# Alternative Forms of Compound Fractional Poisson Processes

Authors: Luisa Beghin; Claudio Macci
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study here different fractional versions of the compound Poisson process. The fractionality is introduced in the counting process representing the number of jumps as well as in the density of the jumps themselves. The corresponding distributions are obtained explicitly and proved to be solution of fractional equations of order less than one. Only in the final case treated in this paper, where the number of jumps is given by the fractional-difference Poisson process defined in Orsingher and Polito (2012), we have a fractional driving equation, with respect to the time argument, with order greater than one. Moreover, in this case, the compound Poisson process is Markovian and this is also true for the corresponding limiting process. All the processes considered here are proved to be compositions of continuous time random walks with stable processes (or inverse stable subordinators). These subordinating relationships hold, not only in the limit, but also in the finite domain. In some cases the densities satisfy master equations which are the fractional analogues of the well-known Kolmogorov one.


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## Body

## 1. Introduction and Preliminary Results

The fractional Poisson process (FPP), which we will denote by𝒩β(t), t>0, β∈(0,1], has been introduced in [1], by replacing, in the differential equation governing the Poisson process, the time derivative with a fractional one. Later, in [2, 3], it was proved to be a renewal process with Mittag-Leffler distributed waiting times (and therefore with infinite mean). In [4] it has been expressed as the composition N(𝒯β(t)) of a standard Poisson process N with the fractional diffusion 𝒯β, independent of N. A full characterization of 𝒩β in terms of its finite multidimensional distributions can be found in [5]. In [6] the coincidence between 𝒩β and the fractal time Poisson process (FTPP) defined as N(ℒβ(t)) has been proved, where ℒβ(t), t≥0 is the inverse of the stable subordinator 𝒜β(t) of index β (with parameters μ=0, θ=1,  σ=(tcosπβ/2)1/β, in the notation of [7], that we will adopt hereafter). Thus, the process 𝒜β is characterized by the following Laplace pairs:
(1.1)𝔼e-k𝒜β(t)=e-kβt,k,t>0,∫0+∞‍e-sthβ(x,t)dt=xβ-1Eβ,β(-sxβ),s,x>0,
where Eβ,δ is the Mittag-Leffler function of parameters β, δ and hβ(x,t) is the density of 𝒜β(t). The inverse stable subordinator ℒβ is defined by the following relation:
(1.2)ℒβ(t):=inf{s:𝒜β(s)=t},z,t>0,
and therefore we get
(1.3)𝔼e-kℒβ(t)=Eβ,1(-ktβ),k,t>0,∫0+∞‍e-stlβ(x,t)dt=sβ-1e-xsβ,s,x>0,
where lβ(x,t) is the density of ℒβ(t).We will make use also of different forms of FPP such as the alternative fractional Poisson process in [8] and the fractional-difference Poisson process in [9].In this paper we study several fractional compound Poisson processes and, to help the reader, we list the acronyms used throughout the paper by the end of the paper.The first form of fractional compound Poisson process has been introduced in [10], in the form of a continuous time random walk with infinite-mean waiting times (see also [11]). This corresponds to the following random walk time changed via the FTPP, that is,
(1.4)Yβ(t)=∑j=1N(ℒβ(t))‍Xj,t≥0,
with Xj,j=1,2,…  are i.i.d. random variables, independent from N and ℒβ. The last assumption (that we will adopt throughout the paper) corresponds to the so-called uncoupled case.In [6] it is proved that subordinating random walk to the fractional Poisson process 𝒩β(t), t≥0, produces the same one-dimensional distribution. The (generalized) density function of Yβ(t) can be expressed as
(1.5)gYβ(y,t):=Eβ,1(-λtβ)δ(y)+fYβ(y,t),y,t≥0,
where the first term refers to the probability mass concentrated in the origin, δ(y) denotes the Dirac delta function, and fYβ denotes the density of the absolutely continuous component. The function gYβ given in (1.5) satisfies the following fractional master equation, that is,
(1.6)∂β∂tβgYβ(y,t)=-λgYβ(y,t)+λ∫-∞+∞‍gYβ(y-x,t)fX(x)dx,
where ∂β/∂tβ is the Caputo fractional derivative of order β∈(0,1] (see, for example, [12]) and the random variables Xj, j=1,2,… have continuous density fX.We also recall the following result proved in [13] for the rescaled version of the time-fractional compound Poisson process (hereafter TFCPP): if the random variables Xj, j=1,2,… are centered and have finite variance, then
(1.7)c-β/2Yβ(ct)⇒W(ℒβ(t)),c→∞,
where W is a standard Brownian motion and ⇒ denotes weak convergence.A detailed exposition of the theory of TFCPP and continuous time random walks can be found in [14, 15], where the density fYβ is expressed in terms of successive derivatives of the Mittag-Leffler function as follows:
(1.8)fYβ(y,t)=∑n=1∞‍fX*n(y)Pr{𝒩β(t)=n}=∑n=1∞‍fX*n(y)(λtβ)nn!∂n∂xnEβ,1(x)|x=-λtβ,t,y≥0,
where fX*n is the nth convolution of the density fX of the r.v.'s Xj.A further asymptotic result has been proved in [15], under the assumption that the density of the jump variables (which we will denote, in this special case, as Xj*) behaves asymptotically as
(1.9)f^X*(hκ):=∫-∞+∞‍eiκhxf^X*(x)dx≃1-hα|κ|α,h→0,α∈(0,1],
where ·^ denotes the Fourier transform. In this case the TFCPP is defined as Yβ*(t)=∑j=1𝒩β(t)‍Xj* and the rescaled version displays the following weak convergence:
(1.10)hYβ*(tr)⇒Z(t),
for h,r→0, s.t. hα/rβ→1. The characteristic function of the limiting process Z(t) is given by
(1.11)Eβ,1(-λtβ|κ|α)
and thus it can be represented as 𝒮α(ℒβ(t)), where 𝒮α is a symmetric α-stable process with parameters μ=0, θ=0, σ=(tcosπα/2)1/α. For β<1, the inverse stable subordinator ℒβ(t) is not Markovian as well as not Lévy (see [10]) and the same is true for 𝒮α(ℒβ(t)), as remarked in [15]; moreover, the density u=u(y,t) of the latter is the solution to the space-time fractional equation:
(1.12)∂βu∂tβ=λ∂αu∂|y|α,u(y,0)=δ(y),y∈ℝ,t>0,
where ∂α/∂|y|α denotes the Riesz-Feller derivative of order α∈(0,1] (see [16]). Thus, in the special case α=1, it reduces to the composition of a Cauchy process with ℒβ.Finally, we recall the following result proved in [17]: under the assumption of heavy tailed r.v.’s representing the jumps, that is,
(1.13)Pr{|Xj|>x}~x-α,x→∞,
the following convergence holds, as c→∞,
(1.14)c-β/αYβ(ct)⇒ℐ(ℒβ(t)).
In (1.14) ℐ is a α-stable Lévy process with density pα(x,t) and characteristic function
(1.15)p^α(κ,t)=etb[q(-iκ)α+(1-q)(iκ)α],for{b<0,0<α<1orb>0,1<α<2,
under the assumption that limx→∞Pr{Xj<-x}/Pr{|Xj|>x}=q∈0,1]. The density of the limiting process is proved to satisfy the following time and space fractional equation:
(1.16)D0+,tβu=qbD-,xαu+(1-q)bD0+,xαu,
where the fractional derivatives are intended in the Riemann-Liouville sense (see [12], formulae (2.2.3) and (2.2.4), page 80).We present, in this paper, different versions of the compound Poisson process (CPP), fractional (under different acceptions) with respect to time and space; we provide for them analytic expressions of the distributions and some composition relationships with stable and inverse-stable processes, holding not only in the scaling limit, but also in the finite domain.Tables1 and 2 provide a summary of these results in the finite and asymptotic domains, respectively.Table 1
Main results in finite domain.
Process
Equation
CPP
Y
(
t
)
ξ
∂
∂
t
=
-
[
λ
+
∂
∂
t
]
∂
∂
y
TFCPP
Y
β
(
t
)
=
d
Y
(
ℒ
β
(
t
)
)
ξ
∂
β
∂
t
β
=
-
[
λ
+
∂
β
∂
t
β
]
∂
∂
y
SFCPP
Y
(
α
)
(
t
)
=
d
𝒮
α
(
Y
(
t
)
)
ξ
∂
∂
t
=
-
[
λ
+
∂
∂
t
]
∂
α
∂
y
α
STFCPP
Y
β
(
α
)
(
t
)
=
d
𝒮
α
(
Y
β
(
t
)
)
ξ
∂
β
∂
t
β
=
-
[
λ
+
∂
β
∂
t
β
]
∂
α
∂
y
α
ΔFCPP
Y
Δ
(
t
)
=
d
Y
(
𝒜
γ
(
t
)
)
ξ
D
-
,
t
1
/
γ
=
[
λ
-
D
-
,
t
1
/
γ
]
∂
∂
yTable 2
Main results in asymptotic domain.
Process
Hypothesis on jumps
Limiting process
Limiting equation
TFCPPYβ
X
j
*
𝒮
α
(
ℒ
β
(
t
)
)
∂
β
u
∂
t
β
=
λ
∂
α
u
∂
|
y
|
α
"
X
j
(
α
)
𝒜
α
(
ℒ
β
(
t
)
)
∂
β
u
∂
t
β
=
-
λ
∂
α
u
∂
y
α
Altern. TFCPPY¯β
X
j
*
𝒮
α
β
(
t
)
∂
u
∂
t
=
λ
1
/
β
β
∂
α
u
∂
|
y
|
α
"
X
j
(
α
)
𝒜
α
λ
,
ξ
(
t
)
∂
u
∂
t
=
-
λ
ξ
∂
α
u
∂
y
α
ΔFCPP YΔ
X
j
*
𝒮
α
γ
(
t
)
∂
u
∂
t
=
λ
γ
∂
α
γ
u
∂
|
y
|
α
γ
"
X
j
(
α
)
𝒜
α
γ
λ
,
ξ
(
t
)
∂
u
∂
t
=
-
λ
γ
ξ
∂
α
γ
u
∂
y
α
γWe assume here exponential jumps (generalized later to Mittag-Leffler), since this allows to obtain explicit equations (fractional in most cases) driving these fractional CPP's for any finite value of the time and space arguments. This kind of explicit formulae, together with the knowledge of the related governing differential equations, is of great importance in many actuarial applications (see, for example, [18], Section 4.2). In risk theory it is related to the Tweedie's compound Poisson model (see [19]). The hypothesis of exponential jumps has been widely applied also in other fields: in natural sciences it leads to the so-called compound Poisson-Gamma model, which is used for rainfall prediction (see, for example, [20]).


