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ROCKY MOUNTAIN JOURNAL
OF MATHEMATICS
Volume 53 (2023), No. 4, 1073–1085
DOI: 10.1216/rmj.2023.53.1073 © Rocky Mountain Mathematics Consortium
A NEW CHARACTERIZATION OF HOMOGENEOUS FUNCTIONS
AND APPLICATIONS
MONCEF ELGHRIBI
We present a new characterization of real homogeneous functions of a negative degree by a new counterpart
of Euler’s homogeneous function theorem using quantum calculus and replacing the classical derivative
operator by a (p,q)-derivative operator. As an application we study the solution of the Cauchy problem
associated to the
(p,q)
-analogue of the Euler operator. Using this solution, a probabilistic interpretation
is given in some details; more specifically, we prove that this solution is a stochastically continuous
markovian transition operator. Finally, we study it’s associated subordinated stochastically markovian
transition operator.
1. Introduction
The relationship between homogeneous functions and its partial derivatives (or Euler’s theorem) is present
in mathematics and physical sciences. In regard to thermodynamics, extensive variables are homogeneous
functions with degree 1 with respect to the number of moles of each component. In this context, Euler’s
theorem is applied in thermodynamics by taking Gibbs free energy; see [15]. In harmonic sense, the
spherical harmonic function and the solid harmonic are defined as homogeneous functions, such that
Euler’s theorem is used in proving that the Hamiltonian is equal to the total energy [9;15;21]. Also,
in microeconomics, they use homogeneous production functions, including the function of Cobb and
Douglas [6], developed in 1928, the degree of such homogeneous functions can be negative which was
interpreted as decreasing returns to scale. The concept of homogeneity in methods for enforcement finds
modeling physical phenomena and, in particular, for directly solving and inverse problems for potential
fields, where the gravity field of a mass point has the potential
V
[15]. The tests of homogeneity for
this potential can be implemented using equation of Euler’s theorem to study the homogeneity for the
gravity potential
V
with a negative degree of homogeneity
(λ = −
1
)
[21]. Also, Gelfand and Shilov [11]
studied the homogeneous distributions of negative integer degree
λ
on
R
. In mathematics, a homogeneous
function is a function
f
with multiplicative scaling behavior, i.e., if the argument is multiplied by a
factor
α
, then the result is multiplied by some power
λ
of this factor. Positive homogeneous functions are
characterized by Euler’s homogeneous function theorem consisting of
f
which is positive homogeneous
of degree λ∈Rif and only if n
X
j=1
xj
∂
∂xj
f(x)=λf(x).
2020 AMS Mathematics subject classification: primary 39A60, 47D09, 81S25; secondary 60J35, 60J45.
Keywords and phrases: quantum calculus, applications of difference equation, Cauchy problem, potential theory, Markovian
transition operator.
Received by the editors on June 12, 2022, and in revised form on August 3, 2022.
1074 MONCEF ELGHRIBI
The operator
Pn
j=1xj∂
∂xj
is called the Euler operator (see [2;9;11;19]). The quantum white noise
[17;18] counterpart of this theorem, as well as the associated Cauchy problem and potential, were studied
in [2].
In the last three decades, applications of
q
-calculus have been studied and investigated intensively
[1;8;10;12;14].
A recent relationship between homogeneous functions and the
q
-analogue Euler’s theorem is given in
some details by Elghribi, Othman and Al-Nashri [8], using the properties of the
q
-calculus such as the
q-difference operator was first studied by Jackson [13]:fis homogeneous of degree λif and only if
1E,qf(x)= [λ]qf(x), q∈(0,1)and x∈R+,
where 1E,qis the q-Euler operator [8].
Inspired and motivated by these applications, many researchers have developed the theory of quantum
calculus based on two-parameter
(p,q)
-integer which is used efficiently in many fields such as difference
equations, Lie group, hypergeometric series, physical sciences, and so on. The
(p,q)
-calculus was
first studied in quantum algebras by Chakrabarti and Jagannathan [5]. For some results on the study of
(p,q)-calculus, we refer to [4;5].
