PSEUDOPROCESSES RELATED TO SPACE-FRACTIONAL
HIGHER-ORDER HEAT-TYPE EQUATIONS
ENZO ORSINGHER AND BRUNO TOALDO
Abstract. In this paper we construct pseudo random walks (symmetric and
asymmetric) which converge in law to compositions of pseudoprocesses stopped
at stable subordinators. We find the higher-order space-fractional heat-type
equations whose fundamental solutions coincide with the law of the limiting
pseudoprocesses. The fractional equations involve either Riesz operators or
their Feller asymmetric counterparts. The main result of this paper is the
derivation of pseudoprocesses whose law is governed by heat-type equations of
real-valued order γ > 2. The classical pseudoprocesses are very special cases
of those investigated here.
1. Introduction
In this paper we consider pseudoprocesses related to different types of fractional
higher-order heat-type equations. Our starting point is the set of higher-order
equations of the form
∂
∂m
um (x, t) = κm m um (x, t),
x ∈ R, t > 0, m ∈ N > 2,
(1.1)
∂t
∂x
whose solutions have been investigated by many outstanding mathematicians such
as Bernstein [2]; Lévy [15]; Pòlya [22] and also, more recently, by means of the
steepest descent method, by Li and Wong [16]. In (1.1) the constant κm is usually
chosen in the form
(
±1,
m = 2n + 1,
κm =
(1.2)
n+1
(−1)
, m = 2n.
In our investigations we assume throughout that κm = (−1)n when m = 2n + 1.
Pseudoprocesses related to (1.1) have been constructed in the same way as for
the Wiener process by Daletsky [4]; Daletsky and Fomin [5]; Krylov [10]; Ladohin
[13]; Miyamoto [18]. More recently pseudoprocesses related to (1.1) have been
considered by Debbi [6]; Lachal [11, 12]; Mazzucchi [17]. For equations of the form
∂
∂γ
uγ (x, t) =
uγ (x, t),
∂t
∂|x|γ
x ∈ R, t > 0,
(1.3)
γ
∂
where 0 < γ ≤ 2, and ∂|x|
γ is the Riesz operator, the fundamental solution has
the form of the density of a symmetric stable process as Riesz himself has shown.
For γ > 2 the equation (1.3) was studied by Debbi (see [6; 7]) who proved the
sign-varying character of the corresponding solutions.
Date: May 28, 2013.
2000 Mathematics Subject Classification. 60K99, 35Q99.
Key words and phrases. Weyl derivatives, Riesz derivatives, Feller fractional operators, pseudoprocesses, stable processes, subordinators, continuous-time random walks.
1
2
ENZO ORSINGHER AND BRUNO TOALDO
For asymmetric fractional operators of the form
sin π2 (γ + θ) − ∂ γ
sin π2 (γ − θ) + ∂ γ
F γ,θ
+
D
= −
sin πγ
∂xγ
sin πγ
∂xγ
(1.4)
the equation
∂
uγ,θ (x, t) = F Dγ,θ uγ,θ (x, t),
x ∈ R, t > 0, 0 < γ ≤ 2,
(1.5)
∂t
was studied by Feller [8] who proved that the fundamental solution to (1.5) is the
law of an asymmetric stable process of order γ. The fractional derivatives appearing
in (1.4) are the Weyl fractional derivatives defined as
Z x
+ γ
∂
1
u(y)
dm
u(x) =
dy
∂xγ
Γ(m − γ) dxm −∞ (x − y)γ+1−m
Z
∞
− γ
1
dm
∂
u(y)
u(x)
=
dy
(1.6)
γ
m
∂x
Γ(m − γ) dx
(y − x)γ+1−m
x
where m − 1 < γ < m. The Riesz fractional derivatives appearing in (1.3) are
combinations of the Weyl’s derivatives (1.6) and are defined as
+ γ
− γ
∂γ
∂
∂
1
= −
+
.
(1.7)
∂|x|γ
2 cos πγ
∂xγ
∂xγ
2
This paper is devoted to pseudoprocesses related to fractional equations of the
form (1.3) and (1.5) when γ > 2. Of course, this implies that Weyl’s fractional
derivatives (1.6) are considered in the case γ > 2. The fundamental solutions of
these equations are sign-varying as in the case of higher-order heat-type equations
(1.1) studied in the literature (compare with [6]).
Fractional equations arise, for example, in the study of thermal diffusion in
fractal and porous media (Nigmatullin [19]; Saichev and Zaslavsky [23]). Other
fields of application of fractional equations can be found in Debbi [6]. Higher-order
equations emerge in many contexts as in trimolecular chemical reactions (Gardiner
[9] page 295) and in the linear approximation of the Korteweg De Vries equation
(see Beghin et al. [1]).
In our paper we study pseudo random walks (for the definitions and properties
of pseudo random walks and variables see Lachal [12]) of the form
N (tγ −2kβ )
W γ,2kβ (t) =
X
Uj2k (1)Qγ,2kβ
j
(1.8)
j=1
where the r.v.’s Qγ,2kβ
are independent from the Poisson process N , from the
j
2k
pseudo r.v.’s Uj (1) and from each other and have distribution for 0 < β < 1,
γ > 0, k ∈ N,
(
n
o
1,
w < γ,
γ,2kβ
(1.9)
Pr Qj
>w =
γ 2kβ
, w ≥ γ.
w
The Uj2k (1) are independent pseudo r.v.’s with law u2k (x, 1) with Fourier transform
Z ∞
2k
eiξx u2k (x, 1) dx = e−|ξ| .
(1.10)
−∞
SPACE-FRACTIONAL HIGHER-ORDER HEAT-TYPE EQUATIONS
3
1