## 2. Time-Fractional Compound Poisson Processes

We consider different forms of TFCPP, starting with the more familiar one given in (1.4) and then comparing the results with those obtained for an alternative definition of FPP.


### 2.1. The Standard Case

In order to get a form of the density of the TFCPP more explicit than (1.8), we assume that the Xj's are exponentially distributed: in this case it can be expressed in terms of the generalized Mittag-Leffler function:
(2.1)Eα,δγ(x)=∑j=0∞‍(γ)(j)j!xjΓ(αj+δ),α,δ,γ∈ℂ,ℛ(α),ℛ(δ)>0,
where (x)(n)=x(x+1)⋯(x+n-1) is the rising factorial (or Pochhammer symbol). Moreover, we can obtain the fractional partial-differential equation satisfied by the density of its absolutely continuous component.Theorem 2.1.
The process(2.2)Yβ(t)=∑j=1𝒩β(t)‍Xj,t≥0,
with Xj, j=1,2…, independent and exponentially distributed with parameter ξ, has the following distribution:
(2.3)Pr{Yβ(t)≤y}=Eβ,1(-λtβ)1[0,+∞)(y)+∫-∞y‍fYβ(z,t)dz,t≥0,y∈ℝ,
where
(2.4)fYβ(y,t)=e-ξyy∑n=1∞‍(λξtβy)n(n-1)!Eβ,βn+1n+1(-λtβ)1[0,+∞)(y),t≥0.
The function fYβ(y,t) given in (2.4) satisfies the following partial differential equation:
(2.5)ξ∂β∂tβfYβ=-[λ+∂β∂tβ]∂∂yfYβ,t,y≥0,
where ∂β/∂tβ denotes the Caputo fractional derivative with the conditions
(2.6)fYβ(y,0)=0,∫0+∞‍fYβ(y,t)dy=1-Eβ,1(-λtβ).Proof.
Formula  (1.8) can be rewritten by considering that fX*n(y)=ξnyn-1e-ξy/(n-1)! and using the expression of Pr{𝒩β(t)=n} in terms of generalized Mittag-Leffler functions (see [21]), that is,
(2.7)Pr{𝒩β(t)=n}=λntnβEβ,βn+1n+1(-λtβ),n≥0.
In order to derive (2.5), we evaluate the following partial derivatives of (2.4):
(2.8)∂β∂tβfYβ=e-ξyytβ∑n=1∞‍(λξtβy)n(n-1)!n!∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn-β+1),∂∂yfYβ(y,t)=-ξe-ξyy∑n=1∞‍(λξtβy)n(n-1)!n!∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn+1)++e-ξyy2∑n=2∞‍(λξtβy)n(n-2)!n!∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn+1),∂∂y∂β∂tβfYβ(y,t)=-ξe-ξyytβ∑n=1∞‍(λξtβy)nn!(n-1)!∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn-β+1)++e-ξyy2tβ∑n=2∞‍(λξtβy)n(n-2)!n!∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn-β+1)=-ξe-ξyytβ∑n=1∞‍(λξtβy)n((n-1)!)2∑j=0∞‍(n+j-1)!(-λtβ)jj!Γ(βj+βn-β+1)+-ξe-ξyytβ∑n=1∞‍(λξtβy)nn!(n-1)!∑j=1∞‍(n+j-1)!(-λtβ)j(j-1)!Γ(βj+βn-β+1)++e-ξyy2tβ∑n=2∞‍(λξtβy)n(n-2)!(n-1)!∑j=0∞‍(n+j-1)!(-λtβ)jj!Γ(βj+βn-β+1)++e-ξyy2tβ∑n=2∞‍(λξtβy)n(n-2)!n!∑j=0∞‍(n+j-1)!(-λtβ)j(j-1)!Γ(βj+βn-β+1).
By inserting (2.8) in (2.5), the equation is satisfied. Finally, it can be easily verified that the initial condition holds. In order to check the second condition in (2.6), we integrate fYβ with respect to y:
(2.9)∫0∞‍e-ξyy∑n=1∞‍(λξtβy)n(n-1)!Eβ,βn+1n+1(-λtβ)dy=∑n=1∞‍(λξtβ)n(n-1)!(n-1)!ξnEβ,βn+1n+1(-λtβ)=∑n=0∞‍(λtβ)nEβ,βn+1n+1(-λtβ)-Eβ,1(-λtβ)=1-Eβ,1(-λtβ),
where, in the last step, we have applied formula (2.30) of [21], for u=1.


#### 2.1.1. The Nonfractional Caseβ=1

From (2.4), we obtain the distribution of the standard CPP, defined as Y(t)=∑n=1N(t)‍Xj, under the assumption of exponential jumps Xj, which reads
(2.10)Pr{Y(t)≤y}=e-λt1[0,+∞)(y)+∫-∞y‍fY(z,t)dz,t≥0,y∈ℝ,
where
(2.11)fY(y,t)=e-ξy-λty∑n=1∞‍(λξty)nn!(n-1)!1[0,+∞)(y)=λξte-ξy-λtW1,2(λξty)1[0,+∞)(y),t≥0,(2.12)Wα,β(z)=∑j=0∞‍zjj!Γ(αj+β),α>-1,β,z∈ℂ,
is the Wright function. Equation (4.2.8) in [18] provides another expression of fY in terms of the modified Bessel function. The density (2.11) satisfies the following equation:
(2.13)ξ∂∂tfY=-[λ+∂∂t]∂∂yfY,
with conditions
(2.14)fY(y,0)=0,∫0+∞‍fY(y,t)dy=1-e-λt,
as can be easily verified directly.Now we recall the following subordination law presented in [6] in a more general setting:
(2.15)Yβ(t)=dY(ℒβ(t)),
where ℒβ(t), t≥0 is the inverse stable subordinator defined by (1.2). We give an explicit proof of (2.15), which will be useful to prove analogous results in the next sections. We start with the evaluation of the Laplace transform (hereafter denoted by  ·~) of Yβ(t) with respect to y: by considering the probability generating function of 𝒩β, that is,
(2.16)𝔼u𝒩β(t)=Eβ,1(-λtβ(1-u)),|u|≤1,
we get
(2.17)g~Yβ(k,t):=𝔼e-kYβ(t)=Eβ,1(-λkk+ξtβ).
Formula (2.17), Laplace transformed with respect to t, gives
(2.18)g~~Yβ(k,s):=∫0+∞‍e-stg~Yβ(k,t)dt=sβ-1(k+ξ)sβ(k+ξ)+kλ,
which can be rewritten as
(2.19)g~~Yβ(k,s)=sβ-1∫0+∞‍e-sβt𝔼e-kY(t)dt=[by(1.3)]=∫0+∞‍𝔼e-kY(z)l~β(z;s)dz,
where l~β(z;s):=∫0+∞‍e-stlβ(z,t)dt. Thus, by inverting the double Laplace transform, we get
(2.20)Pr{Yβ(t)∈dy}=∫0+∞‍Pr{Y(z)∈dy}lβ(z,t)dz.
Now it is also easy to derive (2.5), since we can write in particular from (2.20) that
(2.21)fYβ(y,t)=∫0+∞‍fY(y,z)lβ(z,t)dz
and thus we get
(2.22)∂β∂tβfYβ(y,t)=∫0+∞‍fY(y,z)∂β∂tβlβ(z,t)dz=-∫0+∞‍fY(y,z)∂∂zlβ(z,t)dz.
Indeed, it is well known that ℒβ(t) is governed by the following equation:
(2.23)∂β∂tβlβ(z,t)=-∂∂zlβ(z,t),lβ(z,0)=δ(z),z,t≥0.
By integrating by parts and applying the initial condition, (2.22) becomes
(2.24)∂β∂tβfYβ(y,t)=∫0+∞‍∂∂zfY(y,z)lβ(z,t)dz=[by(2.13)]=-1ξ∂∂y∫0+∞‍∂∂zfY(y,z)lβ(z,t)dz-λξ∂∂y∫0+∞‍fY(y,z)lβ(z,t)dz=-1ξ∂∂y∂β∂tβfYβ(y,t)-λξ∂∂yfYβ(y,t).


