Z
2πi
θ +i∞
sin νπ − ηνπ
sin ηπ
θ −i∞
x −η dη
(4.15)
for some real θ ∈ (0, 1). From (4.13) and (4.15) we can argue that
ν
law
(ν)
(ν)
(ν)
(ν)
= 1 L t / 2 L t , ∀t > 0.
1 τ̃ t / 2 τ̃ t
(4.16)
We notice that the equivalence in law (4.16) is independent of t as the formulae (4.13) and (4.15)
(ν)
entail. The distribution hν ◦ lν (x, t) of the process {τ̃ (ν) , t > 0} has Mellin transform (by making
Lt
use of the formulae (4.7) and (4.9)) given by
M hν ◦ lν (·, t) (η) =M hν (·, 1) (η) × M lν (·, t)
η−1
ν
1 sin πη η−1
+1 =
t
,
ν sin π 1−η
t >0
ν
with ℜ{η} ∈ (0, 1). Thus, we can infer that
(ν) law
τ̃
(ν)
Lt
(ν)
(ν)
= t × 1 τ̃ t / 2 τ̃ t
t >0
and hν ◦ lν (x, t) = t −1 r(x/t) where r(w) is that in (4.13). For the process {L
distribution lν ◦ hν (x, t) we obtain (from (4.9) and (4.7))
(ν)
(ν)
τ̃ t
, t > 0} with
1 sin πν − πην η−1
M lν ◦ hν (·, t) (η) =M lν (·, 1) (η) × M hν (·, t) (ην − ν + 1) =
t
ν
sin πη
with ℜ{η} ∈ (0, 1) and thus
L
(ν) law
(ν)
τ̃ t
(ν)
(ν)
= t × 1 Lt / 2 Lt ,
t > 0.
We have that lν ◦ hν (x, t) = t −1 k(x/t) where k(x) is that in (4.15).
Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution
5
467
Main results
In this section we consider compositions of processes whose governing equations are (generalized)
fractional diffusion equations. When we consider compositions involving Markov processes and
stable subordinators we still have Markov processes. Here we study Markov processes with random
time which is the inverse of a stable subordinator. Such a process does not belong to the family
of stable subordinators (see (4.12)) and the resultant composition is not, in general, a Markov
process. This somehow explains the effect of the fractional derivative appearing in the governing
equation, see Mainardi et al. [19]. Hereafter, we exploit the Mellin convolution of generalized
Gamma densities in order to write explicitly the solutions to fractional diffusion equations. We
first present a new representation of the density law hν by means of the convolution eµ̄?n introduced
in Section 3. To do this we also introduce the time-stretching function ϕm (s) = (s/m)m , m ≥ 1,
s ∈ (0, ∞).
Lemma 2. The Mellin convolution eµ̄?n (x, ϕn+1 (t)) where µ j = j ν, for j = 1, 2, . . . , n is the density
(ν)
law of a ν-stable subordinator {τ̃ t , t > 0} with ν = 1/(n + 1), n ∈ N. Thus, we have
hν (x, t) = eµ̄?n (x, ϕn+1 (t)),
x > 0, t > 0, ν = 1/(n + 1), n ∈ N.
Proof. From (3.13) we have that
i
M eµ̄?n (·, ϕn+1 (t)) (η) =
h
Qn
€
j=1 Γ 1 − η + µ j
€ Š
Qn
j=1 Γ µ j
Š
η−1
ϕn+1 (t)
.
(5.1)
From Gradshteyn and Ryzhik [12, formula 8.335.3] we deduce that
n
Y
Γ
k=1
k
n+1
n
(2π) 2
=p
,
n+1
n∈N
(5.2)
and formula (5.1) reduces to
h
i
M eµ̄?n (·, ϕn+1 (t)) (η) =
Qn
€
Š
1 − η + µj
η−1
ϕn+1 (t)
.
p
n/2
(2π)
ν
j=1 Γ
(5.3)
Furthermore, by making use of the (product theorem) relation
Γ(nx) = (2π)
1−n
2
n
nx−1/2
n−1 Y
k
Γ x+
n
k=0
(see Gradshteyn and Ryzhik [12, formula 3.335]) formula (5.3) becomes
i
Γ
M eµ̄?n (·, ϕn (t)) (η) =
h
€ 1−η Š
n/2
η/ν−n
(2π)
(n
+
1)
Γ
η−1
η−1
ν
ν
t ν
ϕn+1 (t)
=
n/2
νΓ 1−η
Γ 1 − η (2π)
€ 1−η Š
(with ℜ{η} ∈ (0, 1)) which coincides with (4.7). The claimed result is obtained.