The Poisson process N appearing in (1.8) is homogeneous and has rate λ = Γ(1−β)
.
We prove that
law
lim W γ,2kβ (t) = U 2k H β (t)
(1.11)
γ→0
2k
where U is the pseudoprocess of order 2k related to the heat-type equation (1.1)
for m = 2k and H β is a stable subordinator of order β ∈ (0, 1) independent from
U 2k . We show that the law of (1.11) is the fundamental solution to
∂ 2kβ
∂
v2kβ (x, t),
v2kβ (x, t) =
∂t
∂|x|2kβ
x ∈ R, t > 0, β ∈ (0, 1), k ∈ N.
(1.12)
In other words, we are able to construct pseudoprocesses of order γ > 2 in the form
of integer-valued pseudoprocesses stopped at stable distributed times as the limit
of suitable pseudo random walks. We consider also pseudo random walks of the
form
N (tγ −β(2k+1) )
X
γ,β(2k+1)
j Uj2k+1 (1)Qj
(1.13)
j=0
γ,β(2k+1)
Qj
where the
have distribution (1.9) (suitably adjusted), Uj2k+1 (1) is an
odd-order pseudo random variable with law u2k+1 (x, 1) and Fourier transform
Z ∞
2k+1
eiξx u2k+1 (x, 1) dx = e−iξ
(1.14)
−∞
and the j ’s are random variables which take values ±1 with probability p and q.
All the variables in (1.13) are independent from each other and also independent
1
. In this case we are able to show
from the Poisson process N with rate λ = Γ(1−β)
that
law
lim W γ,(2k+1)β (t) = U12k+1 H1β (pt) − U22k+1 H2β (qt)
(1.15)
γ→0
Hjβ ,
where
j = 1, 2, are independent stable subordinators independent also from the
pseudoprocesses U1 , U2 . We prove that the law of (1.15) satisfies the higher-order
fractional equation
∂
wβ(2k+1) (x, t) = Rwβ(2k+1) (x, t),
∂t
x ∈ R, t > 0,
(1.16)
where
+ β(2k+1)
− β(2k+1)
1
iπβk ∂
−iπβk ∂
R = −
pe
+ qe
.
∂xβ(2k+1)
∂xβ(2k+1)
cos βπ
2
(1.17)
The Fourier transform of the fundamental solution of (1.16) reads
w
bβ(2k+1) (ξ, t) = e−t|ξ|
β(2k+1)
(1−i sign(ξ) (p−q) tan βπ
2 )
(1.18)
We note that (1.18) corresponds to the Fourier transform of the law of (1.15) with
a suitable change of the time-scale that is
( "
!!
!!#)
pt
qt
β
β
2k+1
2k+1
E exp iξ U1
H1
− U2
H2
cos βπ
cos βπ
2
2
β(2k+1)
= e−t|ξ|
(1−i sign(ξ) (p−q) tan
βπ
2
(1.19)
4
ENZO ORSINGHER AND BRUNO TOALDO
The mean value here and below must be understood with respect to the signed measure of the pseudoprocess (see for example [6]). We study also the pseudoprocesses
governed by the equation
∂
zβ(2k+1),θ (x, t) =
∂t
F
Dβ(2k+1),θ zβ(2k+1),θ (x, t)
(1.20)
where F Dβ(2k+1),θ is the operator defined in (1.4) with γ replaced by β(2k + 1).
Also in this case we study continuous-time random walks whose limit has Fourier
transform equal to
EeiξZ
β(2k+1),θ
β(2k+1)
= e−t|ξ|
e
iπθ sign(ξ)
2
β ∈ (0, 1), k ≥ 1, −β < θ < β. (1.21)
,
When we take into account pseudo random walks constructed by means of evenorder pseudo random variables we arrive at limits Z 2βk,θ (t), t > 0, with Fourier
transform
π
EeiξZ
2βk,θ
cos
(t)
−t|ξ|2kβ cos
= e
θ
2
πβ
2
(1.22)
which shows the symmetric structure of the limiting pseudoprocess.
1.1. List of symbols. For the reader’s convenience we give a short list of the most
important symbols and definitions appearing in the paper.
• The right Weyl fractional derivative for m − 1 < γ < m, m ∈ N, x ∈ R
Z x
+ γ
∂
1
dm
u(y, t)
u(x, t) =
dy
(1.23)
∂xγ
Γ(m − γ) dxm −∞ (x − y)γ+1−m
• The left Weyl fractional derivative for m − 1 < γ < m, m ∈ N, x ∈ R,
Z ∞
− γ
∂
(−1)m dm
u(y, t)
u(x,
t)
=
dy
(1.24)
γ
m
∂x
Γ(m − γ) dx
(y − x)γ+1−m
x
• The Riesz fractional derivative for m − 1 < γ < m, m ∈ N, x ∈ R,
+ γ
− γ
∂γ
1
∂
∂
= −
+
∂|x|γ
2 cos γπ
∂xγ
∂xγ
2
(1.25)
• We introduce the operator R, for β ∈ (0, 1), k ∈ N, p, q ∈ [0, 1] : p + q = 1,
x ∈ R,
+ β(2k+1)
− β(2k+1)
1
iπβk ∂
−iπβk ∂
R = −
pe
+ qe
(1.26)
∂xβ(2k+1)
∂xβ(2k+1)
cos βπ
2
• The Feller derivative for m − 1 < γ < m, m ∈ N, θ > 0, x ∈ R,
sin π2 (γ − θ) + ∂ γ
sin π2 (γ + θ) − ∂ γ
F γ,θ
D
= −
+
sin πγ
∂xγ
sin πγ
∂xγ
(1.27)
• U m (t), t > 0 is a pseudoprocess of order m ∈ N with law um (x, t), x ∈ R,
t > 0, governed by (1.1)
• H β (t) is a stable subordinator of order β ∈ (0, 1) with probability density
hβ (x, t), x ≥ 0, t ≥ 0.
2. Preliminaries and auxiliary results
In this paper we consider higher-order heat-type equations where the space derivative is fractional in different ways.
SPACE-FRACTIONAL HIGHER-ORDER HEAT-TYPE EQUATIONS
5
2.1. Weyl fractional derivatives. First of all we consider equations of the form
± γ
∂
∂
uγ (x, t),
uγ (x, t) =
∂t
∂xγ
where
x ∈ R, t > 0, γ > 0,
(2.1)
±
∂γ
∂xγ
are the space-fractional Weyl derivatives defined as
Z x
u(z, t) dz
∂
1
dm
uγ (x, t) =
, m − 1 < γ < 1, m ∈ N,
γ
m
γ−m+1
∂x
Γ(m − γ) dx
−∞ (x − z)
(2.2)
+ γ
− γ
∂
(−1)m dm
u
(x,
t)
=
γ
∂xγ
Γ(m − γ) dxm
Z
∞
x
u(z, t) dz
,
(z − x)γ−m+1
m − 1 < γ < m, m ∈ N.
(2.3)