### 2.2. An Alternative Case

We consider now a different model of TFCPP, based on the alternative definition of FPP given in [4], that is,
(2.25)Pr{N¯β(t)=k}=(λtβ)kΓ(βk+1)1Eβ,1(λtβ),t,k≥0.
The process with the above state probabilities plays a crucial role in the evolution of some random motions (see [22]) and can be considered as a fractional version of the Poisson process because its probability generating function (displayed below) satisfies a fractional equation (see formula (4.5) of [4]). The distribution (2.25) can be interpreted as a weighted Poisson distribution (for the general concept of discrete weighted distribution see, e.g., [23], page 90, and the references cited therein) and, as explained in [8], the weights that do not depend on t; actually we have
(2.26)Pr{N¯β(t)=k}=wkpk(tβ)∑j≥0wjpj(tβ),t,k≥0,
where wj=j!/Γ(βj+1),  j=0,1,… (for all t) and pj(t)=((λt)j/j!)e-λt, j=0,1,… are the distribution of the standard Poisson process N with intensity λ. We also recall [24] where one can find a sample path version of the weighted Poisson process.We remark that the corresponding process is not Markovian, as𝒩β, and moreover is not a renewal. Nevertheless, it is, for some aspects, more similar to the standard Poisson process N than 𝒩β. For example, the rate of the asymptotic behavior of its moments is the same as for N.The moment generating function is given by(2.27)𝔼eθN¯β(t)=Eβ,1(λtβeθ)Eβ,1(λtβ),
so that we get
(2.28)𝔼N¯β(t)=λtββEβ,β(λtβ)Eβ,1(λtβ).
By applying the following asymptotic formula of the Mittag-Leffler function
(2.29)Eβ,ν(z)≃1βz(1-ν)/βexp{z1/β},asz→∞,
(see, for example, [25] or [26]) we get
(2.30)𝔼N¯β(t)≃1βλ1/βt,ast→∞,
while for 𝒩β the mean value behaves asymptotically as tβ.We define the alternative TFCPP as(2.31)Y¯β(t)=∑j=1N¯β(t)‍Xj,t≥0,β∈(0,1],
where again Xj’s are i.i.d. with exponential distribution, independent from N¯β. Under this assumption we obtain the following result on the distribution of Y¯β.Theorem 2.2.
The processY¯β defined in (2.31), with Xj, j=1,2,…, independent and exponentially distributed with parameter ξ, has the following distribution:
(2.32)Pr{Y¯β(t)≤y}=1Eβ,1(λtβ)1[0,+∞)(y)+∫-∞y‍fY¯β(z,t)dz,t≥0,y∈ℝ,
where
(2.33)fY¯β(y,t)=λξtβe-ξyEβ,1(λtβ)Wβ,β+1(λξtβy)1[0,+∞)(y),t≥0.Proof.
The density (2.33) can be obtained as follows:
(2.34)fY¯β(y,t)=e-ξyEβ,1(λtβ)∑n=1∞‍(λtβ)nΓ(βn+1)ξnyn-1(n-1)!=λξtβe-ξyEβ,1(λtβ)∑l=0∞‍(λξtβ)ll!Γ(βl+β+1).
Moreover, one can check that
(2.35)∫0∞‍fY¯β(y,t)dy=1-1Eβ,1(λtβ),
and this completes the proof.Remark 2.3.
Forβ=1, formula (2.33) reduces to (2.11). We note that, as happens for the standard case, the density in (2.33) is expressed in terms of a single Wright function instead of an infinite sum of generalized Mittag-Leffler functions (as for the process Yβ). Nevertheless, the presence of a Mittag-Leffler in the denominator does not allow to evaluate the equation satisfied by fY¯β.


#### 2.2.1. Asymptotic Results

The analogy with the standard case is even more evident in the asymptotic behavior of the rescaled version of (2.31). Under the assumption (1.9) for the r.v.'s Xj*, we can prove that, as h,r→0, s.t. hα/r→1 (not depending on β),
(2.36)hY¯β*(tr)=∑j=1N¯β(t/r)‍hXj*⇒𝒮αβ(t),
where 𝒮αβ is a symmetric α-stable Lévy process with μ=θ=0 and σ=((1/β)λ1/βtcos(πα/2))1/α. Indeed, the characteristic function of (2.36) can be written as
(2.37)g^hY¯β*(κ,tr)=𝔼eiκhY¯β*(t/r)=1Eβ,1(λ(tβ/rβ))∑n=0∞‍(λ(tβ/rβ)f^hX(κ))nΓ(βn+1)=Eβ,1(λf^hX(κ)(tβ/rβ))Eβ,1(λ(tβ/rβ))=[bytheassumption(1.9)]≃Eβ,1(λ(tβ/rβ)-λ(tβhα|κ|α/rβ))Eβ,1(λ(tβ/rβ))=[for(2.29)]≃exp{λ1/βtr[(1-hα|κ|α)1/β-1]}.
By considering the generalized binomial theorem, we get from (2.37) that
(2.38)g^hY¯β*(κ,tr)≃exp{λ1/βtr∑j=0∞‍(1βj)(-hα|κ|α)j}=exp{λ1/βtr[1-hα|κ|αβ+o(hα)]}.
Therefore, the limiting process is represented by the α-stable process 𝒮αβ with characteristic function e-(1/β)λ1/βt|κ|α, instead of the subordinated process 𝒮α(ℒβ(t)) obtained in the limit when considering the FPP 𝒩β; note that 𝒮α(ℒβ(t)) coincides with Z(t) in (1.10). It is clear that the dependence on β is limited to the scale parameter; the space-fractional equation satisfied by its density is therefore given by
(2.39)∂u∂t=λ1/ββ∂αu∂|y|α,u(y,0)=δ(y),y∈ℝ,t≥0,
instead of (1.12). For α=1 the density of the limiting process reduces to a Cauchy with scale parameter λ1/βt/β.


## 2.1. The Standard Case

In order to get a form of the density of the TFCPP more explicit than (1.8), we assume that the Xj's are exponentially distributed: in this case it can be expressed in terms of the generalized Mittag-Leffler function:
(2.1)Eα,δγ(x)=∑j=0∞‍(γ)(j)j!xjΓ(αj+δ),α,δ,γ∈ℂ,ℛ(α),ℛ(δ)>0,
where (x)(n)=x(x+1)⋯(x+n-1) is the rising factorial (or Pochhammer symbol). Moreover, we can obtain the fractional partial-differential equation satisfied by the density of its absolutely continuous component.Theorem 2.1.
The process(2.2)Yβ(t)=∑j=1𝒩β(t)‍Xj,t≥0,
with Xj, j=1,2…, independent and exponentially distributed with parameter ξ, has the following distribution:
(2.3)Pr{Yβ(t)≤y}=Eβ,1(-λtβ)1[0,+∞)(y)+∫-∞y‍fYβ(z,t)dz,t≥0,y∈ℝ,
where
(2.4)fYβ(y,t)=e-ξyy∑n=1∞‍(λξtβy)n(n-1)!Eβ,βn+1n+1(-λtβ)1[0,+∞)(y),t≥0.
The function fYβ(y,t) given in (2.4) satisfies the following partial differential equation:
(2.5)ξ∂β∂tβfYβ=-[λ+∂β∂tβ]∂∂yfYβ,t,y≥0,
where ∂β/∂tβ denotes the Caputo fractional derivative with the conditions
(2.6)fYβ(y,0)=0,∫0+∞‍fYβ(y,t)dy=1-Eβ,1(-λtβ).Proof.
Formula  (1.8) can be rewritten by considering that fX*n(y)=ξnyn-1e-ξy/(n-1)! and using the expression of Pr{𝒩β(t)=n} in terms of generalized Mittag-Leffler functions (see [21]), that is,
(2.7)Pr{𝒩β(t)=n}=λntnβEβ,βn+1n+1(-λtβ),n≥0.
In order to derive (2.5), we evaluate the following partial derivatives of (2.4):
(2.8)∂β∂tβfYβ=e-ξyytβ∑n=1∞‍(λξtβy)n(n-1)!n!∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn-β+1),∂∂yfYβ(y,t)=-ξe-ξyy∑n=1∞‍(λξtβy)n(n-1)!n!∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn+1)++e-ξyy2∑n=2∞‍(λξtβy)n(n-2)!n!∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn+1),∂∂y∂β∂tβfYβ(y,t)=-ξe-ξyytβ∑n=1∞‍(λξtβy)nn!(n-1)!∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn-β+1)++e-ξyy2tβ∑n=2∞‍(λξtβy)n(n-2)!n!∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn-β+1)=-ξe-ξyytβ∑n=1∞‍(λξtβy)n((n-1)!)2∑j=0∞‍(n+j-1)!(-λtβ)jj!Γ(βj+βn-β+1)+-ξe-ξyytβ∑n=1∞‍(λξtβy)nn!(n-1)!∑j=1∞‍(n+j-1)!(-λtβ)j(j-1)!Γ(βj+βn-β+1)++e-ξyy2tβ∑n=2∞‍(λξtβy)n(n-2)!(n-1)!∑j=0∞‍(n+j-1)!(-λtβ)jj!Γ(βj+βn-β+1)++e-ξyy2tβ∑n=2∞‍(λξtβy)n(n-2)!n!∑j=0∞‍(n+j-1)!(-λtβ)j(j-1)!Γ(βj+βn-β+1).
By inserting (2.8) in (2.5), the equation is satisfied. Finally, it can be easily verified that the initial condition holds. In order to check the second condition in (2.6), we integrate fYβ with respect to y:
(2.9)∫0∞‍e-ξyy∑n=1∞‍(λξtβy)n(n-1)!Eβ,βn+1n+1(-λtβ)dy=∑n=1∞‍(λξtβ)n(n-1)!(n-1)!ξnEβ,βn+1n+1(-λtβ)=∑n=0∞‍(λtβ)nEβ,βn+1n+1(-λtβ)-Eβ,1(-λtβ)=1-Eβ,1(-λtβ),
where, in the last step, we have applied formula (2.30) of [21], for u=1.