In particular, the relationship between the homogeneous functions and the
(p,q)
-Euler operator have
not been studied. The gap mentioned above is the motivation for this research. The aim of this paper is to
introduce new concepts of
(p,q)
-Euler operator which play an important role on a characterization of
homogeneous positive functions and applications.
In this paper we give a response to the above question as follows: positive homogeneous functions
f
on
R
of a negative degree
λ
are characterized by a new counterpart of Euler’s homogeneous function
theorem using the (p,q)-Euler operator, i.e., fis homogeneous of degree λif and only if
1E,p,qf(x)= [λ]p,qf(x)for 0 <q<p<1 and x∈R+.
As an application the solution of the Cauchy problem associated to the
(p,q)
-Euler operator is given.
Using this solution we study the associated
ν
-potential. Its markovianity property is treated. Finally the
associated subordinated of this solution for the Cauchy problem is presented in some details. This work
is a continuation of [8].
2. Preliminary definitions and properties
In this section, we provide basic definitions, notations that will be used in this study and known results on
(p,q)-calculus as given in [4;5]. Let 0 <q<p<1. Then the twin-basic number [n]p,qis defined by
[n]p,q:= pn−qn
p−q,n≥1,
with
[0]p,q:= 1.
The twin-basic number is a natural generalization of the q-number [n]q, that is,
lim
p→1[n]p,q= [n]q:= 1−qn
1−q.
A NEW CHARACTERIZATION OF HOMOGENEOUS FUNCTIONS AND APPLICATIONS 1075
Generally a (p,q)-complex number is defined by
[α]p,q:= pα−qα
p−q, α ∈C.
Let fbe a function defined on the set of the complex number. The (p,q)-derivative is defined by
(2-1) Dp,qf(x):= f(px )−f(qx )
(p−q)x,x= 0
and Dp,qf(0):= f′(0), provided that fis differentiable at 0. It happens clearly that
Dp,qxn= [n]p,qxn−1,x= 0,n≥1,
and
Dp,qxα= [α]p,qxα−1,x= 0,
where
α
is a complex number. Note also that for
p→
1, the
(p,q)
-derivative operator reduces to the
q-derivative operator given by
Dqf(x):= f(x)−f(qx)
(1−q)x,x= 0 and Dqf(0):= f′(0),
such that
lim
q→1Dqf(x)=d f (x)
dx
if
f
is differentiable at
x
(see [13] for more details). The derivative (2
-
1) verifies the following
(p,q)
-
derivation property:
(2-2) Dp,q(f.g)(x)=f(px)Dp,qg(x)+g(q x )Dp,qf(x)=g(p x )Dp,qf(x)+f(q x)Dp,qg(x).
And the high (p,q)-derivatives are
D0
p,qf:= f,Dn
p,qf:= Dp,q(Dn−1
p,qf), n≥1.
Notice that a continuous function on an interval, which does not include 0, is continuous
(p,q)
-
differentiable. Note that the
(p,q)
-derivative operator
Dp,q
and the operator
X
defined by
X f (x)=x f (x)
are linear operators on some Hilbert space. It is obvious that
X Dp,q
is the
(p,q)
-deformation of the
operator
Xd
dx
verifying: for
p
and
q
tend to 1,
X Dp,q
tends to
Xd
dx
. Now, if we replace
Xd
dx
by
X Dp,q
,
what is the (p,q)-analogues of Euler’s theorem?
3. Main results
Definition 3.1. Let 0 <q<p<1. We define the (p,q)-Euler operator by
1E,p,q=X Dp,q.
Proposition 3.2. Let f,g:R+→R,then for 0<q<p<1, we have
(3-1) 1E,p,q(f.g)(x)=g(px) 1E,p,qf(x)+f(qx) 1E,p,qg(x).