In our analysis the following result on the Fourier transforms of Weyl derivatives is
very important.
Theorem 2.1 ([24], page 137). The Fourier transforms of (2.2) and (2.3) read
Z ∞
+ γ
iπγ
∂
b(ξ, t),
(2.4)
b(ξ, t) = |ξ|γ e− 2 sign(ξ) u
dx eiξx γ u(x, t) = (−iξ)γ u
∂x
−∞
Z ∞
− γ
iπγ
∂
dx eiξx γ u(x, t) = (iξ)γ u
b(ξ, t) = |ξ|γ e 2 sign(ξ) u
b(ξ, t).
(2.5)
∂x
−∞
Clearly u
b(ξ, t) is the x-Fourier transform of u(x, t).
Proof. We give a sketch of the proof of (2.4) with some details.
Z ∞
Z x
Z ∞
+ γ
1
∂m
u(z, t)
iξx
iξx ∂
u(x, t) =
dx e
dz
dx e
∂xγ
Γ(m − γ) ∂xm −∞ (x − z)γ−m+1
−∞
−∞
Z ∞
Z ∞
∂ m u(x − z, t)
1
iξx
=
dz m γ−m+1
dx e
Γ(m − γ) 0
∂x
z
−∞
Z ∞
Z ∞
m
∂
1
=
dw eiξw
u(w, t)
dz eiξz z m−γ−1
m
∂w
Γ(m
− γ) 0
−∞
Z ∞
Z ∞
1
m
iξw
= (−iξ)
dz eiξz z m−γ−1 .
e u(w, t) dw
Γ(m − γ) 0
−∞
(2.6)
The result
Z ∞
iπ
(−iξ)m
(2.7)
dz eiξz z m−γ−1 = |ξ|γ e− 2 sign(ξ)
Γ(m − γ) 0
can be obtained for example by applying the Cauchy integral Theorem (see Samko
et al. [24] page 138).
2.2. Riesz fractional derivatives. By means of the Weyl fractional derivatives
we arrive at the Riesz fractional derivative, for m − 1 < γ < m, m ∈ N,
Z x
Z ∞
∂m
∂γ
u(y, t) dy
(−1)m u(y, t) dy
∂xm
u(x,
t)
=
−
+
γ−m+1
∂|x|γ
2 cos πγ
(y − x)γ−m+1
−∞ (x − y)
x
2 Γ(m − γ)
+ γ
− γ
1
∂
∂
=−
+
u(x, t)
(2.8)
πγ
γ
2 cos 2
∂x
∂xγ
6
ENZO ORSINGHER AND BRUNO TOALDO
In view of Theorem 2.1 we have that, for γ > 0, γ ∈
/ N,
Z ∞
h
γ
iπγ
∂
1
γ − iπγ
sign(ξ)
2
dx eiξx
u(x, t) = −
+ |ξ|γ e 2
πγ |ξ| e
γ
∂|x|
2
cos
−∞
2
= − |ξ|γ u
b(ξ, t).
sign(ξ)
Remark 2.2. The general fractional higher-order heat equation
∂
∂γ
uγ (x, t),
x ∈ R, t > 0,
uγ (x, t) =
∂t
∂|x|γ
i
u
b(ξ, t)
(2.9)
(2.10)
has solution whose Fourier transform reads
γ
u
cγ (ξ, t) = e−t|ξ| .
(2.11)
For 0 < γ < 2, (2.11) corresponds to the characteristic function of the symmetric
stable processes (this is a classical result due to M. Riesz himself ).
3. From pseudo random walks to fractional pseudoprocesses
We consider in this section continuous-time pseudo random walks with steps
which are pseudo random variables, that is measurable functions endowed with
signed measures, and with total mass equal to one (see [12]). In order to obtain in
the limit pseudoprocesses whose signed law satisfies higher-order fractional equations we must construct sums of the form
N (tγ −β(2k+1) )
X
γ,β(2k+1)
j Uj2k+1 (1) Qj
,
β ∈ (0, 1), k ∈ N, γ > 0,
(3.1)
j=0
where
(
j =
1,
−1,
with probability p,
with probability q,
p + q = 1.
(3.2)
γ,β(2k+1)
The r.v.’s Qj
have probability distributions, for β ∈ (0, 1), k ∈ N,
( β(2k+1)
n
o
γ
,
for w > γ
γ,β(2k+1)
w
Pr Qj
>w =
1,
for w < γ.
(3.3)
The Poisson process N (t), t > 0, appearing in (3.1) is homogeneous with rate
1
λ=
,
β ∈ (0, 1).
(3.4)
Γ(1 − β)
The pseudo random variables (see Lachal [12]) Uj2k+1 (1) have law with Fourier
transform
Z ∞
2k+1
dx eiξx u2k+1 (x, 1) = e−iξ
(3.5)
−∞
and the function u2k+1 (x, t), x ∈ R, t > 0, is the fundamental solution to the
odd-order heat-type equation, for k ∈ N,
(
∂
k ∂ 2k+1
x ∈ R, t > 0,
∂t u2k+1 (x, t) = (−1) ∂x2k+1 u2k+1 (x, t),
(3.6)
u2k+1 (x, 0) = δ(x).
There is a vast literature devoted to odd-order heat-type equations of the form (3.6),
to the behaviour of their solutions, and to the related pseudoprocesses (Beghin et
al. [1]; Lachal [11]; Orsingher [20]; Orsingher and D’Ovidio [21]).
SPACE-FRACTIONAL HIGHER-ORDER HEAT-TYPE EQUATIONS
7
The r.v.’s and pseudo r.v.’s appearing in (3.1) are independent and also independent from each other. We say that two pseudo r.v.’s (or pseudoprocesses) with
signed density u1m , u2m , are independent if the Fourier transform F of the convolution u1m ∗ u2m factorizes, that is
F u1m ∗ u2m (ξ) = F u1m (ξ) F u2m (ξ).
(3.7)
We are now able to state the first theorem of this section.
Theorem 3.1. The following limit in distribution holds true
N (tγ −β(2k+1) )
lim
γ→0
X
γ,β(2k+1)
j Uj2j+1 (1) Qj
= U12k+1 H1β (pt) − U22k+1 H2β (qt) ,
law
j=0
(3.8)
H1β
and H2β are independent
while U12k+1 and U22k+1 are
where
positively-skewed stable processes of order 0 <
β < 1
independent pseudoprocesses of order 2k + 1.
γ,β(2k+1)
All the random variables N (t), t > 0, j , Qj
are independent and also
2k+1
independent from the pseudo random variables Uj
(1). The Fourier transform
of the limiting pseudoprocess reads
β
β
2k+1
2k+1
βπ
β(2k+1)
(cos βπ
2 −i sign(ξ) (p−q) sin 2 ) .
(3.9)
EeiξU1 (H1 (pt))−U2 (H2 (qt)) = e−t|ξ|
Proof. In view of the independence of the r.v’s and pseudo random variables appearing in (3.8) we have that
PN (tγ −β(2k+1) )
γ,β(2k+1)
j Uj2k+1 (1) Qj
Eeiξ j=0
N (tγ −β(2k+1) ) 2k+1
(1)Qγ,β(2k+1)
= E E eiξU
2k+1
λt
(1)Qγ,β(2k+1)