### 2.1.1. The Nonfractional Caseβ=1

From (2.4), we obtain the distribution of the standard CPP, defined as Y(t)=∑n=1N(t)‍Xj, under the assumption of exponential jumps Xj, which reads
(2.10)Pr{Y(t)≤y}=e-λt1[0,+∞)(y)+∫-∞y‍fY(z,t)dz,t≥0,y∈ℝ,
where
(2.11)fY(y,t)=e-ξy-λty∑n=1∞‍(λξty)nn!(n-1)!1[0,+∞)(y)=λξte-ξy-λtW1,2(λξty)1[0,+∞)(y),t≥0,(2.12)Wα,β(z)=∑j=0∞‍zjj!Γ(αj+β),α>-1,β,z∈ℂ,
is the Wright function. Equation (4.2.8) in [18] provides another expression of fY in terms of the modified Bessel function. The density (2.11) satisfies the following equation:
(2.13)ξ∂∂tfY=-[λ+∂∂t]∂∂yfY,
with conditions
(2.14)fY(y,0)=0,∫0+∞‍fY(y,t)dy=1-e-λt,
as can be easily verified directly.Now we recall the following subordination law presented in [6] in a more general setting:
(2.15)Yβ(t)=dY(ℒβ(t)),
where ℒβ(t), t≥0 is the inverse stable subordinator defined by (1.2). We give an explicit proof of (2.15), which will be useful to prove analogous results in the next sections. We start with the evaluation of the Laplace transform (hereafter denoted by  ·~) of Yβ(t) with respect to y: by considering the probability generating function of 𝒩β, that is,
(2.16)𝔼u𝒩β(t)=Eβ,1(-λtβ(1-u)),|u|≤1,
we get
(2.17)g~Yβ(k,t):=𝔼e-kYβ(t)=Eβ,1(-λkk+ξtβ).
Formula (2.17), Laplace transformed with respect to t, gives
(2.18)g~~Yβ(k,s):=∫0+∞‍e-stg~Yβ(k,t)dt=sβ-1(k+ξ)sβ(k+ξ)+kλ,
which can be rewritten as
(2.19)g~~Yβ(k,s)=sβ-1∫0+∞‍e-sβt𝔼e-kY(t)dt=[by(1.3)]=∫0+∞‍𝔼e-kY(z)l~β(z;s)dz,
where l~β(z;s):=∫0+∞‍e-stlβ(z,t)dt. Thus, by inverting the double Laplace transform, we get
(2.20)Pr{Yβ(t)∈dy}=∫0+∞‍Pr{Y(z)∈dy}lβ(z,t)dz.
Now it is also easy to derive (2.5), since we can write in particular from (2.20) that
(2.21)fYβ(y,t)=∫0+∞‍fY(y,z)lβ(z,t)dz
and thus we get
(2.22)∂β∂tβfYβ(y,t)=∫0+∞‍fY(y,z)∂β∂tβlβ(z,t)dz=-∫0+∞‍fY(y,z)∂∂zlβ(z,t)dz.
Indeed, it is well known that ℒβ(t) is governed by the following equation:
(2.23)∂β∂tβlβ(z,t)=-∂∂zlβ(z,t),lβ(z,0)=δ(z),z,t≥0.
By integrating by parts and applying the initial condition, (2.22) becomes
(2.24)∂β∂tβfYβ(y,t)=∫0+∞‍∂∂zfY(y,z)lβ(z,t)dz=[by(2.13)]=-1ξ∂∂y∫0+∞‍∂∂zfY(y,z)lβ(z,t)dz-λξ∂∂y∫0+∞‍fY(y,z)lβ(z,t)dz=-1ξ∂∂y∂β∂tβfYβ(y,t)-λξ∂∂yfYβ(y,t).


## 2.1.1. The Nonfractional Caseβ=1

From (2.4), we obtain the distribution of the standard CPP, defined as Y(t)=∑n=1N(t)‍Xj, under the assumption of exponential jumps Xj, which reads
(2.10)Pr{Y(t)≤y}=e-λt1[0,+∞)(y)+∫-∞y‍fY(z,t)dz,t≥0,y∈ℝ,
where
(2.11)fY(y,t)=e-ξy-λty∑n=1∞‍(λξty)nn!(n-1)!1[0,+∞)(y)=λξte-ξy-λtW1,2(λξty)1[0,+∞)(y),t≥0,(2.12)Wα,β(z)=∑j=0∞‍zjj!Γ(αj+β),α>-1,β,z∈ℂ,
is the Wright function. Equation (4.2.8) in [18] provides another expression of fY in terms of the modified Bessel function. The density (2.11) satisfies the following equation:
(2.13)ξ∂∂tfY=-[λ+∂∂t]∂∂yfY,
with conditions
(2.14)fY(y,0)=0,∫0+∞‍fY(y,t)dy=1-e-λt,
as can be easily verified directly.Now we recall the following subordination law presented in [6] in a more general setting:
(2.15)Yβ(t)=dY(ℒβ(t)),
where ℒβ(t), t≥0 is the inverse stable subordinator defined by (1.2). We give an explicit proof of (2.15), which will be useful to prove analogous results in the next sections. We start with the evaluation of the Laplace transform (hereafter denoted by  ·~) of Yβ(t) with respect to y: by considering the probability generating function of 𝒩β, that is,
(2.16)𝔼u𝒩β(t)=Eβ,1(-λtβ(1-u)),|u|≤1,
we get
(2.17)g~Yβ(k,t):=𝔼e-kYβ(t)=Eβ,1(-λkk+ξtβ).
Formula (2.17), Laplace transformed with respect to t, gives
(2.18)g~~Yβ(k,s):=∫0+∞‍e-stg~Yβ(k,t)dt=sβ-1(k+ξ)sβ(k+ξ)+kλ,
which can be rewritten as
(2.19)g~~Yβ(k,s)=sβ-1∫0+∞‍e-sβt𝔼e-kY(t)dt=[by(1.3)]=∫0+∞‍𝔼e-kY(z)l~β(z;s)dz,
where l~β(z;s):=∫0+∞‍e-stlβ(z,t)dt. Thus, by inverting the double Laplace transform, we get
(2.20)Pr{Yβ(t)∈dy}=∫0+∞‍Pr{Y(z)∈dy}lβ(z,t)dz.
Now it is also easy to derive (2.5), since we can write in particular from (2.20) that
(2.21)fYβ(y,t)=∫0+∞‍fY(y,z)lβ(z,t)dz
and thus we get
(2.22)∂β∂tβfYβ(y,t)=∫0+∞‍fY(y,z)∂β∂tβlβ(z,t)dz=-∫0+∞‍fY(y,z)∂∂zlβ(z,t)dz.
Indeed, it is well known that ℒβ(t) is governed by the following equation:
(2.23)∂β∂tβlβ(z,t)=-∂∂zlβ(z,t),lβ(z,0)=δ(z),z,t≥0.
By integrating by parts and applying the initial condition, (2.22) becomes
(2.24)∂β∂tβfYβ(y,t)=∫0+∞‍∂∂zfY(y,z)lβ(z,t)dz=[by(2.13)]=-1ξ∂∂y∫0+∞‍∂∂zfY(y,z)lβ(z,t)dz-λξ∂∂y∫0+∞‍fY(y,z)lβ(z,t)dz=-1ξ∂∂y∂β∂tβfYβ(y,t)-λξ∂∂yfYβ(y,t).


## 2.2. An Alternative Case

We consider now a different model of TFCPP, based on the alternative definition of FPP given in [4], that is,
(2.25)Pr{N¯β(t)=k}=(λtβ)kΓ(βk+1)1Eβ,1(λtβ),t,k≥0.
The process with the above state probabilities plays a crucial role in the evolution of some random motions (see [22]) and can be considered as a fractional version of the Poisson process because its probability generating function (displayed below) satisfies a fractional equation (see formula (4.5) of [4]). The distribution (2.25) can be interpreted as a weighted Poisson distribution (for the general concept of discrete weighted distribution see, e.g., [23], page 90, and the references cited therein) and, as explained in [8], the weights that do not depend on t; actually we have
(2.26)Pr{N¯β(t)=k}=wkpk(tβ)∑j≥0wjpj(tβ),t,k≥0,
where wj=j!/Γ(βj+1),  j=0,1,… (for all t) and pj(t)=((λt)j/j!)e-λt, j=0,1,… are the distribution of the standard Poisson process N with intensity λ. We also recall [24] where one can find a sample path version of the weighted Poisson process.We remark that the corresponding process is not Markovian, as𝒩β, and moreover is not a renewal. Nevertheless, it is, for some aspects, more similar to the standard Poisson process N than 𝒩β. For example, the rate of the asymptotic behavior of its moments is the same as for N.The moment generating function is given by(2.27)𝔼eθN¯β(t)=Eβ,1(λtβeθ)Eβ,1(λtβ),
so that we get
(2.28)𝔼N¯β(t)=λtββEβ,β(λtβ)Eβ,1(λtβ).
By applying the following asymptotic formula of the Mittag-Leffler function
(2.29)Eβ,ν(z)≃1βz(1-ν)/βexp{z1/β},asz→∞,
(see, for example, [25] or [26]) we get
(2.30)𝔼N¯β(t)≃1βλ1/βt,ast→∞,
while for 𝒩β the mean value behaves asymptotically as tβ.We define the alternative TFCPP as(2.31)Y¯β(t)=∑j=1N¯β(t)‍Xj,t≥0,β∈(0,1],
where again Xj’s are i.i.d. with exponential distribution, independent from N¯β. Under this assumption we obtain the following result on the distribution of Y¯β.Theorem 2.2.
The processY¯β defined in (2.31), with Xj, j=1,2,…, independent and exponentially distributed with parameter ξ, has the following distribution:
(2.32)Pr{Y¯β(t)≤y}=1Eβ,1(λtβ)1[0,+∞)(y)+∫-∞y‍fY¯β(z,t)dz,t≥0,y∈ℝ,
where
(2.33)fY¯β(y,t)=λξtβe-ξyEβ,1(λtβ)Wβ,β+1(λξtβy)1[0,+∞)(y),t≥0.Proof.
The density (2.33) can be obtained as follows:
(2.34)fY¯β(y,t)=e-ξyEβ,1(λtβ)∑n=1∞‍(λtβ)nΓ(βn+1)ξnyn-1(n-1)!=λξtβe-ξyEβ,1(λtβ)∑l=0∞‍(λξtβ)ll!Γ(βl+β+1).
Moreover, one can check that
(2.35)∫0∞‍fY¯β(y,t)dy=1-1Eβ,1(λtβ),
and this completes the proof.Remark 2.3.
Forβ=1, formula (2.33) reduces to (2.11). We note that, as happens for the standard case, the density in (2.33) is expressed in terms of a single Wright function instead of an infinite sum of generalized Mittag-Leffler functions (as for the process Yβ). Nevertheless, the presence of a Mittag-Leffler in the denominator does not allow to evaluate the equation satisfied by fY¯β.