(5.4)
468
Electronic Communications in Probability
In light of the last result we are able to write explicitly the density law of a stable subordinator.
For ν = 1/2, Lemma 2 says that
t2
h1/2 (x, t) =
eµ̄?1 (x, ϕ2 (t))
x −1/2−1 e− 4x
= e1/2 (x, (t/2) ) =
p € Š,
t −1 4 Γ 12
2
x > 0, t > 0
(5.5)
which is the well-known density law of the 1/2-stable
subordinator or the first-passage time of a
p
standard Brownian motion trough the level t/ 2. For ν = 1/3, from (3.7), we obtain
‚
Œ
1 t 3/2
2 t 3/2
?2
3
h1/3 (x, t) = eµ̄ (x, ϕ3 (t)) = e1/3 ? e2/3 (x, (t/3) ) =
K1
, x > 0, t > 0.
p
3π x 3/2 3 33/2 x
For ν = 1/4, by (3.7) (and the commutativity under ?, see Lemma 1), we have
h1/4 (x, t) = eµ̄?3 (x, ϕ4 (t)) = e1/4 ? e2/4 ? e3/4 (x, (t/4)4 ) = e1/2 ? (e1/4 ? e3/4 )(x, (t/4)4 )
p
where K1/2 (z) = π/2z exp{−z} (see [12, formula 8.469]). We notice that
i
h
”
—
M eµ̄?3 (·, ϕ4 (t)) (η) = M h1/2 ◦ h1/2 (·, t) (η)
which is in line with the well-known fact that
n
o
(ν1 )
(ν )
E exp −λ 1 τ̃ (ν2 ) = E exp −λν1 2 τ̃ t 2 = exp{−tλν1 ν2 },
2 τ̃ t
0 < νi < 1, i = 1, 2. For ν = 1/5, by exploiting twice (3.7) (and the commutativity under ?), we
can write
h1/5 (x, t) = eµ̄?4 (x, (t/5)5 ) =(e1/5 ? e2/5 ) ? (e3/5 ? e4/5 )(x, (t/5)5 )
(5.6)
‚ Ç Œ
‚
Œ
Z∞
7/2
5/2
t
2 t
s
= 3 2 3/10+1
K1
ds
s−2/5−1 K 1 2
p
5/2
5
5
x
s
5 π x
5
0
or equivalently
(5.7)
h1/5 (x, t) = eµ̄?4 (x, (t/5)5 ) =(e1/5 ? e3/5 ) ? (e2/5 ? e4/5 )(x, (t/5)5 )
‚
Œ
‚
Œ
Z
Ç
∞
t3
s
2 t 5/2
= 5/2 2 2/5+1
s−1/5−1 K 2 2
K2
ds.
p
5
5
x
5 π x
55/2 s
0
For ν = 1/(2n + 1), n ∈ N, by using repeatedly (3.7) we arrive at
hν (x, t) =
x ν/2 t 1/ν−3/2
ν 2−1/ν π1/2ν−1/2
where
Kν◦ n (x, t)
=
Z
∞
Z
x > 0, t > 0
∞
...
0
€
Š
Kν◦ n x, (ν t)1/ν ,
Kν (x, s1 ) . . . Kν (sn−1 , t) ds1 . . . dsn−1
0
is the integral
p(2.5) (as the symbol "◦ n" denote) where n functions are involved and Kν (x, t) =
−2ν−1
x
Kν 2 t/x , x > 0, t > 0. We state a similar result for the density law lν and the
γ,?n
convolution gµ̄ (see Section 3). Let us consider the time-stretching function ψm (s) = m s1/m ,
s ∈ (0, ∞), m ∈ N, (ψ = ϕ −1 where ϕ has been introduced in the previous Lemma).
Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution
(n+1),?n
Lemma 3. The Mellin convolution gµ̄
469
(x, ψn+1 (t)) where µ j = j ν, j = 1, 2, . . . , n and ν =
(ν)
1/(n + 1), n ∈ N, is the density law of a ν-inverse process {L t , t > 0}. Thus, we have
(n+1),?n
lν (x, t) = gµ̄
(x, ψn+1 (t)),
x > 0, t > 0, ν = 1/(n + 1), n ∈ N.
Proof. The proof can be carried out as the proof of Lemma 2.
x2 p
2
(x, 2t 1/2 ) = e− 4t / πt, x > 0, t > 0. Moreover, by making use of
We obtain that l1/2 (x, t) = g1/2
(3.7) and (5.2), we have that
‚
Œ
Ç
1 x
2 x 3/2
3
3
1/3
1
l1/3 (x, t) = g2/3 ? g1/3 (x, 3t ) =
K
, x > 0, t > 0
p
π t 3 33/2 t
4
4
4
and l1/4 (x, t) = g3/4
? g2/4
? g1/4
(x, 4t 1/4 ) follows (thank to the commutativity under ?) from
4
g3/4
?
4
g2/4
?
4
g1/4
(x, t)
4
=g1/2
4
? (g3/4
?
4
g1/4
)(x, t)
=
23/2 x
Z
π t
∞
2
4
exp −(sx) −
0
(st)2
ds
p
4
4
where g3/4
? g1/4
(x, t) is given by (3.7) and K1/2 (z) = π/2z exp{−z} (see [12, formula 8.469]).
In a more general setting, by making use of (3.7) we can write down
1/ν,?(1/ν−1)
gµ̄
(x, t) =
 x ‹ 1−ν
4ν
1
ν 1/2ν
π2 t 3
Q ◦n
1−ν (x, t),
ν = 1/(2n + 1), n ∈ N
(5.8)
2
where the symbol ”◦ n ” stands for the integral (2.5) where n functions Q 1−ν are involved and
2
p
2 (x/t)1/ν ,
Q 1−ν (x, t) = K 1−ν
2
2
x > 0, t > 0.
(5.9)
Now, we present the main result of this paper concerning the explicit solutions to (generalized)
fractional diffusion equations. We study a generalized problem which leads to fractional diffusion
equations involving the adjoint operators of both Bessel and squared Bessel processes. Let us
γ
γ
γ
1/γ
) and the Mellin transform of
introduce the distribution ũγ,µ
ν = g̃ µ ◦ l ν where g̃ µ (x, t) = g µ (x, t
γ,µ
ũν which reads
η−1
η−1
Γ γ + µ Γ γ + 1 η−1
”
—
ν
M ũγ,µ
t γ , 1 − γµ < ℜ{η} < 1 + γ/ν − γ. (5.10)
ν (·, t) (η) =
η−1
Γ(µ)Γ γ ν + 1
We state the following result.
Theorem 1. Let the previous setting prevail. For ν = 1/(2n + 1), n ∈ N ∪ {0}, the solutions to
γ,µ
Dνt ũγ,µ
ν = Gγ,µ ũν ,
x > 0, t > 0
(5.11)
can be represented in terms of generalized Gamma convolution as
ũγ,µ
ν (x, t)
=γ
x 2µ−1 ν
1−3ν
4ν
(π2 t 3ν )
1−ν
4ν
Z
∞
0
1
1
s 4ν − 4 −µ e−x
γ
/s
vν (s, t) ds,
x ≥ 0, t > 0
(5.12)
470
Electronic Communications in Probability
where Gγ,µ is the operator appearing in (3.4),
Z∞ Z∞
vν (s, t) =
Q 1−ν (s, s1 ) . . . Q 1−ν (sn−1 , t ν /ν)ds1 . . . dsn−1
...