= exp − β(2k+1) 1 − EeiξU
γ
λt
iξU 2k+1 (1)Qγ,β(2k+1)
−iξU 2k+1 (1)Qγ,β(2k+1)
− qEe
= exp − β(2k+1) 1 − pEe
γ
λt
iξU 2k+1 (1)Qγ,β(2k+1)
−iξU 2k+1 (1)Qγ,β(2k+1)
= exp − β(2k+1) p + q − pEe
− qEe
γ
λt
iξU 2k+1 (1)Qγ,β(2k+1)
−iξU 2k+1 (1)Qγ,β(2k+1)
= exp − β(2k+1) p 1 − Ee
+ q 1 − Ee
.
γ
(3.10)
We observe that
2k+1
2k+1
(1)Qγ,β(2k+1)
(1)Qγ,β(2k+1)
p 1 − EeiξU
+ q 1 − Ee−iξU
Z ∞
γ β(2k+1) β(2k + 1) iξ2k+1 w2k+1
e
=p
dw 1 −
wβ(2k+1)+1
γ
Z ∞
γ β(2k+1) β(2k + 1) −iξ2k+1 w2k+1
+q
dw 1 −
e
wβ(2k+1)+1
γ
and therefore
PN (tγ −β(2k+1) )
γ,β(2k+1)
j Uj2k+1 (1) Qj
Eeiξ j=0
=
(3.11)
8
ENZO ORSINGHER AND BRUNO TOALDO
∞
γ β(2k+1) β(2k + 1) iξ2k+1 w2k+1
dw
1
−
e
γ β(2k+1)
wβ(2k+1)+1
γ
Z ∞
γ β(2k+1) β(2k + 1) −iξ2k+1 w2k+1
e
+q
dw 1 −
wβ(2k+1)+1
γ
"
(
Z ∞
2k+1
dw γ β(2k+1) ei(ξw)
−λt
i(ξγ)2k+1
2k+1
p
1
−
e
−
p
iξ
(2k
+
1)
= exp
γ β(2k+1)
wβ(2k+1)−2k
γ
#)
Z ∞
2k+1
dw γ β(2k+1) e−i(ξw)
−i(ξγ)2k+1
2k+1
+q 1 − e
+ q iξ
(2k + 1)
.
wβ(2k+1)−2k
γ
(3.12)
= exp −
Z
p
λt
By taking the limit we get that
lim Eeiξ
γ→0
"
PN (tγ −β(2k+1) )
j=0
γ,β(2k+1)
j Uj2k+1 (1) Qj
= exp −λt(2k + 1) −piξ
=e
2k+1
Z
∞
0
h
i
β
−λtΓ(1−β) p(−iξ 2k+1 ) +q(iξ 2k+1 )β
By setting λ =
lim Eeiξ
1
Γ(1−β)
=
2k+1
dw ei(ξw)
+ qiξ 2k+1
wβ(2k+1)−2k
Z
∞
0
2k+1
dw e−i(ξw)
wβ(2k+1)−2k
.
!#
(3.13)
we obtain
PN (tγ −β(2k+1) )
j=0
γ,β(2k+1)
j Uj2k+1 (1) Qj
γ→0
=e
iπβ
−t p|ξ|β(2k+1) e− 2
= e−t|ξ|
β(2k+1)
sign(ξ)
+q|ξ|β(2k+1) e
iπβ
sign(ξ)
2
πβ
(cos πβ
2 −i sign(ξ) (p−q) sin 2 )
(3.14)
Now we consider the Fourier transform of the law of the pseudoprocess
V (2k+1)β (t) = U12k+1 H1β (pt) − U22k+1 H2β (qt)
(3.15)
which reads
EeiξV
(2k+1)β
(t)
β
β
2k+1
2k+1
= EeiξU1 (H1 (pt)) Ee−iξU2 (H2 (qt))
Z ∞
Z ∞
2k+1
2k+1
s 1
z 2
=
ds eiξ
hβ (s, pt)
dz e−iξ
hβ (z, qt)
0
0
β
=e
−t p(−iξ 2k+1 )
=e
e
−t q (iξ 2k+1 )
β(2k+1) −
−t p|ξ|
β(2k+1)
= e−t|ξ|
e
β
iβπ
iβπ
sign(ξ)
2
+q|ξ|β(2k+1) e 2 sign(ξ)
πβ
(cos πβ
2 −i sign(ξ) (p−q) sin 2 ) ,
and coincides with (3.14).
Remark 3.2. If β =
Ee
iξU12k+1
(
H1β (pt)
1
2k+1
(3.16)
the Fourier transform (3.16) becomes
π
π
+itξ sin 2(2k+1)
) Ee−iξU22k+1 (H2β (qt)) = e−t|ξ| cos 2(2k+1)
(3.17)
which corresponds to the characteristic function of a Cauchy r.v. with position
β
π
parameter equal to t(p−q) sin 2(2k+1)
and scale parameter t cos 2(2k+1)
. This slightly
generalizes result 1.4 of Orsingher and D’Ovidio [21].
SPACE-FRACTIONAL HIGHER-ORDER HEAT-TYPE EQUATIONS
9
For even-order pseudoprocesses we have the following limit in distribution.
Theorem 3.3. If U 2k (t), t > 0, is an even-order pseudoprocess and N (t), t > 0,
is a homogeneous Poisson process, independent from U 2k (t), t > 0, we have that
N (tγ −2kβ )
lim
γ→0
X
Uj2k (1)Qγ,2kβ
j
law
= U 2k H β (t) ,
t > 0,
(3.18)
j=0
are i.i.d. random
where H β is a stable subordinator of order β ∈ (0, 1) and Qγ,2kβ
j
variables with distribution
(
o
n
1,
w < γ,
γ,2kβ
>w =
(3.19)
Pr Qj
γ 2kβ
,
w
> γ.
w
The pseudoprocess U 2k (t) is governed by the equation
∂
∂ 2k
u2k (x, t) = (−1)k+1 2k u2k (x, t),
x ∈ R.
∂t
∂x
Proof. We start by evaluating the Fourier transform
PN (tγ −2kβ ) 2k
Uj (1)Qjγ,2kβ
Eeiξ j=0
−2kβ )
2k
γ,2kβ N (tγ
= E E eiξU (1)Q
2k
γ,2kβ
λt = exp − 2kβ 1 − EeiξU (1)Q
γ
Z ∞ 2kβγ 2kβ
λt
−|ξ|2k y 2k
= exp − 2kβ
dy 1 − e
γ
y 2kβ+1
γ
Z ∞
λt
−|ξ|2k γ 2k
−|ξ|2k y 2k 2k−1−2kβ
2kβ
= exp − 2kβ 1 − e
+
dy e
y
2kγ
γ
γ
(3.20)
(3.21)
By taking the limit we have that
R ∞ −|ξ|2k y2k 2k(1−β)−1
PN (tγ −2kβ ) 2k
2k
Uj (1)Qγ,2kβ
y
dy
j
= e−λt|ξ| 2k 0 e
lim eiξ j=0
γ→0
2kβ
= e−λt|ξ|
2kβ
= e−λt|ξ|
R∞
0
e−w w−β dw
Γ(1−β)
(3.22)
which coincides with
Ee
iξU 2k (H β (t))
Z
=
∞
e−sξ
2k
2kβ
Pr H β (t) ∈ ds = e−t|ξ|
(3.23)
0
since λ =
1
Γ(1−β) .
Remark 3.4. For β = k1 the composition U 2k H β (t) has Gaussian distribution.
1
1
For β = 2k
we have instead the Cauchy distribution and for β = 4k
we extract
the inverse Gaussian corresponding to the distribution of the first passage time of
1
a Brownian motion. The case β = 6k
yields the stable law with distribution
t
t
√
f 13 (x) = √
Ai
(3.24)
3
x 3 3x
3x
where Ai denotes the Airy function (see [21]).
10
ENZO ORSINGHER AND BRUNO TOALDO
In order to arrive at asymmetric higher-order fractional pseudoprocesses we construct pseudo random walks by adapting the Feller approach (used for asymmetric
stable laws) to our context. This means that we combine independent pseudo random walks with suitable trigonometric weights as in (1.4).
γ,(2k+1)β
γ,(2k+1)β
Theorem 3.5. Let Xj
and Yj
n
o
Pr X γ,(2k+1)β > w =
(
be i.i.d. r.v.’s with distribution
γ (2k+1)β
w
,
w>γ
w < γ,
1,
(3.25)
and let U 2k+1 (t), t > 0, be a pseudoprocess of odd-order 2k + 1, k ∈ N. For
0 < β < 1 and −β < θ < β we have that