1076 MONCEF ELGHRIBI
Proof. By definition, we have, for x= 0
1E,p,q(f.g)(x)=x D p,q(f.g)(x)=xf(px )g(px )−f(q x )g(qx )
(p−q)x
=xf(px )−f(qx )
(p−q)xg(px )+x f (q x )g(p x )−g(q x)
(p−q)x
=x(Dp,qf)(x)g(px)+x f (q x)(Dp,qg)(x)
=g(px ) 1E,p,qf(x)+f(q x) 1E,p,qg(x).
Now, taking x=0, we know that
Dp,q(f.g)(x)|x=0=(f.g)′(0)=f′(0)g(0)+g′(0)f(0).
Then, applying the multiplication operator x, we obtain
1E,p,q(f.g)(0)=g(p.0)(1E,p,qf)(0)+f(q.0)(1E,p,qg)(0). □
Remark. Note that, our new operator 1E,p,qcan be easily written in terms of 1E,rsuch that
1E,p,qf(x)=1
p1E,rf(px ), r=p
q∈(0,1).
So, it is possible to reformulate results from [8] for the operator 1E,p,q.
Definition 3.3. Let
f:R+→R
where
f
is said to be a homogeneous function of degree
λ∈R
, if for all
x∈R+and for all α > 0f(αx)=αλf(x).
Theorem 3.4. Let 0<q<p<1, x∈R+and λ < 0. If we have
1E,p,qf(x)= [λ]p,qf(x),
then f is homogeneous of degree λ.
Proof. Let f:R+→Rsuch that 1E,p,qf(x)= [λ]p,qf(x). For all x∈R+, define 8:R+→Rby
8(α) =f(α x)−αλf(x).
Then, for α > 0, we get
Dp,q(8)(α) =f(pαx)−f(qαx)
(p−q) α − [λ]p,qαλ−1f(x)
=x
α
f(pαx)−f(qαx)
(p−q)x− [λ]p,qαλ−1f(x)=1
α1E,p,qf(αx)− [λ]p,qαλ−1f(x).
Immediately, it becomes
αDp,q8(α) =1E,p,qf(α x)− [λ]p,qαλf(x)= [λ]p,q(f(α x)−αλf(x)) = [λ]p,q8(α).
Since αis arbitrary, 8satisfies the following (p,q)-differential equation:
(3-2) Dp,q8(α) −[λ]p,q
α8(α) =0.
A NEW CHARACTERIZATION OF HOMOGENEOUS FUNCTIONS AND APPLICATIONS 1077
Equation (3-2) is equivalent to
8( pα) −8(qα)
(p−q) α =[λ]p,q
α8(α),
which yields
8( pα) −8(qα) =(pλ−qλ) 8(α).
As a consequence, we also obtain the representation
8( pα) −8(qnα) =(pλ−(qn)λ) 8(α).
Since
lim
n→∞(8( pα) −8(qnα)) =8 ( pα) −8(0)and lim
n→∞(pλ−(qn)λ) 8(α) =pλ8(α),
we have
8( pα) =pλ8(α) +8(0).
By a simple iteration, for n≥1, we have
8( pnα) =pλ8 ( pn−1α) +8(0)
such that
lim
n→∞ 8( pnα) =8(0)and lim
n→∞ pλ8( pn−1α) +8(0)=pλ8(0)+8(0).
Then
8(0)=0 and f(0)=0,
which is equivalent to
8( pα) =pλ8(α).
As a consequence, by a simple iteration, we obtain
8( pnα) =(pn)λ8(α),
or equivalently
8(α) =p−nλ8 ( pnα).
Then, for n→ ∞, we get 8(α) =0 which is equivalent to
f(αx)=αλf(x). □
Theorem 3.5. If f is homogeneous of degree λ∈R,then we have
1E,p,qf(x)= [λ]p,qf(x), 0<q<p<1.