### 2.2.1. Asymptotic Results

The analogy with the standard case is even more evident in the asymptotic behavior of the rescaled version of (2.31). Under the assumption (1.9) for the r.v.'s Xj*, we can prove that, as h,r→0, s.t. hα/r→1 (not depending on β),
(2.36)hY¯β*(tr)=∑j=1N¯β(t/r)‍hXj*⇒𝒮αβ(t),
where 𝒮αβ is a symmetric α-stable Lévy process with μ=θ=0 and σ=((1/β)λ1/βtcos(πα/2))1/α. Indeed, the characteristic function of (2.36) can be written as
(2.37)g^hY¯β*(κ,tr)=𝔼eiκhY¯β*(t/r)=1Eβ,1(λ(tβ/rβ))∑n=0∞‍(λ(tβ/rβ)f^hX(κ))nΓ(βn+1)=Eβ,1(λf^hX(κ)(tβ/rβ))Eβ,1(λ(tβ/rβ))=[bytheassumption(1.9)]≃Eβ,1(λ(tβ/rβ)-λ(tβhα|κ|α/rβ))Eβ,1(λ(tβ/rβ))=[for(2.29)]≃exp{λ1/βtr[(1-hα|κ|α)1/β-1]}.
By considering the generalized binomial theorem, we get from (2.37) that
(2.38)g^hY¯β*(κ,tr)≃exp{λ1/βtr∑j=0∞‍(1βj)(-hα|κ|α)j}=exp{λ1/βtr[1-hα|κ|αβ+o(hα)]}.
Therefore, the limiting process is represented by the α-stable process 𝒮αβ with characteristic function e-(1/β)λ1/βt|κ|α, instead of the subordinated process 𝒮α(ℒβ(t)) obtained in the limit when considering the FPP 𝒩β; note that 𝒮α(ℒβ(t)) coincides with Z(t) in (1.10). It is clear that the dependence on β is limited to the scale parameter; the space-fractional equation satisfied by its density is therefore given by
(2.39)∂u∂t=λ1/ββ∂αu∂|y|α,u(y,0)=δ(y),y∈ℝ,t≥0,
instead of (1.12). For α=1 the density of the limiting process reduces to a Cauchy with scale parameter λ1/βt/β.


## 2.2.1. Asymptotic Results

The analogy with the standard case is even more evident in the asymptotic behavior of the rescaled version of (2.31). Under the assumption (1.9) for the r.v.'s Xj*, we can prove that, as h,r→0, s.t. hα/r→1 (not depending on β),
(2.36)hY¯β*(tr)=∑j=1N¯β(t/r)‍hXj*⇒𝒮αβ(t),
where 𝒮αβ is a symmetric α-stable Lévy process with μ=θ=0 and σ=((1/β)λ1/βtcos(πα/2))1/α. Indeed, the characteristic function of (2.36) can be written as
(2.37)g^hY¯β*(κ,tr)=𝔼eiκhY¯β*(t/r)=1Eβ,1(λ(tβ/rβ))∑n=0∞‍(λ(tβ/rβ)f^hX(κ))nΓ(βn+1)=Eβ,1(λf^hX(κ)(tβ/rβ))Eβ,1(λ(tβ/rβ))=[bytheassumption(1.9)]≃Eβ,1(λ(tβ/rβ)-λ(tβhα|κ|α/rβ))Eβ,1(λ(tβ/rβ))=[for(2.29)]≃exp{λ1/βtr[(1-hα|κ|α)1/β-1]}.
By considering the generalized binomial theorem, we get from (2.37) that
(2.38)g^hY¯β*(κ,tr)≃exp{λ1/βtr∑j=0∞‍(1βj)(-hα|κ|α)j}=exp{λ1/βtr[1-hα|κ|αβ+o(hα)]}.
Therefore, the limiting process is represented by the α-stable process 𝒮αβ with characteristic function e-(1/β)λ1/βt|κ|α, instead of the subordinated process 𝒮α(ℒβ(t)) obtained in the limit when considering the FPP 𝒩β; note that 𝒮α(ℒβ(t)) coincides with Z(t) in (1.10). It is clear that the dependence on β is limited to the scale parameter; the space-fractional equation satisfied by its density is therefore given by
(2.39)∂u∂t=λ1/ββ∂αu∂|y|α,u(y,0)=δ(y),y∈ℝ,t≥0,
instead of (1.12). For α=1 the density of the limiting process reduces to a Cauchy with scale parameter λ1/βt/β.


## 3. Space-Fractional Compound Poisson Process

We define now a space-fractional version of the compound Poisson process (which we will indicate hereafter by SFCPP): indeed, its distribution satisfies (2.5), but with integer time derivative and fractional space derivative. We consider the standard CPP
(3.1)Y(α)(t)=∑j=1N(t)‍Xj(α),α∈(0,1],
where, as usual, N(t),t>0 is a standard Poisson process with parameter λ and the random variables Xj(α) have the following heavy tail distribution:
(3.2)fX(α)(x)=ξxα-1Eα,α(-ξxα),x>0,α∈(0,1]
for ξ>0. The Laplace transform of (3.2) is f~X(α)(k)=ξ/(kα+ξ). The distribution of Xj(α) given in (3.2) is usually called Mittag-Leffler and coincides with the geometric-stable law of index α (hereafter 𝒢𝒮α) with parameters μ=0,θ=1,andσ=[cos(πα/2)/ξ]1/α (see [27]). The density of ∑j=1n‍Xj(α) is given by
(3.3)fX(α)*n(y)=ξnyαn-1Eα,αnn(-ξyα)
with Laplace transform
(3.4)f~X(α)*n(k)=ξn(kα+ξ)n.
Note that (3.3) coincides with the density of the nth event waiting time for the fractional Poisson process 𝒩α (see [21]). It is easy to check that the variable Xj(α) displays the asymptotic behavior (1.13).Theorem 3.1.
The processY(α) defined in (3.1), with Xj(α),  j=1,2,…, independent and distributed according to (3.2), has the following distribution:
(3.5)Pr{Y(α)(t)≤y}=e-λt1[0,+∞)(y)+∫-∞y‍fY(α)(z,t)dz,t≥0,y∈ℝ,
where
(3.6)fY(α)(y,t)=e-λty∑n=1∞‍(ξλtyα)nn!Eα,αnn(-ξyα)1(0,+∞)(y),t≥0.
The density (3.6) satisfies the following equation:
(3.7)ξ∂∂tfY(α)=-[λ+∂∂t]∂α∂yαfY(α),t≥0,y>0,
with conditions
(3.8)fY(α)(y,0)=0,∫0+∞‍fY(α)(y,t)dy=1-e-λt.
The following composition rule holds for the one-dimensional distribution of (3.1):
(3.9)Y(α)(t)=d𝒜α(Y(t)),
where 𝒜α(t) is the stable subordinator defined in (1.1) and Y is the standard CPP.Proof.
We start by noting that the absolutely continuous part of the distribution is defined in(0,∞), with the exclusion of y=0, where only the discrete component gives some contribution.
In order to check (3.7) we evaluate the following fractional derivatives, arguing as in the proof of Theorem 2.1:
(3.10)∂∂tfY(α)(y,t)=-λe-λty∑n=1∞‍(λξtyα)n(n-1)!n!∑j=0∞‍(n+j-1)!(-ξyα)jj!Γ(αj+αn)++e-λtyt∑n=1∞‍(λξtyα)n((n-1)!)2∑j=0∞‍(n+j-1)!(-ξyα)jj!Γ(αj+αn),∂α∂yαfY(α)(y,t)=e-λty1+α∑n=1∞‍(λξtyα)nn!(n-1)!∑j=0∞‍(n+j-1)!(-ξyα)jj!Γ(αj+αn-α),∂α∂yα∂∂tfY(α)(y,t)=-e-λty1+α∑n=2∞‍(λξtyα)nn!(n-2)!∑j=0∞‍(n+j-2)!(-ξyα)jj!Γ(αj+αn-α)++e-λty1+αt∑n=1∞‍(λξtyα)n((n-1)!)2∑j=1∞‍(n+j-2)!(-ξyα)j(j-1)!Γ(αj+αn-α)++e-λty1+αt∑n=2∞‍(λξtyα)n(n-1)!(n-2)!∑j=0∞‍(n+j-2)!(-ξyα)jj!Γ(αj+αn-α)++e-λty1+α∑n=1∞‍(λξtyα)nn!(n-1)!∑j=1∞‍(n+j-2)!(-ξyα)j(j-1)!Γ(αj+αn-α).
The initial condition is immediately satisfied by (3.6), while the second condition in (3.8) can be verified as follows:
(3.11)∫0∞‍e-kyfY(α)(y,t)dy=e-λt∑n=1∞‍(ξλt)nn!kαn∑j=0∞‍(n+j-1)!j!kαj=e-λt∑n=1∞‍1n!(ξλtkα+ξ)n=e-λt(eλtξ/(kα+ξ)-1),
which, for k=0, becomes 1-e-λt. The composition rule given in (3.9) can be verified by taking the Laplace transform of  Y(α),
(3.12)g~Y(α)(k,t):=𝔼e-kY(α)(t)=e-(λkα/(kα+ξ))t,
which Laplace transformed with respect to t gets
(3.13)g~~Y(α)(k,s):=∫0∞‍e-stg~Y(α)(k,t)dt=kα+ξkα(λ+s)+sξ=∫0+∞‍𝔼e-kαY(z)e-szdz.
Thus,
(3.14)g~Y(α)(k,t)=𝔼e-kαY(t)=∫0+∞‍e-kαvPr{Y(t)∈dv},
so that, by (1.1), we get
(3.15)Pr{Y(α)(t)∈dy}=∫0+∞‍hα(y,v)Pr{Y(t)∈dv}dy,
and formula (3.9) follows.Remark 3.2.
Equation (3.15) yields an alternative proof of (3.7) noting that the density of 𝒜αλ,ξ satisfies the following equation (where the space-fractional derivative is defined now in the Caputo sense):
(3.16)∂u∂t=-∂αu∂yα,u(y,0)=δ(y),y,t≥0.
Indeed, we get
(3.17)∂α∂yαfY(α)(y,t)=∫0+∞‍∂α∂yαhα(y,v)fY(v,t)dv=-∫0+∞‍∂∂vhα(y,v)fY(v,t)dv=∫0+∞‍hα(y,v)∂∂vfY(v,t)dv=[by(2.5)]=-ξλ∂∂t∫0+∞‍hα(y,v)fY(v,t)dv-1λ∂∂t∫0+∞‍hα(y,v)∂∂vfY(v,t)dv=-ξλ∂∂tfY(α)(y,t)-1λ∂∂t∂α∂yαfY(α)(y,t).
By considering (3.9) together with (1.2), we can write the following relationship:
(3.18)FY(α)(t)(z)=Pr{Y(α)(t)≤z}=Pr{𝒜α(Y(t))≤z}=Pr{Y(t)≥ℒα(z)},
while for the first version of TFCPP we had, from (2.15), that FYβ(t)(z)=Pr{Y(ℒβ(t))≤z}.
We finally note that the processY(α) is still a Markovian and Lévy process, since it is substantially a special case of CPP.