)
1 N (tγ −(2k+1)β
π
X
 sin 2 (β − θ) (2k+1)β
γ,(2k+1)β 2k+1
Xj
Uj
(1)
lim 
γ→0
sin πβ
j=0
−
sin
π
2 (β
+ θ)
sin πβ

N (tγ −(2k+1)β )
1
(2k+1)β
γ,(2k+1)β
X
Yj
 law
Uj2k+1 (1) =
Z β(2k+1),θ
j=0
(3.26)
where
EeiξZ
β(2k+1),θ
= e−t|ξ|
(2k+1)β
e
iπθ
2
(3.27)
Proof. The Fourier transform of (3.26) is written as
iξ
Ee
sin π (β−θ)
2
sin πβ
× Ee
−iξ
1
(2k+1)β
PN (tγ −(2k+1)β )
j=0
sin π (β+θ)
2
sin πβ
1
(2k+1)β
γ,(2k+1)β
Xj
PN (tγ −(2k+1)β )
j=0
Uj2k+1 (1)
γ,(2k+1)β
Yj
Uj2k+1 (1)
(3.28)
where the first member is given by
iξ
Ee
sin π (β−θ)
2
sin πβ
1
(2k+1)β
PN (tγ −(2k+1)β )
j=0



γ,(2k+1)β
Xj
sin π (β−θ)
2
sin πβ
1
(2k+1)β
Uj2k+1 (1)
2k+1
=
(2k+1)β




iξ
U
(1)X
λt
1 − Ee
−
 γ (2k+1)β




1
Z ∞
β 2k+1
sin π (β−θ)


2k+1
(2k+1)β
2
iξ
y
λt
sin πβ
 γ
1 − e
= exp − (2k+1)β
(2k + 1)β
(2k+1)β+1
 γ

y
γ
Z ∞
sin π2 (β − θ)
γ→0
e−w w−β dw
−→ exp −λt i−β ξ (2k+1)β
sin πβ
0
= exp
=e
−λt|ξ|(2k+1)β e−
iπβ sign(ξ)
2
sin π (β−θ)
2
sin πβ
Γ(1−β)
.
(3.29)
The second member of (3.26) becomes, by performing a similar calculation,
−iξ
Ee
sin π (β+θ)
2
sin πβ
1
(2k+1)β
PN (tγ −(2k+1)β )
j=0
γ,(2k+1)β
Yj
Uj2k+1 (1)
SPACE-FRACTIONAL HIGHER-ORDER HEAT-TYPE EQUATIONS
(2k+1)β
e
γ→0 −λt|ξ|
iπβ sign(ξ)
2
−→ e
and thus for λ =
1
Γ(1−β)
Ee
×Ee
−iξ
(2k+1)β −
e
γ→0 −t|ξ|
1
(2k+1)β
1
(2k+1)β
Γ(1−β)
.
(3.30)
iπβ sign(ξ)
2
γ,(2k+1)β
Xj
PN (tγ −(2k+1)β )
j=0
−→ e
= e−t|ξ|
PN (tγ −(2k+1)β )
j=0
sin π (β+θ)
2
sin πβ
(2k+1)β
we obtain that
sin π (β−θ)
2
sin πβ
iξ
sin π (β+θ)
2
sin πβ
11
sin π (β−θ)
2
sin πβ
Uj2k+1 (1)
γ,(2k+1)β
Yj
−t|ξ|(2k+1)β e
Uj2k+1 (1)
iπβ sign(ξ)
2
e
sin π (β+θ)
2
sin πβ
iπθ
e 2 sign(ξ)
(3.31)
By considering symmetric pseudo random walks with the Feller construction we
arrive in the next theorem at symmetric pseudoprocesses with time scale equal to
cos
sin
πβ
2
πβ
2
, 0 < β < 1, −β < θ < β.
Theorem 3.6. Let Xjγ,2βk and Yjγ,2βk be i.i.d. r.v.’s with distribution
( 2βk
γ
γ,2βk
,
w>γ
w
Pr X
>w =
1,
w < γ,
(3.32)
and let U 2k (t), t > 0, be a pseudoprocess of order 2k, k ∈ N. If N (t) is a homoge1
neous Poisson process, with parameter λ = Γ(1−β)
, independent from Xjγ,2βk and
Yjγ,2βk we have that

1
π
 sin 2 (β − θ) 2βk
lim 
γ→0
sin πβ
+
sin
π
2 (β
+ θ)
sin πβ
N (tγ −2βk )
1
2βk
X
Xjγ,2βk Uj2k (1)
j=0
N (tγ −2βk )

X
 law
Yjγ,2βk Uj2k (1) = Z 2kβ,θ ,
t > 0,
j=0
(3.33)
for 0 < β < 1 and −β < θ < β and
Ee
iξZ 2kβ,θ
cos π θ
2
πβ
2
−t|ξ|2kβ cos
= e
(3.34)
Proof. The Fourier transform of (3.33) is written as
iξ
Ee
sin π (β−θ)
2
sin πβ
1
2βk
PN (tγ −2βk )
j=0
Xj Uj2k (1)
iξ
Ee
sin π (β+θ)
2
sin πβ
1
2βk
PN (tγ −2βk )
j=0
Yj Uj2k (1)
(3.35)
where the first member is given by
iξ
sin π (β−θ)
2
sin πβ
1
2βk
PN (tγ −2βk )
j=0
Xjγ,2βk Uj2k (1)
=
sin π (β−θ) 2k
γ,2βk
λt
2
= exp − 2kβ 1 − Eeiξ sin πβ U (1)X
γ
Ee
12
ENZO ORSINGHER AND BRUNO TOALDO



1 2k
π (β−θ)
2kβ 

2k
2
 λt Z ∞ −ξ sinsin

2kβ
y
πβ
γ


e
(2kβ) 2kβ+1 dy 
= exp − 2kβ 


y
γ
 γ



1
β1
Z ∞
β 2k
sin π (β−θ)