Proof. Let fbe a homogenous function of degree λ∈R. Then for x= 0, we have
Dp,q(f)(x)=f(px)−f(q x )
(p−q)x=pλf(x)−qλf(x)
(p−q)x=(pλ−qλ)
(p−q)
f(x)
x= [λ]p,q
f(x)
x,
which yields
1E,p,qf(x)= [λ]p,qf(x).
1078 MONCEF ELGHRIBI
Now, for x=0 and λ∈R\ {0}, we have
(X Dp,qf)(0)=0 and f′(0)=0.
But, we know that
f
is homogeneous of degree
λ
, then
f(αx)=αλf(x)
. In particular, for
x=
0, we
have f(0)=αλf(0)for α > 0, which implies that f(0)=0 for λ∈R∗. Then, we get
(X Dp,qf)(0)=0= [λ]p,qf(0).
Now, for λ=0 and x=0 we have
(X Dp,qf)(0)=0= [0]p,qf(0).
Conclusively, for x∈R+and λ∈R, we obtain
1E,p,qf(x)= [λ]p,qf(x). □
Combining Theorems 3.4 with 3.5 above, we get the following theorem, which will be called
(p,q)
-
analogues Euler’s theorem.
Theorem 3.6. Let λ < 0. Then,f is homogeneous of degree λif and only if
1E,p,qf(x)= [λ]p,qf(x)
for 0<q<p<1and x ∈R+.
Remark. It is obvious from Theorem 3.6 that, as
p
and
q
tend to 1, we refined the classical Euler’s
theorem for λ < 0.
4. Applications of the (p,q)-Euler operator
Cauchy problem associated to the
(p,q)
-Euler operator. Let
f:R+→R
be a homogeneous function
of degree λwhere 0 < λ ≤1. For 0 <q<p<1, consider the following Cauchy problem:
(4-1) ∂
∂tU(t,x)=1E,p,qU(t,x), t>0,x∈R+,
U(0,x)=f(x).
Theorem 4.1. The Cauchy problem (4-1) admits an unique solution given by
(4-2) U(t,x)=f(x)+
∞
X
n=1
n−1
X
k=0
(−1)ktnfqk
pkpx −fqk
pkqx
n(n−1−k)!k!(p−q)n.
Proof. We start by verifying that
U(t,x):= f(x)+
∞
X
n=1
n−1
X
k=0
(−1)ktnfqk
pkpx −fqk
pkqx
n(n−1−k)!k!(p−q)n
is a solution of the system (4-1). On the one hand, we have
∂U(t,x)
∂t=
∞
X
n=1
n−1
X
k=0
(−1)ktn−1fqk
pkpx −fqk
pkqx
(n−1−k)!k!(p−q)n.
A NEW CHARACTERIZATION OF HOMOGENEOUS FUNCTIONS AND APPLICATIONS 1079
On the other hand, we have
1E,p,qU(t,x)=x Dp,qU(t,x)=U(t,px )−U(t,qx )
p−q
=f(px )−f(qx )
p−q+
∞
X
n=1
n−1
X
k=0
(−1)ktnfqk
pkpx −fqk
pkqx
n(n−1−k)!k!(p−q)n+1
−
∞
X
n=1
n−1
X
k=0
(−1)ktnfqk+1
pkpx −fqk+1
pkqx
n(n−1−k)!k!(p−q)n+1.