### 3.1. Special Cases

Forα=1, since the Xj's reduce to exponential r.v.'s, from (3.6) and (3.7) we retrieve the results (2.11) and (2.13) valid for the standard CPP, under the exponential assumption for Xj's. As a direct check of (3.9), we can consider the special case α=1/2, so that the law h1/2(·,z) can be written explicitly as the density of the first passage time of a standard Brownian motion through the level  z>0. Then by considering (3.15) we can write
(3.19)Pr{Y1/2(t)∈dy}=∫0+∞‍h1/2(y,v)fY(v,t)dvdy=∫0+∞‍ze-z2/2y2πy3e-ξz-λtz∑n=1∞‍(λξtz)nn!(n-1)!dzdy=e-λty∑n=1∞‍(λξt)nn!(n-1)!(-1)ndndξn∫0+∞‍e-z2/2y2πye-ξzdzdy=e-λt2y∑n=1∞‍(λξt)nn!(n-1)!(-1)ndndξnE1/2,1(-ξy1/2)dy,
where the last equality holds by (2.11)-( 2.12) in [28]; then, by (1.10.3) in [12], we get
(3.20)Pr{Y1/2(t)∈dy}=e-λt2y∑n=1∞‍(λξt)n(n-1)!yn/2n!∑j=0∞‍(n+j)!(-ξy1/2)jj!Γ(j/2+n/2+1)dy=e-λty∑n=1∞‍(λξty1/2)nn!E1/2,n/2n(-ξy1/2)dy.


### 3.2. Asymptotic Results

We study now the asymptotic behavior of the rescaled version ofY(α) defined as
(3.21)hY(α)(tr)=∑j=1N(t/r)‍hXj(α),
for h,r→0. The Fourier transform of the r.v.'s, Xj(α), for any α∈(0,1), is given by
(3.22)f^X(α)(κ)=11+(1/ξ)cos(πα/2)|κ|α(1-isgn(κ)tan(πα/2))
(see [27], formula (2.4.1)), which, in the limit, behaves as
(3.23)f^X(α)(hκ)≃1-Ahα|κ|α,h→0,
where A=(1/ξ)cos(πα/2)(1-isgn(κ)tan(πα/2)). Thus, the characteristic function of (3.21) can be written as
(3.24)g^hY(α)(κ,tr)=eλ(t/r)[f^X(α)(hκ)-1]≃e-(λt/ξ)cos(πα/2)|κ|α(1-isgn(κ)tan(πα/2)),α∈(0,1),
for h,r→0, s.t. hα/r→1.We can conclude that
(3.25)hY(α)(tr)⇒𝒜αλ,ξ(t),
where the limiting process is represented, in this case, by an α-stable subordinator 𝒜αλ,ξ(t) with parameters μ=0,  θ=1, σ=((λt/ξ)cosπα/2)1/α, whose density satisfies
(3.26)∂u∂t=-λξ∂αu∂yα,u(y,0)=δ(y),y>0,t>0.


## 3.1. Special Cases

Forα=1, since the Xj's reduce to exponential r.v.'s, from (3.6) and (3.7) we retrieve the results (2.11) and (2.13) valid for the standard CPP, under the exponential assumption for Xj's. As a direct check of (3.9), we can consider the special case α=1/2, so that the law h1/2(·,z) can be written explicitly as the density of the first passage time of a standard Brownian motion through the level  z>0. Then by considering (3.15) we can write
(3.19)Pr{Y1/2(t)∈dy}=∫0+∞‍h1/2(y,v)fY(v,t)dvdy=∫0+∞‍ze-z2/2y2πy3e-ξz-λtz∑n=1∞‍(λξtz)nn!(n-1)!dzdy=e-λty∑n=1∞‍(λξt)nn!(n-1)!(-1)ndndξn∫0+∞‍e-z2/2y2πye-ξzdzdy=e-λt2y∑n=1∞‍(λξt)nn!(n-1)!(-1)ndndξnE1/2,1(-ξy1/2)dy,
where the last equality holds by (2.11)-( 2.12) in [28]; then, by (1.10.3) in [12], we get
(3.20)Pr{Y1/2(t)∈dy}=e-λt2y∑n=1∞‍(λξt)n(n-1)!yn/2n!∑j=0∞‍(n+j)!(-ξy1/2)jj!Γ(j/2+n/2+1)dy=e-λty∑n=1∞‍(λξty1/2)nn!E1/2,n/2n(-ξy1/2)dy.


## 3.2. Asymptotic Results

We study now the asymptotic behavior of the rescaled version ofY(α) defined as
(3.21)hY(α)(tr)=∑j=1N(t/r)‍hXj(α),
for h,r→0. The Fourier transform of the r.v.'s, Xj(α), for any α∈(0,1), is given by
(3.22)f^X(α)(κ)=11+(1/ξ)cos(πα/2)|κ|α(1-isgn(κ)tan(πα/2))
(see [27], formula (2.4.1)), which, in the limit, behaves as
(3.23)f^X(α)(hκ)≃1-Ahα|κ|α,h→0,
where A=(1/ξ)cos(πα/2)(1-isgn(κ)tan(πα/2)). Thus, the characteristic function of (3.21) can be written as
(3.24)g^hY(α)(κ,tr)=eλ(t/r)[f^X(α)(hκ)-1]≃e-(λt/ξ)cos(πα/2)|κ|α(1-isgn(κ)tan(πα/2)),α∈(0,1),
for h,r→0, s.t. hα/r→1.We can conclude that
(3.25)hY(α)(tr)⇒𝒜αλ,ξ(t),
where the limiting process is represented, in this case, by an α-stable subordinator 𝒜αλ,ξ(t) with parameters μ=0,  θ=1, σ=((λt/ξ)cosπα/2)1/α, whose density satisfies
(3.26)∂u∂t=-λξ∂αu∂yα,u(y,0)=δ(y),y>0,t>0.


## 4.  Compound Poisson Processes Fractional in Time and Space

We consider now together the results obtained in the previous sections, by defining a CPP fractional both in space and time (STFCPP), that is,(4.1)Yβ(α)(t)=∑j=1𝒩β(t)‍Xj(α),t>0,
where Xj(α)'s are i.i.d. with density (3.2) and 𝒩β(t), t>0 is again the FPP.Theorem 4.1.
The processYβ(α)(t), t>0, defined in (4.1) has the following distribution:
(4.2)Pr{Yβ(α)(t)≤y}=Eβ,1(-λtβ)1[0,+∞)(y)+∫-∞y‍fYβ(α)(z,t)dz,t≥0,y∈ℝ,
where
(4.3)fYβ(α)(y,t)=1y∑n=1∞‍(λξtβyα)nEβ,βn+1n+1(-λtβ)Eα,αnn(-ξyα)1(0,+∞)(y),t≥0.
The density fYβ(α) solves the following equation:
(4.4)ξ∂β∂tβfYαβ=-[λ+∂β∂tβ]∂α∂yαfYβ(α),t≥0,y>0,
with conditions
(4.5)fYβ(α)(y,0)=0,∫0+∞‍fYβ(α)(y,t)dy=1-Eβ,1(-λtβ).
The following equality of the one-dimensional distributions holds:
(4.6)Yβ(α)(t)=d𝒮α(Yβ(t)).Proof.
In order to check (4.4) we evaluate the following fractional derivatives:
(4.7)∂∂tβfYβ(α)(y,t)=1ytβ∑n=1∞‍(λξtβyα)n(n-1)!n!(∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn-β+1))(∑r=0∞‍(n+r-1)!(-ξyα)rr!Γ(αr+αn)),∂α∂yαfYβ(α)(y,t)=1y1+α∑n=1∞‍(λξtβyα)nn!(n-1)!(∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn+1))(∑r=0∞‍(n+r-1)!(-ξyα)rr!Γ(αr+αn-α)),∂α∂yα∂∂tβfYβ(α)(y,t)=1y1+αtβ∑n=1∞‍(λξtβyα)nn!(n-1)!(∑j=0∞‍(n+j)!(-λtβ)jj!Γ(βj+βn-β+1))×(∑r=0∞‍(n+r-1)!(-ξyα)rr!Γ(αr+αn-α)).
By some algebraic manipulations we finally get (4.4). While the initial condition is trivially satisfied, the second condition in (4.5) can be checked as follows:
(4.8)∫0∞‍e-kyfYβ(α)(y,t)dy=∑n=1∞‍(λξtβ)nkαnEβ,βn+1n+1(-λtβ)∑r=0∞‍(n+r-1r)(-ξkα)r=∑n=1∞‍(λξtβkα+ξ)nEβ,βn+1n+1(-λtβ)=[by(2.30)of[21]]=Eβ,1(-λξtβkαkα+ξ)-Eβ,1(-λtβ),
which, for k=0, becomes 1-Eβ,1(-λtβ).
The relationship (4.6) can be checked by evaluating the double Laplace transform of Yβ(α) as follows:
(4.9)g~~Yβ(α)(k,s)=∫0+∞‍𝔼e-kYβ(α)(t)e-stdt=sβ-1(kα+ξ)sβ(kα+ξ)+λkα.
We then rewrite formula (4.9) as
(4.10)g~~Yβ(α)(k,s)=sβ-1∫0+∞‍e-sβz𝔼e-kαY(z)dz
and we follow the same lines which lead to (2.15) to get the conclusion.Remark 4.2.
Forα=1 formulae (4.3) and (4.4) coincide with (2.4) and (2.5), while for α=β=1 we get (3.6) and (3.7).
From (4.6), by considering (1.2), we get the following relation:
(4.11)FYβ(α)(t)(z)=Pr{Yβ(α)(t)≤z}=Pr{𝒮α(Yβ(t))≤z}=Pr{Yβ(t)≥ℒα(z)},
where ℒα is the inverse stable subordinator.