0
2
0
2
and Q 1−ν is that in (5.9). Moreover, for ν ∈ (0, 1], we have
2
 γ
γ
2,0  x (1, ν);
γ,µ
ũν (x, t) = ν/γ H2,2
t ν (1, 1);
xt

(µ, 0) 
(µ, 1)
(5.13)
in terms of H Fox functions.
Proof. By exploiting the property (2.6) of the Mellin transform and the fact that
Z∞
x η−1 x θ f (x)d x = M f (·) (η + θ ),
0
for the operator (3.4) we have that
”
—
M Gγ,µ ũγ,µ
ν (·, t) (η)
”
—
∂ γ,µ
1
1
= − 2 (η − 1)M
ũν (·, t) (η − γ + 1) + 2 (γµ − 1)(η − 1)M ũνγ,µ (·, t) (η − γ)
∂x
γ
γ
”
—
”
—
1
1
γ,µ
= 2 (η − 1)(η − γ)M ũγ,µ
(·,
t)
(η
−
γ)
+
(γµ
−
1)(η
−
1)M
ũ
(·,
t)
(η − γ)
ν
ν
γ
γ2
”
—
1
(5.14)
= 2 (η − 1)(η − 1 + γµ − γ)M ũνγ,µ (·, t) (η − γ)
γ
”
—
where M ũγ,µ
ν (·, t) (η) is that in (5.10). We obtain
η−γ−1
η−γ−1
+
µ
Γ
+
1
Γ
—
”
η−γ−1
1
γ
γ
ν
M Gγ,µ ũγ,µ
t γ
ν (·, t) (η) = 2 (η − 1)(η − 1 + γµ − γ)
η−γ−1
γ
Γ(µ)Γ
ν +1
γ
η−1
η−1
Γ γ +µ Γ γ
η−1
1
ν−ν
t γ
= (η − 1)
η−1
γ
Γ(µ)Γ γ ν − ν + 1
η−1
η−1
Γ γ + µ Γ γ + 1 η−1
”
—
ν−ν
t γ
=
= Dνt M ũνγ,µ (·, t) (η)
η−1
Γ(µ)Γ γ ν − ν + 1
and ũγ,µ
ν (x, t) solves (5.11) for ν ∈ (0, 1). In view of Lemma 3 we can write
Z∞
ũγ,µ
ν (x, t) =
g̃µγ (x, s) gµ̄
1/ν,?(1/ν−1)
(s, ψ1/ν (t)) ds
0
and by means of (5.8) result (5.12) appears. Formula (5.13) follows directly from (2.2) by considering formula (3.10) and the fact that
– – ™
™
(a , α )
(a + cα , α )
1 m,n
i
i i=1,..,p
i
i
i i=1,..,p
m,n
H p,q x = c H p,q x (5.15)
(b j , β j ) j=1,..,q
(b j + cβ j , β j ) j=1,..,q
x
for all c ∈ R (see Mathai and Saxena [20]).
Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution
471
We specialize the previous result by keeping in mind formula (2.10) and the operator (3.4).
Corollary 1. For ν = 1, Theorem 1 says that
∂
∂t
γ,µ
ũ1
=
1
γ2
∂
∂x
x
2−γ
∂
∂x
− (γµ − 1)
∂
∂x
x
1−γ
γ,µ
ũ1 ,
x > 0, t > 0, γ 6= 0
γ,µ
where ũ1 = g̃µγ is the distribution of the GGP.
a.s.
This is because L 1t = t. Indeed, for ν = 1, L νt is the elementary subordinator (see [4]).
Proof. If ν = 1, then the equation (5.10) takes the form
Ψ t (η) =
M [g̃µγ (·, t)](η)
=Γ
η−1
+µ
γ
t
η−1
γ
Γ(µ)
,
ℜ{η} > 1 − γµ
(5.16)
where g̃µγ (x, t) = gµγ (x, t 1/γ ). For γ > 0, we perform the time derivative of (5.16) and obtain
∂
∂t
η−γ−1
η−1
Γ
+µ t γ
γ
γ
η−γ−1
η − 1 η − γ − 1 + γµ
η−γ−1
=
Γ
+µ t γ
γ
γ
γ
1
= 2 (η − 1)(η − γ − 1 + γµ)Ψ t (η − γ)
γ
Ψ t (η) =
η−1
γ,µ
which coincides with (5.14) and ũ1 (x, t) = g̃µγ (x, t), γ > 0. Similar calculation must be done for
γ < 0 and the proof is completed.