By changing the indices in the right sums of the above equation, we obtain
1E,p,qU(t,x)=f(px )−f(qx )
p−q+
∞
X
n=1
n−1
X
k=0
(−1)ktnfqk
pkpx −fqk
pkqx
n(n−1−k)!k!(p−q)n+1
+
∞
X
n=1
n
X
k=1
(−1)ktnfqk
pkpx −fqk
pkqx
n(n−k)!(k−1)!(p−q)n+1
=x Dp,qf(x)+
∞
X
n=1
n
X
k=0
(−1)ktnfqk
pkpx −fqk
pkqx
(n−k)!k!(1−q)n+1
=
∞
X
n=0
n
X
k=0
(−1)ktnfqk
pkpx −fqk
pkqx
(n−k)!k!(p−q)n+1
=
∞
X
n=1
n−1
X
k=0
(−1)ktn−1[f(qkx)−f(qk+1x)]
(n−1−k)!k!(p−q)n=∂U(t,x)
∂t,
which shows that U(t,x)is a solution of (4-1). Let us show by recursion on n≥1 that
(4-3) 1n
E,p,qf(x)=1
(p−q)n
n−1
X
k=0n−1
k(−1)kfqk
pkpx −fqk
pkqx .
For n=1 we have
1E,p,qf(x):= x Dp,qf(x)=1
p−q(f(px )−f(qx )), x∈R+.
Now, suppose that (4-2) is verified, then we get
1n+1
E,p,qf(x)=1E,p,q(1n
E,qf(x)) =x Dp,q(1n
E,p,qf(x))
=1
p−q[1n
E,p,qf(px )−1n
E,p,qf(qx)]
=1
(p−q)n
n−1
X
k=0n−1
k(−1)kqk
pkpx D p,qfqk
pkpx
−1
(p−q)n
n−1
X
k=0n−1
k(−1)kqk
pkqx D p,qfqk
pkqx .
1080 MONCEF ELGHRIBI
By changing the indices in the right sums of the above equation, we obtain
(4-4) 1n+1
E,p,qf(x)=1
(p−q)n
n−1
X
k=0n−1
k(−1)kqk
pkx Dp,qfqk
pkx
+1
(p−q)n
n
X
k=1n−1
k−1(−1)kqk
pkx Dp,qfqk
pkx
=1
(p−q)nx Dp,qf(x)+(−1)nqn
pnx Dp,qfqn
pnx
+1
(p−q)n
n−1
X
k=1n−1
k+n−1
k−1(−1)kqk
pkx Dp,qfqk
pkx
=1
(p−q)nx Dp,qf(x)+(−1)nqn
pnx Dp,qfqn
pnx
+1
(p−q)n
n−1
X
k=1n
k(−1)kqk
pkx Dp,qfqk
pkx
=1
(p−q)n
n
X
k=0n
k(−1)kqk
pkx Dp,qfqk
pkx
=1
(p−q)n+1
n
X
k=0n
k(−1)kfqk
pkpx −fqk
pkqx .
This shows (4-1) for all n≥0. Then, using (4-1) we get
(4-5) Qtf(x):=
∞
X
n=0
tn
n!1n
E,p,qf(x)=f(x)+
∞
X
n=1
tn
n!1n
E,p,qf(x)
=f(x)+
∞
X
n=1
n−1
X
k=0
(−1)ktnfqk
pkpx −fqk
pkqx
n(n−1−k)!k!(p−q)n=U(t,x).
Finally, we show the uniqueness of the above solution. Let
V(t,x)
another solution of the equation (4
-
1),
we set W(t,x)=Q−tV(t,x). Then
∂W(t,x)
∂t= −1E,p,qW(t,x)+Q−t(1E,p,qV(t,x)) = −1E,p,qW(t,x)+1E,p,qQ−tV(t,x)=0,
from which, we deduce that
W(t,x)=W(0,x)=V(0,x)=f(x).
This implies that
V(t,x)=Qtf(x)=U(t,x). □
ν-(p,q)-potential. Using the semigroup Q:= {Qt}t>0we come to the following definition.
A NEW CHARACTERIZATION OF HOMOGENEOUS FUNCTIONS AND APPLICATIONS 1081
Definition 4.2. For ν > 0, we define the ν-(p,q)-potential by
Hν, p,qf(x)=Z∞
0
e−νt(Qt(f)(x)−f(x)) dt,
Theorem 4.3. The ν-(p,q)-potential is the unique solution of the following (p,q)-Poisson equation:
(ν I−1E,p,q)F=1
νDp,q,
where I is the identical operator.