### 4.1. Asymptotic Results

For the rescaled version ofYβ(α) we obtain the following asymptotic result, which agrees with (1.14) and (1.15) proved in [17]: the characteristic function of the process
(4.12)hYβ(α)(tr)=∑j=1𝒩β(t/r)‍hXj(α)
can be written as
(4.13)g^hYβ(α)(κ,tr)=Eβ,1(λtβrβ[f^hX(α)(κ)-1]).
By applying formula (3.23) we conclude that (4.13) converges, for h,r→0 s.t. hα/rβ→1, to
(4.14)Eβ,1(-λtβξ|κ|αcosπα2(1-isgn(κ)tanπα2))
so that the process hYβ(α)(t/r) converges weakly to the α-stable subordinator 𝒜αλ,ξ(t), composed with the inverse β-stable subordinator ℒβ. Indeed, the characteristic function of 𝒜αλ,ξ(ℒβ(t)) can be evaluated as follows:
(4.15)∫0+∞‍e-stf^𝒜αλ,ξ(ℒβ)(κ,t)dt=∫0+∞‍e-stdt∫0+∞‍eiκydy∫0+∞‍pαλ,ξ(y;z)lβ(z,t)dz=sβ-1∫0+∞‍e-zλ|κ|αAe-zsβdz=sβ-1λ|κ|αA+sβ,
where hαλ,ξ(y,z) is the law of 𝒜αλ,ξ(z) and A=(1/ξ)cos(πα/2)(1-isgn(κ)tan(πα/2)). By inverting the Laplace transform in (4.15) we get (4.14). The density of 𝒜αλ,ξ(ℒβ(t)) satisfies the following equation:
(4.16)∂βu∂tβ=-λξ∂αu∂yα,y,t>0,
as can be easily seen from (4.15) (see also [29]). A relevant special case of this result can be obtained by taking α=β=ν, so that the composition 𝒜νλ,ξ(ℒν(t)) is proved to display a Lamperti-type law (see on this topic [30, 31]); therefore, the latter can be seen as the weak limit of the STFCPP.We note that in the particular caseβ=1, the Fourier transform (4.14) reduces to (3.24) and correspondingly (4.16) coincides with (3.26).Finally, we consider the case where we haveN¯β(t) in place of Nβ(t). If the jumps are Mittag-Leffler distributed, we get the following space-time fractional CPP:
(4.17)Y¯α,β(t)=∑j=1N¯β(t)‍Xj(α),
whose distribution is given by
(4.18)Pr{Y¯α,β≤y}=1Eβ,1(-λtβ)1[0,+∞)(y)+∫-∞y‍fY¯α,β(z,t)dz,t≥0,y∈ℝ,
where
(4.19)fY¯α,β(y,t)=1Eβ,1(λtβ)∑n=1∞‍(λξtβyα)nΓ(βn+1)Eα,αnn(-ξyα)1[0,+∞)(y),t≥0.
The rescaled version of (4.17) is defined as
(4.20)hY¯α,β(tr)=∑j=1N¯β(t/r)‍hXj(α)⇒𝒜αλ,ξ(t)
for h,r→0, s.t. hα/r→1, where again 𝒜αλ,ξ denotes the α-stable subordinator with characteristic function given in (3.24) (last line). Thus, in the limit, the fractional nature of the counting process N¯β does not exert any influence, in analogy with the result given in (3.25).


## 4.1. Asymptotic Results

For the rescaled version ofYβ(α) we obtain the following asymptotic result, which agrees with (1.14) and (1.15) proved in [17]: the characteristic function of the process
(4.12)hYβ(α)(tr)=∑j=1𝒩β(t/r)‍hXj(α)
can be written as
(4.13)g^hYβ(α)(κ,tr)=Eβ,1(λtβrβ[f^hX(α)(κ)-1]).
By applying formula (3.23) we conclude that (4.13) converges, for h,r→0 s.t. hα/rβ→1, to
(4.14)Eβ,1(-λtβξ|κ|αcosπα2(1-isgn(κ)tanπα2))
so that the process hYβ(α)(t/r) converges weakly to the α-stable subordinator 𝒜αλ,ξ(t), composed with the inverse β-stable subordinator ℒβ. Indeed, the characteristic function of 𝒜αλ,ξ(ℒβ(t)) can be evaluated as follows:
(4.15)∫0+∞‍e-stf^𝒜αλ,ξ(ℒβ)(κ,t)dt=∫0+∞‍e-stdt∫0+∞‍eiκydy∫0+∞‍pαλ,ξ(y;z)lβ(z,t)dz=sβ-1∫0+∞‍e-zλ|κ|αAe-zsβdz=sβ-1λ|κ|αA+sβ,
where hαλ,ξ(y,z) is the law of 𝒜αλ,ξ(z) and A=(1/ξ)cos(πα/2)(1-isgn(κ)tan(πα/2)). By inverting the Laplace transform in (4.15) we get (4.14). The density of 𝒜αλ,ξ(ℒβ(t)) satisfies the following equation:
(4.16)∂βu∂tβ=-λξ∂αu∂yα,y,t>0,
as can be easily seen from (4.15) (see also [29]). A relevant special case of this result can be obtained by taking α=β=ν, so that the composition 𝒜νλ,ξ(ℒν(t)) is proved to display a Lamperti-type law (see on this topic [30, 31]); therefore, the latter can be seen as the weak limit of the STFCPP.We note that in the particular caseβ=1, the Fourier transform (4.14) reduces to (3.24) and correspondingly (4.16) coincides with (3.26).Finally, we consider the case where we haveN¯β(t) in place of Nβ(t). If the jumps are Mittag-Leffler distributed, we get the following space-time fractional CPP:
(4.17)Y¯α,β(t)=∑j=1N¯β(t)‍Xj(α),
whose distribution is given by
(4.18)Pr{Y¯α,β≤y}=1Eβ,1(-λtβ)1[0,+∞)(y)+∫-∞y‍fY¯α,β(z,t)dz,t≥0,y∈ℝ,
where
(4.19)fY¯α,β(y,t)=1Eβ,1(λtβ)∑n=1∞‍(λξtβyα)nΓ(βn+1)Eα,αnn(-ξyα)1[0,+∞)(y),t≥0.
The rescaled version of (4.17) is defined as
(4.20)hY¯α,β(tr)=∑j=1N¯β(t/r)‍hXj(α)⇒𝒜αλ,ξ(t)
for h,r→0, s.t. hα/r→1, where again 𝒜αλ,ξ denotes the α-stable subordinator with characteristic function given in (3.24) (last line). Thus, in the limit, the fractional nature of the counting process N¯β does not exert any influence, in analogy with the result given in (3.25).