π
2k
2
−|ξ|
y
sin
(β
−
θ)
γ→0
sin πβ
2
2k
e
y 2k−1−2kβ dy
−→ exp −λt|ξ|2k


sin πβ
0
=e
sin π (β−θ)
2
−λt|ξ|2kβ Γ(1−β)
sin πβ
(3.36)
and by similar calculations the second member becomes
iξ
Ee
sin π (β+θ)
2
sin πβ
1
2βk
PN (tγ −2βk )
j=0
Yjγ,2βk Uj2k (1) γ→0
−→ e
sin π (β+θ)
2
−λt|ξ|2kβ Γ(1−β)
sin πβ
. (3.37)
Thus we have that
iξ
Ee
sin π (β−θ)
2
sin πβ
1
2βk
PN (tγ −2βk )
j=0
Xjγ,2βk Uj2k (1)
iξ
Ee
sin π (β+θ)
2
sin πβ
1
2βk
PN (tγ −2βk )
j=0
Yjγ,2βk Uj2k (1)
sin π (β+θ)
sin π (β−θ)
2kβ
2
2
+ sin
Γ(1−β)
γ→0 −λt|ξ|
sin πβ
πβ
−→ e
=e
cos π θ
2
πβ
2
−t|ξ|2βk cos
(3.38)
4. Governing equations
In the previous section we obtained fractional pseudoprocesses as limit of suitable
pseudo random walks. In this section we will show that the limiting fractional pseudoprocesses obtained before have signed density satisfying space-fractional heattype equations of higher-order with Riesz or Feller fractional derivatives. The order of fractionality of the governing equations is a positive real number and this is
the major difference with respect to the pseudoprocesses considered so far in the
literature.
We start by examining space fractional higher-order equations of order 2kβ,
β ∈ (0, 1), k ∈ N, which interpolate equations of the form (1.1).
Theorem 4.1. The solution to the initial-value problem
(
β
∂ β
∂ 2kβ
x ∈ R, t > 0, k ∈ N, β ∈ (0, 1)
∂t v2k (x, t) = ∂|x|2kβ v2k (x, t),
β
v2k
(x, 0) = δ(x)
(4.1)
can be written as
1
1
E sin xG2k
πx
H β (t)
1
1
=
E sin xG2kβ
πx
t
β
v2k
(x, t) =
and coincides with the law of the pseudoprocess
V 2kβ (t) = U 2k H β (t) ,
t > 0,
(4.2)
(4.3)
SPACE-FRACTIONAL HIGHER-ORDER HEAT-TYPE EQUATIONS
13
where U 2k is related to equation (1.1) for m = 2k and H β is a stable subordinator
independent from U 2k . Gγ (t) is a gamma r.v. with density
xγ−1 − xγ
x > 0, t > 0, γ > 0.
e t ,
t
Proof. The Fourier transform of (4.1) leads to the Cauchy problem
(
β
∂ β
b2k (ξ, t) = −|ξ|2kβ vb2k
(ξ, t)
∂t v
β
vb2k
(ξ, 0) = 1,
g γ (x, t) = γ
(4.4)
(4.5)
whose unique solution reads
EeiξV
2kβ
(t)
Z
dx eiξx
=
ZR∞
=
∞
Z
ds u2k (x, s) hβ (s, t)
0
2k
2kβ
ds e−sξ hβ (s, t) = e−t|ξ|
.
(4.6)
0
2k
In (4.6) u2k is the density of U and hβ (x, t) is the probability density of the
subordinator H β . Now we show that the Fourier transform of (4.2) coincides with
(4.6). We have that
Z
1
1
β
E sin xG2k
vb2k
(ξ, t) =
dxeiξx
πx
H β (t)
R
Z ∞ Z ∞
Z
1
sin xy
Pr G2k
=
dx eiξx
∈ dy Pr H β (t) ∈ ds
πx
s
R
0
0
Z
Z ∞Z ∞ 1
sin xy
=
Pr G2k
∈ dy Pr H β (t) ∈ ds
.
dx eiξx
s
πx
0
0
R
(4.7)
By considering that the Heaviside function
(
1,
Hα (z) =
0,
z > α,
z<α
(4.8)
can be represented as
1
Hα (z) =
2π
Z
iwz e
dw e
R
−iαw
iw
1
= −
2π
Z
dw e−iwz
R
eiαw
,
iw
(4.9)
we obtain that formula (4.7) becomes
β
vb2k
(ξ, t) =
Z ∞Z ∞ 1
=
Pr G2k
∈ dy Pr H β (t) ∈ ds [H−y (ξ) − Hy (ξ)]
s
0
0
Z ∞Z ∞ 1
=
Pr G2k
∈ dy Pr H β (t) ∈ ds I[−ξ,+∞] (y) − I[−∞,ξ] (y)
s
Z0 ∞ Z0 ∞
2k
=
dy ds 2ksy 2k−1 e−sy
I[0,∞] (y) I[−ξ,+∞] (y) − I[−∞,ξ] (y) hβ (s, t).
0
0
(4.10)
For ξ > 0 (4.10) becomes
β
vb2k
(ξ, t)
Z
∞
"
Z
ds 1 −
=
0
#
ξ
dy 2ksy
0
2k−1 −y 2k s
e
hβ (s, t)
14
ENZO ORSINGHER AND BRUNO TOALDO
∞
Z
ds e−ξ
=
2k
s
2kβ
hβ (s, t) = e−t|ξ|
,
(4.11)
0
and for ξ < 0 (4.10) is
β
vb2k
(ξ, t)
Z
∞
=
Z
∞
2ksy
ds
=
e
hβ (s, t)
−ξ
0
Z
2k−1 −y 2k s
∞
2k
ds e−|ξ|
s
2kβ
hβ (s, t) = e−t|ξ|
.
(4.12)
0
Since
Pr G2k
1
H β (t)
Z ∞
2k
se−sy hβ (s, t) ds
∈ dy /dy = 2ky 2k−1
Z ∞0
2k
∂
=−
e−sy hβ (s, t) ds
∂y 0
∂ −y2kβ t
e
=−
∂y
1
2kβ
= Pr G
∈ dy /dy
t
(4.13)
the second form of the solution (4.2) follows immediately.
For k ≥ 1, β ∈ 0, k1 the solutions (4.2) are densities of symmetric random
variables, while for β > k1 the functions (4.2) are sign-varying. Clearly for β = 1
we obtain the solution of even-order heat-type equations discussed in [21]. As far
as space-fractional higher-order heat-type equations we have the result of the next
theorem where the governing fractional operator R is obtained as a suitable combination of Weyl derivatives. The operator R governing the fractional pseudoprocesses appearing in Theorem 3.1 is explicitely written for {p, q ∈ [0, 1] : p + q = 1},
{β ∈ (0, 1), k ∈ N : m − 1 < β(2k + 1) < m, m ∈ N} as
β
R v2k+1
(x, t) =
+ β(2k+1)
− β(2k+1)
1
β
iπβk ∂
−iπβk ∂
=−
pe
+qe
v2k+1
(x, t)
β(2k+1)
β(2k+1)
∂x
∂x
cos πβ
2
"
Z x
β
v2k+1
(y, t)
1
∂m
iπβk
e
p
=−
dy
πβ
m
(2k+1)β−m+1
cos 2 Γ(m − (2k + 1)β)) ∂x
−∞ (x − y)
#
Z ∞
β
v
(y,
t)
2k+1
+q e−iπβk (−1)m
dy ,
(4.14)
(y − x)(2k+1)β−m+1
x
where the left and right Weyl fractional derivatives appear.
Theorem 4.2. The solution to the problem
(
β
∂ β
x ∈ R, t > 0, β ∈ (0, 1), k ∈ N,
∂t v2k+1 (x, t) = R v2k+1 (x, t),
β
v2k+1 (x, 0) = δ(x),
is given by the signed law of the pseudoprocess
!!
pt
β
2k+1
β(2k+1)
V̄
(t) = U1
− U22k+1
H1
cos βπ
2
H2β
qt
cos βπ
2
(4.15)
!!
,
(4.16)
SPACE-FRACTIONAL HIGHER-ORDER HEAT-TYPE EQUATIONS
15
where U12k+1 , U22k+1 are independent odd-order pseudoprocesses and H1β , H2β , are
independent stable subordinators.
Proof. The Fourier transform of (4.14) is written as
h
i
β
F R v2k+1
(x, t) (ξ) =
"
#
+ β(2k+1)
− β(2k+1)
1
β
iπβk ∂
−iπβk ∂
=F −
+qe
v
(x, t) (ξ)
pe
∂xβ(2k+1)
∂xβ(2k+1) 2k+1
cos πβ
2
i
1 h iπβk
β
β(2k+1)
−iπβk
β(2k+1)
p
e
vb2k+1
(ξ, t)
(−iξ)
+
q
e
(iξ)
=−
βπ
cos 2
h
i
iπβ
1
β
β(2k+1)
− iπβ
sign(ξ)
sign(ξ)
2
2
pe
=−
|ξ|
+
qe
vb2k+1
(ξ, t)
cos βπ
2
πβ
β
= − |ξ|β(2k+1) 1 − i sign(ξ) (p − q) tan
vb2k+1
(ξ, t)
(4.17)
2
and therefore we have that
β(2k+1)
(1−i(p−q) sign(ξ) tan βπ
2 )
(4.18)