Proof. By Definition 4.2 and (4-5) we have
Hν, p,qf(x)=Z∞
0
e−νt
∞
X
n=1
n−1
X
k=0
(−1)ktnfqk
pkpx −fqk
pkqx
n(n−1−k)!k!(p−q)ndt
=
∞
X
n=1
n−1
X
k=0
(−1)kfqk
pkpx −fqk
pkqx
n(n−1−k)!k!(p−q)nZ∞
0
e−νttndt.
One can show easily that
Z∞
0
e−νttndt =n!
νn+1.
Then,
(4-6) Hν, p,qf(x)=
∞
X
n=1
n−1
X
k=0
(−1)ktnfqk
pkpx −fqk
pkqx
(n−k)!k!(p−q)n(n−1)!
νn+1.
On the other hand, we have
X Dp,qQtf(x)=1E,p,qet1E,p,qf(x)=
∞
X
n=0
tn
n!1n+1
E,p,qf(x)=1E,p,qf(x)+
∞
X
n=1
tn
n!1n+1
E,p,qf(x).
Using (4-4), we get
1E,p,qQtf(x)=1E,p,qf(x)+
∞
X
n=1
tn
n!
1
(p−q)n+1
n
X
k=0n
k(−1)kfqk
pkpx −fqk
pkqx ,
from which we obtain
1E,p,q(Qtf−f)(x)=
∞
X
n=1
n
X
k=0
tn(−1)kfqk
pkpx −fqk
pkqx
k!(n−k)!(p−q)n+1.
Then, from Definition 4.2, we get
(4-7) 1E,p,qHν, p,qf(x)=
∞
X
n=1
n
X
k=0
n!(−1)kfqk
pkpx −fqk
pkqx
νn+1k!(n−k)!(p−q)n+1.
1082 MONCEF ELGHRIBI
At this point, by the change indices (n−1=j)in the right sums of (4-6), we obtain
Hν, p,qf(x)=
∞
X
j=0
j
X
k=0
j!(−1)k
νn+2k!(j−k)!(p−q)j+1fqk
pkpx −fqk
pkqx .
Using (4-7) we get
Hν, p,qf(x)
=1
ν2(p−q)f(px )−f(qx )+
∞
X
j=0
j
X
k=0
j!(−1)k
νn+2k!(j−k)!(p−q)j+1fqk
pkpx −fqk
pkqx
=1
ν2(p−q)f(px )−f(qx )+1
ν1E,p,qHν, p,qf(x)
.
As a consequence we have
νHν, p,qf(x)−1E,p,qHν, p,qf(x)=1
νDp,qf(x),
which is equivalent to
(ν I−1E,p,q)Hν, p,q=1
νDp,q.□
Markovianity property. Let
(E,E)
be a measurable space. Let
B(E)
be the set of all Borel bounded
functions defined on
E
and let
C(E)
be the set of all continuous and bounded functions defined on
E
. We
will refer to [3;7;16;20] for the following notations. Recall that from
P:= {Pt}t≥0
is called a Markovian
transition operator on Eif it satisfies the following axioms:
(a) P0=I.
(b) Pt+s=PtPsfor s,t≥0.
(c) Strong continuity: Ptf→fas t→0, f∈C(E).
(d) Ptf≥0 for f≥0.
(e) Pt1=1 for t≥0.
Remarks. (i) Property (b) is equivalent to the so called Chapman–Kolmogorov equation
(4-8) Pt+s(x,A)=ZPs(y,A)Pt(x,dx ), t,s≥0,x∈E,A∈E
(ii) Conditions (d) and (e) imply contraction property
∥Ptf∥≤∥f∥,f∈C(E), t>0.
(iii) If, for each s≥0
(4-9) lim
t→sPtf(x)=Psf(x), f∈C(E), x∈E,
then the Markovian transition operator P:= {Pt}t≥0is said to be stochastically continuous on E.