## 5. Fractional-Difference Compound Poisson Process

We present now a final version of the fractional CPP, where the fractionality of the counting process is referred to the difference operator involved in the recursive equation governing its distribution. LetB denote the standard backward shift operator, Δ=1-B, and let γ be a fractional parameter in (0,1], then the fractional recursive differential equation
(5.1)ddtpkΔ(t)=-λγΔγpkΔ(t),pkΔ(0)=1[k=0],
has been introduced in [9]. In (5.1) the following definition of the fractional difference operator Δγ of a function f(n) has been used (see [12], formula (2.8.2), page121):
(5.2)Δγf(n)=∑j=0∞‍(-1)j(γ)jj!f(n-j),
where (x)n=x(x-1)⋯(x-(n-1)) is the falling factorial. We use the notation pkΔ(t):=Pr{NΔ(t)=k},k≥0,t>0, and we have
(5.3)pkΔ(t)=(-1)kk!∑r=0∞‍(-λγt)rr!(γr)k,γ∈(0,1].
It can be proved that NΔ is not a renewal process, by verifying that the density of the kth event waiting time cannot be expressed as kth convolution of i.i.d. random variables. Nevertheless, NΔ(t) is a Lévy process, with infinite expected value for any t. Moreover, by (5.3), one can check that (as h→0)
(5.4)Pr{NΔ(h)=k}=(-1)k+1λγ(γ)kk!h+o(h),∀k≥1
instead of o(h) for k≥2, as for the standard or the time-fractional Poisson process. We can obtain (5.1) from (5.4) by taking into account that the increments are independent and stationary.Let us define the corresponding fractional-difference compound Poisson process (hereafterΔFCPP) as
(5.5)YΔ(t)=∑j=1NΔ(t)‍Xj,t≥0,γ∈(0,1]
so that we can obtain, under the assumption of i.i.d. exponential Xj's, the distribution of YΔ together with the differential equation which is satisfied by its absolutely continuous component.Theorem 5.1.
Forγ∈(0,1], the distribution of the process YΔ defined in (5.5), with Xj, j=1,2,…, independent and exponentially distributed with parameter ξ, is given by
(5.6)Pr{YΔ(t)<y}=e-λγt1[0,+∞)(y)+∫0y‍fYΔ(z,t)dz,t,y≥0,
where
(5.7)fYΔ(y,t)=e-ξyy∑n=1∞‍(-ξy)nn!(n-1)!∑r=0∞‍(-λγt)rr!(γr)n1[0,+∞)(y),t≥0.
The density fYΔ solves the differential equation:
(5.8)ξD-,t1/γfYΔ=[λ-D-,t1/γ]∂∂yfYΔ,
where D0-;t1/γ is the right-sided fractional Riemann-Liouville derivative on the half-axis ℝ+, with conditions
(5.9)fYΔ(y,0)=0,D-,trfYΔ(y,t)|t=0=Φr(y),∫0+∞‍fYΔ(y,t)dy=1-e-λγt,
where Φr(y)=(λγre-ξy/y)∑n=1∞‍((-ξy)n(γr)n/n!(n-1)!).
The following subordinating relationship holds for (5.5):
(5.10)YΔ(t)=dY(𝒜γ(t)),
where, as usual, 𝒜γ denotes the γ-stable subordinator.Proof.
Formula (5.7) can be easily derived by (5.3) and can be checked by verifying that, for γ=1, it reduces to (2.11):
(5.11)fYΔ(y,t)|γ=1=e-ξyy∑n=1∞‍(-ξy)nn!(n-1)!∑r=n∞‍(-λt)rr!(r)n=e-ξyy∑n=1∞‍(-ξy)nn!(n-1)!(-λt)ne-λt.
We now prove the relationship (5.10) as follows. The Laplace transform of YΔ(t) is given by
(5.12)g~YΔ(k,t)=𝔼e-kYΔ(t)=∑n=0∞‍(-ξ)nn!1(k+ξ)n∑r=0∞‍(-λγt)rr!(γr)n=∑r=0∞‍(-λγt)rr!∑n=0∞‍(γr)nn!(-ξk+ξ)n=∑r=0∞‍(-λγt)rr!(1-ξk+ξ)γr=e-λγkγt/(k+ξ)γ;
moreover, (5.12) can be rewritten as
(5.13)g~YΔ(k,t)=∫0+∞‍e-λzeλξz/(k+ξ)hγ(z,t)dz=∑n=0∞‍(λξ)nn!(ξ+k)n∫0+∞‍zne-λzhγ(z,t)dz=∫0+∞‍𝔼e-kY(z)hγ(z,t)dz,
which gives (5.10). Thus, we also have that
(5.14)fYΔ(y,t)=∫0+∞‍fY(y,z)hγ(z,t)dz.
In order to prove that (5.7) satisfies (5.8), we recall the following result proved in [32]: the density hν/n(y,t) of the stable subordinator 𝒜ν/n is governed by the following equation (as well as by (3.16) for α=ν/n):
(5.15)D-,tnhν/n=∂ν∂yνhν/n,y,t≥0,ν∈(0,1],
for n∈ℕ, with conditions
(5.16)hν/n(0,t)=0,hν/n(y,0)=δ(y),D-,trhν/n(y,t)|t=0=y-(νr/n)-1Γ(-rν/n),r=1,…,n-1.
We can prove that the slightly different result holds:
(5.17)D-,t1/γhγ=∂∂yhγ,y,t≥0,γ∈(0,1],n=⌊1γ⌋+1
with the following conditions
(5.18)hγ(0,t)=0,hγ(y,0)=δ(y),D-,trhγ(y,t)|t=0=y-γr-1Γ(-γr),r=1,…,n-1.
Equation (5.17) can be checked by resorting to the Laplace transform with respect to y as follows:
(5.19)D-,t1/γh~γ(k,t)=D-,t1/γe-kγt=[by(2.2.15)of[12]]=ke-kγt=∫0+∞‍e-ky∂∂yhγ(y,t)dy.
Analogously, we can check (5.18): in particular we get
(5.20)D-,trh~γ(k,t)|t=0=(-1)r∂r∂trh~γ(k,t)|t=0=kγr=∫0+∞‍e-kyy-γr-1Γ(-γr)dy.
We now take the derivative of  (5.14) of order 1/γ with respect to t:
(5.21)D-,t1/γfYΔ(y,t)=∫0+∞‍fY(y,z)D-,t1/γhγ(z,t)dz=∫0+∞‍fY(y,z)∂∂zhγ(z,t)dz=[byconsidering(2.11)]=-∫0+∞‍∂∂zfY(y,z)hγ(z,t)dz=1ξ∂∂y∫0+∞‍∂∂zfY(y,z)hγ(z,t)dz+λξ∂∂yfYΔ(y,t)=-1ξ∂∂yD-,t1/γfYΔ(y,t)+λξ∂∂yfYΔ(y,t).
We remark that, for γ=1,  D-,t1/γfYΔ=-∂fYΔ/∂t, and therefore the previous equation reduces to (2.13). Finally, we have to check (5.9): the first initial condition is trivially satisfied, while the second condition can be checked either directly by taking the derivatives of (5.7) or by noting that
(5.22)D-,trfYΔ(y,t)=∫0+∞‍fY(y,z)D-,trhγ(z,t)dz=[by(5.18)]=e-ξyy∑n=1∞‍(λξy)nn!(n-1)!∫0+∞‍e-λzzn-rγ-1Γ(-rγ)dz=λγne-ξyy∑n=1∞‍(ξy)nn!(n-1)!Γ(n-rγ)Γ(-rγ)=λγne-ξyy∑n=1∞‍(ξy)nn!(n-1)!(-γr)(n)=λγne-ξyy∑n=1∞‍(-ξy)nn!(n-1)!(γr)n,
where, in the last step, we have applied the following relationship between falling and rising factorial (x)n=(-1)n(x)(n). The last condition in (5.9) holds, since g~YΔ(0,t)=1 by (5.12), so that f~YΔ(0,t)=g~YΔ(0,t)-e-λγt=1-e-λγt.Remark 5.2.
We show that the distribution of the ΔFCPP satisfies a fractional master equation of order1/γ greater than one, when the jumps have an arbitrary continuous density fX. If we consider the (generalized) density function of YΔ(t),
(5.23)gYΔ(y,t):=e-λγtδ(y)+fYΔ(y,t),y,t≥0,
then we get
(5.24)D-,t1/γgYΔ(y,t)=λgYΔ(y,t)-λ∫-∞+∞‍gYΔ(y-x,t)fX(x)dx,
which is analogue to (1.6) for the TFCPP Yβ. Indeed, by (5.10), we can write (5.23) as
(5.25)gYΔ(y,t)=∫0+∞‍gY(y,z)hγ(z,t)dz,
where gY(y,t)=e-λtδ(y)+fY(y,t) and fY(y,t) are the density of the standard CPP. By taking the fractional time-derivative of (5.25) we get
(5.26)D-,t1/γfYΔ(y,t)=∫0+∞‍gY(y,z)D-,t1/γhγ(z,t)dz=[by(5.17)]=∫0+∞‍gY(y,z)∂∂zhγ(z,t)dz=[by(5.18)]=-∫0+∞‍∂∂zgY(y,z)hγ(z,t)dz=[bytheKolmogorovmasterequation]=λ∫0+∞‍gY(y,z)hγ(z,t)dz-λ∫-∞+∞‍fX(x)∫0+∞‍gY(y-x,z)hγ(z,t)dzdx,
which coincides with (5.24). For γ=1 (5.24) reduces to the well-known master equation of the standard CPP, by considering again that D-,t1/γfYΔ=-∂fYΔ/∂t.


### 5.1. Asymptotic Results

We study the asymptotic behavior of the rescaled version of (5.5) under the two alternative assumptions on the r.v.'s representing the jumps: for Xj* distributed according to (1.9) and for Xj(α) with density (3.2). In the first case, we have that
(5.27)hYΔ(tr)=∑j=1NΔ(t/r)‍hXj*⇒𝒮αγ(t)
for h,r→0, s.t. hαγ/r→1, where 𝒮αγ(t) is a symmetric stable process of index αγ (which is strictly less than one) and parameters μ=0, θ=0, and  σ=(tcosπαγ/2)1/αγ. Indeed, the characteristic function of (5.27) can be evaluated, by considering that the probability generating function of NΔ is G(u,t)=e-λγt(1-u)γ (see [9]) as follows:
(5.28)f^hYΔ(κ,tr)=e-(λγt/r)(1-f^hX*(κ))γ≃e-λγt|κ|αγ.
Under the assumption of Mittag-Leffler distributed Xj(α)'s, we get instead the following result: the rescaled process
(5.29)hYΔ(α)(tr)=∑j=1NΔ(t/r)‍hXj(α)⇒𝒜αγλ,ξ(t)
can be written as
(5.30)f^hYΔ(α)(κ,tr)≃exp{-λγtξγr(cosπα2)γhαγ|κ|αγ(1-isgn(κ)tanπα2)γ}≃exp{-λγtξγ|κ|αγexp{-isgn(κ)παγ2}},
for h,r→0, s.t. hαγ/r→1. The last line of (5.30) corresponds to the Fourier transform of a stable subordinator 𝒜αγλ,ξ of index αγ and with parameters μ=0,   θ=1,  and σ=((λγt/ξγ)cosπαγ/2)1/αγ. Therefore, in both cases, the limiting processes are simply the stable symmetric process and the stable subordinator of index αγ, respectively, instead of their compositions with the inverse stable subordinator as happened when 𝒩β was used as counting process.


## 5.1. Asymptotic Results

We study the asymptotic behavior of the rescaled version of (5.5) under the two alternative assumptions on the r.v.'s representing the jumps: for Xj* distributed according to (1.9) and for Xj(α) with density (3.2). In the first case, we have that
(5.27)hYΔ(tr)=∑j=1NΔ(t/r)‍hXj*⇒𝒮αγ(t)
for h,r→0, s.t. hαγ/r→1, where 𝒮αγ(t) is a symmetric stable process of index αγ (which is strictly less than one) and parameters μ=0, θ=0, and  σ=(tcosπαγ/2)1/αγ. Indeed, the characteristic function of (5.27) can be evaluated, by considering that the probability generating function of NΔ is G(u,t)=e-λγt(1-u)γ (see [9]) as follows:
(5.28)f^hYΔ(κ,tr)=e-(λγt/r)(1-f^hX*(κ))γ≃e-λγt|κ|αγ.
Under the assumption of Mittag-Leffler distributed Xj(α)'s, we get instead the following result: the rescaled process
(5.29)hYΔ(α)(tr)=∑j=1NΔ(t/r)‍hXj(α)⇒𝒜αγλ,ξ(t)
can be written as
(5.30)f^hYΔ(α)(κ,tr)≃exp{-λγtξγr(cosπα2)γhαγ|κ|αγ(1-isgn(κ)tanπα2)γ}≃exp{-λγtξγ|κ|αγexp{-isgn(κ)παγ2}},
for h,r→0, s.t. hαγ/r→1. The last line of (5.30) corresponds to the Fourier transform of a stable subordinator 𝒜αγλ,ξ of index αγ and with parameters μ=0,   θ=1,  and σ=((λγt/ξγ)cosπαγ/2)1/αγ. Therefore, in both cases, the limiting processes are simply the stable symmetric process and the stable subordinator of index αγ, respectively, instead of their compositions with the inverse stable subordinator as happened when 𝒩β was used as counting process.



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*Source: 747503-2012-10-30.xml*