vbβ (ξ, t) = e−t|ξ|
2k+1
In view of (3.16) we get
Ee
iξ V̄ (2k+1)β (t)
iξV (2k+1)β
= Ee
β(2k+1)
= e−t|ξ|
t
βπ
cos
2
(1−i sign(ξ) (p−q) tan πβ
2 )
(4.19)
and this confirms that the solution to (4.15) is given by the law of the pseudoprocess
(4.16).
kβ
kβ
±iπkβ
1
iπkβ
Remark 4.3. Since ecos βπ = cos β(2k+1)
= eiπ
= e−iπ
=
π (because e
2
2
e−iπkβ ) the operator (4.14) takes the form of the Riesz fractional derivative of order
β(2k + 1) when p = q = 12 .
We now pass to the derivation of the governing equation of the fractional pseudoprocesses studied in Theorem 3.5. We first recall the definition of the Feller
space-fractional derivative which is
sin π2 (β − θ) + ∂ β
sin π2 (β + θ) − ∂ β
F β,θ
+
D u(x) = −
u(x).
(4.20)
sin(πβ) ∂xβ
sin(πβ) ∂xβ
We recall that
iπθ
F F Dβ,θ u(x) (ξ) = −|ξ|β e 2 sign(ξ) u
b(ξ),
(4.21)
as can be shown by means of the following calculation
Z
sin π2 (β − θ)
sin π2 (β + θ)
iξx F β,θ
β
β
dx e
D u(x) = −
(−iξ) +
(iξ) u
b(ξ)
sin(πβ)
sin(πβ)
R
iπ
iπ
iπ
|ξ|β h iπ β − iπ θ
=−
e 2 e 2 − e− 2 β e 2 θ e− 2 β sign(ξ) +
2i sin πβ
iπ iπ
iπ
i
iπ
iπ
+ e 2 β e 2 θ − e− 2 β e− 2 θ e 2 β sign(ξ) u
b(ξ)
(
iπθ
−ξ β e 2 u
b(ξ),
ξ > 0,
=
β − iπθ
2
−(−ξ) e
u
b(ξ),
ξ<0
16
ENZO ORSINGHER AND BRUNO TOALDO
= − |ξ|β e
iπθ
2
sign(ξ)
u
b(ξ)
(4.22)
where we used the results of Theorem 2.1. The explicit form of the Fourier transform
of the solution to
∂
u(x, t) = F Dβ,θ u(x, t),
u(x, 0) = δ(x),
x ∈ R, t > 0,
(4.23)
∂t
is written as
β
u
b(ξ, t) = e−|ξ|
te
iπθ sign(ξ)
2
(4.24)
and for β ∈ (0, 2], 4m − 1 < θ < 4m + 1, m ∈ N, represents the characteristic
function of a stable r.v.. The last condition on θ is due to the fact that
θπ
∈ (0, 1].
(4.25)
|b
u(ξ, t)| ≤ 1 if and only if cos
2
The condition (4.25) must be assumed also for β > 2 where (4.24) however fails
to be the characteristic function of a genuine r.v.. For θ = β < 1 (4.24) becomes
totally negatively skewed. By interchanging sin(β − θ) π2 with sin(β + θ) π2 we obtain
instead
u
b(ξ, t) = e−|ξ|
β
te−
iπ θ sign(ξ)
2
(4.26)
which is totally positively skewed for θ = β < 1.
We are now ready to prove the following Theorem.
Theorem 4.4. Let Z β(2k+1),θ (t), t > 0, be the limiting fractional pseudoprocess
studied in Theorem 3.5. The signed density of Z β(2k+1),θ (t) is the solution to
(
∂ β(2k+1),θ
(x, t) = F Dβ(2k+1),θ z β(2k+1),θ (x, t)
∂t z
(4.27)
β(2k+1),θ
z
(x, 0) = δ(x)
and coincide with the signed distribution of the composition for β ∈ (0, 1), −β <
θ < β,
sin π2 (β − θ)
sin π2 (β + θ)
t
− U22k+1 H2β
t
,
Zβ(2k+1),θ (t) = U12k+1 H1β
sin πβ
sin πβ
(4.28)
where Hjβ , j = 1, 2 are independent stable r.v.’s and the independent pseudoprocesses Uj2k+1 , j = 1, 2, are related to the odd-order heat-type equation
∂ 2k+1
∂
u2k+1 (x, t) = (−1)k 2k+1 u2k+1 (x, t).
∂t
∂x
The positivity of the time scales in (4.28) implies that −β < θ < β.
(4.29)
Proof. By profiting from the result (4.21) we note that the Fourier transform of
(4.27) is written as
(
iπ
∂ β(2k+1),θ
(ξ, t) = −|ξ|β(2k+1) e 2 θ sign(ξ) zbβ(2k+1),θ (ξ, t)
b
∂t z
(4.30)
zbβ(2k+1),θ (ξ, 0) = 1.
which is satisfied by the Fourier transform
β(2k+1)
zbβ(2k+1),θ (ξ, t) = e−t|ξ|
e
iπ θ sign(ξ)
2
.
(4.31)
SPACE-FRACTIONAL HIGHER-ORDER HEAT-TYPE EQUATIONS
17
We now prove that the Fourier transform of (4.28) coincides with (4.31). In view
of the independence of the r.v.’s and pseudo r.v.’s involved we write that
iξZβ(2k+1),θ (t)
Ee
Z
= Ee
sin π (β+θ)
sin π (β−θ)
2
2
iξ U12k+1 H1β
t
−U22k+1 H2β
t
sin πβ
sin πβ
∞
sin π2 (β + θ)
=
dx e
s,
t
sin πβ
0
R
Z
Z ∞
sin π2 (β − θ)
2
2
−iξx
×
ds u2k+1 (x, s)hβ s,
dx e
t
sin πβ
0
R
Z ∞
Z
∞
sin π2 (β + θ)
sin π2 (β + θ)
−iξ 2k+1 s 1
iξ 2k+1 s 2
e
hβ s,
e
hβ s,
=
t ds
t ds
sin πβ
sin πβ
0
0
= e−t
=e
Z
iξx
sin π (β+θ)
2
sin πβ
ds u12k+1 (x, s)h1β
β
(iξ2k+1 ) e−t
sin π (β−θ)
2
sin πβ
−
t|ξ|β(2k+1)
sin πβ
h
sin
−
t|ξ|β(2k+1)
2i sin πβ
h iπ
iπ
iπ
iπ iπ
e 2 β e 2 θ −e− 2 β e− 2 θ e 2 β
=e
β(2k+1)
= e−t|ξ|
e
β
(−iξ2k+1 )
i
iπ β sign(ξ)
− iπ β sign(ξ)
π
2
2
+sin π
2 (β+θ)e
2 (β−θ)e
sign(ξ)
iπ
iπ
iπ
iπ iπ
+ e 2 β e− 2 θ −e− 2 β e 2 θ e− 2 β
iπθ sign(ξ)
2
sign(ξ)
i
(4.32)
which coincides with (4.31).
5. Some remarks
We give various forms for the density v γ (x, t) of symmetric pseudoprocesses of
arbitrary order γ > 0. For integer values of γ = 2n or γ = 2n + 1 the analysis of
the structure of these densities is presented in Orsingher and D’Ovidio [21]. We
give here an analytical representation of v γ (x, t) for non-integer values of γ, which
is an alternative to (4.2), as a power series and in integral form (involving the
Mittag-Leffler functions). Furthermore, in Figure 1 we give some curves for special
values of γ. We also give the distribution of the sojourn time of compositions of
pseudoprocesses with stable subordinators (totally positively skewed stable r.v.’s).
Proposition 5.1. For γ > 1 the inverse of the Fourier transform
γ
vbγ (ξ, t) = e−t|ξ|
(5.1)
can also be written as
Z
γ
1 ∞
cos(ξx) e−tξ dξ
v γ (x, t) =
π 0
2k+1
∞
k 2k Γ
X
γ
1
(−1) x
=
2k+1
πγ
(2k)!
t γ
k=0
1
2k+1
∞
,
(2k
+
1)
1
−
k 2k B
X
γ
γ
1
(−1) x
=
2k+1
1
πγ
γ
t
Γ (2k + 1) 1 −
k=0
γ
Z 1
∞
2k+1
1
1 X (−1)k x2k
1
=
dy y γ −1 (1 − y)(2k+1)(1− γ )−1
2k+1
1
πγ
0
t γ
Γ (2k + 1) 1 − γ
k=0
18
ENZO ORSINGHER AND BRUNO TOALDO
1
2k
1
∞ (−1)k xy γ (1 − y)1− γ
X
1
1
1
y γ −1 (1 − y)− γ
=
dy
2k+1
1
πγ 0
γ
Γ (2k + 1) 1 − γ
k=0 t
1 Z
2 1 1
1
1
t− γ 1
−1
dy E2(1− 1 ),1− 1 − xy γ (1 − y)1− γ t− γ y γ −1 (1 − y) γ
=
γ
γ
πγ 0