µ
Corollary 2. Let us write ũµν (x, t) = ũ1,µ
ν (x, t). The distribution ũν (x, t), x > 0, t > 0 µ > 0,
ν ∈ (0, 1], solves the following fractional equation
‚
Œ
∂ν µ
∂2
∂
u = x
− (µ − 2)
uµν .
(5.17)
∂ tν ν
∂x
∂ x2
In particular, for ν = 1/2, we have
µ
ũ1/2 (x, t)
x µ−1
=p
πtΓ(µ)
Z
∞
¨
s
0
−µ
exp −
x
−
s
s2
4t
«
ds,
x > 0, t > 0, µ > 0
1,µ
which can be seen as the distribution of the process {G|B(2t)| , t > 0} where B is a standard
γ,µ
Brownian motion run at twice its usual speed and G t
a squared Bessel process starting from zero.
1,µ
is a GGP. We notice that the process G t
is
2,µ
Corollary 3. The distribution ũ2,µ
ν = ũν (x, t), x > 0, t > 0, µ > 0, ν ∈ (0, 1] solves the following
fractional equation
‚ 2
Œ
∂ ν 2,µ
1
∂
∂ (2µ − 1)
ũ = 2
−
ũ2,µ
ν .
∂ tν ν
x
2
∂ x2 ∂ x
472
Electronic Communications in Probability
In particular, for ν = 1/3, we have
2,µ
ũ1/3 (x, t)
2 x 2µ−1
=
p
πΓ(µ) t
Z
∞
0
e−
x2
s
‚
sµ−1/2
K1
3
2 s3/2
p
33/2 t
Œ
ds,
x > 0, t > 0, µ > 0
and for µ = 1/2 we obtain
2,1/2
ũ1/3 (x, t)
=
2
p
π3/2 t
Z
∞
e
0
‚
2
− xs
K1
3
2 s3/2
p
3/2
3
t
Œ
ds,
x > 0, t > 0
1/3
which is the distribution of |B(L t )| where |B(t)| is a folded Brownian motion with variance t/2.
Acknowledgement The author is grateful to the anonymous referee for careful checks and comments.
References
1 B. Baeumer and M. Meerschaert. Stochastic solutions for fractional cauchy problems. Fract.
Calc. Appl. Anal., 4(4):481 – 500, 2001. MR1874479
2 B. Baeumer, M. Meerschaert, and E. Nane. Space-time duality for fractional diffusion. J. Appl.
Probab., 46:1100 – 1115, 2009. MR2582709
3 L. Beghin and E. Orsingher. The telegraph process stopped at stable-distributed times and its
connection with the fractional telegraph equation. Fract. Calc. Appl. Anal., 6:187 – 204, 2003.
MR2035414
4 J. Bertoin. Lévy Processes. Cambridge University Press, 1996. MR1406564
5 L. Bondesson, G. K. Kristiansen, and F. W. Steutel. Infinite divisibility of random variables and
their integer parts. Stat. Prob. Lett, 28:271 – 278, 1996. MR1407001
6 R. D. DeBlassie. Higher order PDEs and symmetric stable process. Probab. Theory Rel. Fields,
129:495 – 536, 2004. MR2078980
7 M. D’Ovidio and E. Orsingher. Composition of processes and related partial differential equations. J. Theor. Probab. (published on line), 2010.
8 M. D’Ovidio and E. Orsingher. Bessel processes and hyperbolic Brownian motions stopped
at different random times. Under revision for Stochastic Processes and their Applications,
(arXiv:1003.6085v1), 2010.