Theorem 4.4. The family
Q:= {Qt}t≥0
is a stochastically continuous Markovian transition operator
on E.
A NEW CHARACTERIZATION OF HOMOGENEOUS FUNCTIONS AND APPLICATIONS 1083
Proof. (a) It is obvious that Q0=I.
(b) Let s,t≥0, then
Qt+s=e(s+t) 1E,p,q=es1E,p,qet1E,p,q=QtQs.
(c) Let t>0, then
∥Qtf−f∥ ≤
∞
X
n=1
tn∥1E,p,q∥n∥f∥
n!=(et∥1E,p,q∥−1)∥f∥ → 0,as t→0.
(d) Using Theorem 3.4, we get
1E,p,qf(x)= [λ]p,qf(x).
Similarly using Theorem 3.4, we obtain
(1E,p,q)nf(x)=([λ]p,q)nf(x).
Then,
Qtf(x)=
∞
X
n=o
tn
n![λ]n
p,qf(x)=et[λ]p,qf(x).
Hence, when f≥0, we obtain
Qtf≥0.
(e) Using (4-5), we get
Qt1=1 for all t≥0.
Then Q:= {Qt}t≥0is a Markovian transition operator on E.
Moreover, for s≥0 we have
lim
t→sQtf(x):= lim
t→set1E,p,qf(x)=es1E,p,qf(x)=Qsf(x), f∈C(E), x∈E.
Then Q:= {Qt}t≥0is stochastically continuous on E.□
Subordination. We consider
R
endowed with its Borel
σ
-field
A
, we denote by
supp(µ)
the support
of the measure
µ
defined on
(R,A)
and by
ϵx
the Dirac measure at point
x∈R
. We refer to [3] for the
following notions. A Bochner subordinator is a semigroup of probability measures
β:= {βt}t>0
on
(R,A)
such that
for each t>0, the measure βt= ϵ0and supp(βt)⊂ [0,∞[,
βs∗βt=βs+t,s,t>0,(4-10)
lim
t→0βt=ϵ0,vaguely.(4-11)
Theorem 4.5. Let
β:= {βt}t>0
on
(R,A)
be a Bochner subordinator. Then for every
t>
0the following
kernel Qβ:= {Qβ
t}t>0defined on (E,E)by
(4-12) Qβ
tf:= Z∞
0
Qsfβt(ds), f∈B(E)
is a stochastically continuous Markovian transition operator on E.
1084 MONCEF ELGHRIBI
Proof. Using Chapman–Kolmogorov equation given by the relation (4
-
8) and using (4
-
10) and (4
-
12), it is
easy to see that
Qβ:= {Qβ
t}t>0
is a Markovian transition operator on
E
. Moreover, since
Q:= {Qt}t>0
is stochastically continuous on
E
and by (4
-
9),(4
-
11) and (4
-
12) it is clear that
Qβ
is stochastically
continuous on
E
. In this case,
Qβ
is said to be subordinated to
Q
in the sense of Bochner by means
of β[3].□
Example. The most important example of Bochner subordinator is the following: Let
ηα:= {ηα
t}t≥0
“fractional powers” be the one-sided stable subordinator of order
α∈]
0
,
1
[
such that the associated Laplace
transform
L(ηα
t)(x)=exp(−t x α)
for
x>
0. In this case, using equation (4
-
12),
Qηα:= {Qηα
t}t>0
the
subordinated transition operator of Qby means of ηαis given by
(4-13) Qηα
tf(x):= Z∞
0
Qsf(x) ηα
t(ds)=Z∞
0
es[λ]p,qf(x) ηα
t(ds), x,t>0,f∈B(E).
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MONCEF ELGHRIBI:melghribi@yahoo.com
Department of Mathematics, University College of Taymaa, University of Tabuk, Tabuk, Saudi Arabia
RMJ — prepared by msp for the
Rocky Mountain Mathematics Consortium