!2
1
1
− γ1 Z ∞
γ
γ
w
1
w=y/(1−y) t
− γ1  w
2

=
dw E2(1− 1 ),1− 1 −x
t
(5.2)
γ
γ
πγ 0
1+w
1+ww
Z
1
and for γ < 2 coincides with the characteristic function of symmetric stable processes.
Formula (5.2) is an alternative to the probabilistic representation (4.2) for γ =
2kβ. For 1 < γ < 2 it represents the density of a symmetric stable r.v..
Remark 5.2. We note that
1
1
t− γ
Γ 1+
v (0, t) =
π
γ
γ
(5.3)
as can be inferred from (5.2). In the neighbourhood of x = 0 the density v γ (x, t)
can be written as


3
2 Γ γ
1
x
1 1
γ

−
Γ
v (x, t) ≈
πγ t γ1
γ
2 t γ3
Cγ
(5.4)
= v γ (0, t) 1 − x2 2
2t γ
where
3 1
Γ γ1 + 31 Γ γ1 + 32 3 γ − 2
Cγ =
.
(5.5)
2π
In the above calculation the triplication formula of the Gamma function (see Lebedev
[14] page 14) has been applied
1
2
2π
Γ(z)Γ z +
(5.6)
Γ z+
= 3z− 1 Γ(3z).
3
3
2
3
Remark 5.3. For even-order pseudoprocesses U 2k (t), t > 0, the distribution of the
sojourn time
Z t
2k
Γt U
=
I[0,∞) U 2k (s) ds
(5.7)
0
follows the arcsine law for all n ≥ 1 (see Krylov [10]). Therefore the distribution
of the sojourn time of U 2k H β (t) , t > 0, β ∈ (0, 1), reads
Z ∞
Pr Γt U 2k H β ∈ dx =
Pr Γs U 2k ∈ dx Pr H β (t) ∈ ds
0
Z
dx ∞
1
p
=
Pr H β (t) ∈ ds .
(5.8)
π x
x(s − x)
In the odd-order case the distribution of the sojourn time
Z t
2k+1
Γt U
=
I[0,∞) U 2k+1 (s) ds
0
(5.9)
SPACE-FRACTIONAL HIGHER-ORDER HEAT-TYPE EQUATIONS
19
is written as (see Lachal [11])
π
sin 2k+1
1
− 2k
Pr Γt U 2k+1 ∈ dx = dx
x− 2k+1 (t − x) 2k+1 I(0,t) (x)
(5.10)
π
and thus we get
Z ∞
π
dx sin 2k+1
1
p
Pr H β (t) ∈ ds .
Pr Γt U 2k+1 H β ∈ dx =
2k+1
π
x(s − x)2k
x
(5.11)
For β =
1
2
the integral (5.8) can be evaluated explicitly
Z
t2
o
1 n dx ∞
1
te− 2s
2k
p
√
∈ dx =
H2
Pr Γt U
ds
π x
x(s − x) 2πs3
Z x1
t2
dx t
e− 2 y
√
dy
= √
1 − xy
π 2πx 0
Z 1 − t2 w
dx t
e 2x
√
= √
dw
1−w
π 2πx3 0
k Z 1
∞ dx t X
t2
1
−1
= √
−
wk (1 − w) 2 dw
3
2x
k! 0
π 2πx k=0
2
dx t
t
E1, 23 −
,
x > 0, t > 0,
(5.12)
= √
3
2x
π 2x
where
Eν,µ (x) =
∞
X
j=0
xj
,
Γ(jν + µ)
ν, µ > 0,
is the Mittag-Leffler function.
Figure 1. The density v γ (x, t) for γ > 2 displays an oscillating
behaviour similar to that of the fundamental solution of even-order
heat equations.
(5.13)
20
ENZO ORSINGHER AND BRUNO TOALDO
Ackwnoledgements
The authors have benefited from fruitful discussions on the topics of this paper
with Dr. Mirko D’Ovidio.
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Department of Statistical Sciences, Sapienza University of Rome
E-mail address: [email protected]
E-mail address: [email protected]