9 W. Feller. An introduction to probability theory and its applications, volume 2. 2 edition, 1971.
10 C. Fox. The G and H functions as symmetrical Fourier kernels. Trans. Amer. Math. Soc., 98:395
– 429, 1961. MR0131578
11 R. Gorenflo and F. Mainardi. Fractional calculus: integral and differential equations of frational
order, in A. Carpinteri and F. Mainardi (Editors). Fractals and Fractional Calculus in Continuum
Mechanics, pages 223 – 276, 1997. Wien and New York, Springher Verlag. MR1611585
Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution
12 I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series and products. Academic Press, 2007.
Seventh edition. MR2360010
13 A. Kilbas, H. Srivastava, and J. Trujillo. Theory and applications of fractional differential
equations (North-Holland Mathematics Studies), volume 204. Elsevier, Amsterdam, 2006.
MR2218073
14 A. N. Kochubei. The Cauchy problem for evolution equations of fractional order. Differential
Equations, 25:967 – 974, 1989. MR1014153
15 A. N. Kochubei. Diffusion of fractional order. Lecture Notes in Physics, 26:485 – 492, 1990.
MR1061448
16 S. Kotz, N. Balakrishnan, and N. L. Johnson. Continuous multivariate distributions, volume 1.
New York, Wiley, 2nd edition, 2000. MR1788152
17 F. Mainardi, Y. Luchko, and G. Pagnini. The fundamental solution of the space-time fractional
diffusion equation. Fract. Calc. Appl. Anal., 4(2):153 – 192, 2001. MR1829592
18 F. Mainardi, G. Pagnini, and R. Gorenflo. Mellin transform and subordination laws in fractional
diffusion processes. Fract. Calc. Appl. Anal., 6(4):441 – 459, 2003. MR2044309
19 F. Mainardi, G. Pagnini, and R. Gorenflo. Some aspects of fractional diffusion equations of
single and distributed order. Applied Mathematics and Computing, 187:295 – 305, 2007.
MR2323582
20 A. Mathai and R. Saxena. Generalized Hypergeometric functions with applications in statistics
and physical sciences. Lecture Notes in Mathematics, n. 348, 1973. MR0463524
21 M. Meerschaert and H. P. Scheffler. Triangular array limits for continuous time random walks.
Stoch. Proc. Appl., 118:1606 – 1633, 2008. MR2442372
22 M. Meerschaert, E. Nane, and P. Vellaisamy. Fractional cauchy problems on bounded domains.
Ann. of Probab., 37(3):979 – 1007, 2009. MR2537547
23 E. Nane. Higher order PDE’s and iterated processes . Trans. Amer. Math. Soc., (5):681–692,
2008. MR2373329
24 E. Nane. Fractional cauchy problems on bounded domains: survey of recent results.
arXiv:1004.1577v1, 2010.
25 R. Nigmatullin. The realization of the generalized transfer in a medium with fractal geometry.
Phys. Status Solidi B, 133:425 – 430, 1986.
26 E. Orsingher and L. Beghin. Time-fractional telegraph equations and telegraph processes with
Brownian time. Probab. Theory Rel. Fields, 128(1):141 – 160, 2004. MR2027298
27 E. Orsingher and L. Beghin. Fractional diffusion equations and processes with randomly varying time. Ann. Probab., 37:206 – 249, 2009. MR2489164
28 S. Samko, A. A. Kilbas, and O. I. Marichev. Fractional Integrals and Derivatives: Theory and
Applications. Gordon and Breach, Newark, N. J., 1993. MR1347689
473
474
Electronic Communications in Probability
29 W. Schneider and W. Wyss. Fractional diffusion and wave equations. J. Math. Phys., 30:134 –
144, 1989. MR0974464
30 W. Wyss. The fractional diffusion equations.
MR0861345
J. Math. Phys., 27:2782 – 2785, 1986.
31 G. Zaslavsky. Fractional kinetic equation for Hamiltonian chaos. Phys. D, 76:110 – 122, 1994.
MR1295881
32 V. M. Zolotarev. One-dimensional stable distributions, volume 65 of Translations of Mathematical
Monographs. American Mathematical Society, 1986. ISBN 0-8218-4519-5. Translated from the
Russian by H. H. McFaden, Translation edited by Ben Silver. MR0854867