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in PROBABILITY

EXPLICIT SOLUTIONS TO FRACTIONAL DIFFUSION EQUATIONS

VIA GENERALIZED GAMMA CONVOLUTION

MIRKO D’OVIDIO

Department of Statistics, Probability and Applied Statistics. Sapienza University of Rome, P.le Aldo Moro, 5 - 00185 Rome (Italy).

email: mirko.dovidio@uniroma1.it

SubmittedApril 18, 2010, accepted in final formSeptember 20, 2010 AMS 2000 Subject classification: 60J65, 60J60, 26A33

Keywords: Mellin convolution formula, generalized Gamma r.v.’s, Stable subordinators, Fox func-tions, Bessel processes, Modified Bessel functions.

Abstract

In this paper we deal with Mellin convolution of generalized Gamma densities which leads to integrals of modified Bessel functions of the second kind. Such convolutions allow us to explicitly write the solutions of the time-fractional diffusion equations involving the adjoint operators of a square Bessel process and a Bessel process.

1 Introduction and main result

In the last years, the analysis of the compositions of processes and the corresponding governing equations has received the attention of many researchers. Many of them are interested in compo-sitions involving subordinators, in other words, subordinated processesY(T(t)),t>0 (according to[9]) where T(t), t >0 is a random time with non-negative, independent and homogeneous increments (see[4]). If the random time is a (symmetric or totally skewed) stable process we have results which are strictly related to the Bochner’s subordination and the p.d.e.’s connections have been investigated, e.g., in[6; 7; 8; 22; 23]. If the random time is an inverse stable subordinator we shall refer to the governing equation ofY(T(t))as a fractional equation considering that a frac-tional time-derivative must be taken into account. In the literature, several authors have studied the solutions to space-time fractional equations. In the papers by Wyss[30], Schneider and Wyss

[29], the authors present solutions of the fractional diffusion equation∂_tλT =∂2

xT in terms of

Fox’s functions (see Section 2). In the works by Mainardi et al., see e.g.[17; 18]the authors have shown that the solutions to space-time fractional equation xDαθu= tD

∗ucan be represented by

means of Mellin-Barnes integral representations (or Fox’s functions) and M-Wright functions (see e.g. Kilbas et al.[13]). The fractional Cauchy problemDα_tu=L uhas been thoroughly studied by yet other authors and several representations of the solutions have been carried out, but an explicit form of the solutions has never been obtained. Nigmatullin[25]gave a physical derivation when Lis the generator of some continuous Markov process. Zaslavsky[31]introduced the space-time fractional kinetic equation for Hamiltonian chaos. Kochubei[14, 15]first introduced a

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ical approach while Baeumer and Meerschaert[1]established the connections between fractional problem and subordination by means of inverse stable subordinator when Lis an infinitely divis-ible generator on a finite dimensional vector space. In particular, if ∂tp = L p is the governing

equation of X(t), then under certain conditions, ∂_tβq = Lq+δ(x)t−β/Γ(1−β)is the equation governing the processX(Vt)whereVtis the inverse or hitting time process to theβ-stable

subordi-nator,β_∈(0, 1). Orsingher and Beghin[26, 27]found explicit representations of the solutions to ∂ν

t u=λ

2_∂2

xuonly in some particlular cases: ν= (1/2)

n_,1/n,_n_∈_N_and_ν₌_{1/3, 2/3, 4/3. Also,}

they represented the solutions to the fractional telegraph equations in terms of stable densities, see[3; 26]. In general, the solutions to fractional equations represent the probability densities of certain subordinated processes obtained by using a time clock (in the following we will refer to it asLν

t) which is an inverse stable subordinator (see Section 4). For a short review on this field, see

also Nane[24]and the references therein.

We will present the role of the Mellin convolution formula in finding solutions of fractional dif-fusion equations. In particular, our result allows us to write the distribution of both stable sub-ordinator and its inverse process whose governing equations are respectively space-fractional or time-fractional equations. This result turns out to be useful for representing the solutions to the following fractional diffusion equation

Dν_t ˜uγ,µ_ν =_G_γ,µu˜γ,µ_ν (1.1) where ˜uγ,µ

ν =˜uγ,µν (x,t),x>0,t>0,Dνt is the Riemann-Liouville fractional derivative,ν∈(0, 1]

and_Gγ,µis an operator to be defined below (see formula (3.4)). We present, forν=1/(2n+1), n_∈N_{∪ {}₀_}_{, the explicit solutions to (1.1) in terms of integrals of modified Bessel functions of}

the second kind (Kν) whereas, forν ∈(0, 1], we obtain the solutions to (1.1) in terms of Fox’s functions. After some preliminaries in Section 2, in Section 3 we recall the generalized Gamma densityQγ_µstarting from which we define the distributiongγ_µof the (generalized Gamma) process Gγ,µ_t and the distribution eγ_µ of the processEγ,µ_t . The latter can be seen as the reciprocal Gamma process, indeed Eγ,µ_t = 1/G_tγ,µ, or in a more striking interpretation, as the hitting time process for which (Eγ,µ_t < x) = (Gγ,µ_x > t). We shall refer to E_tγ,µ as the reciprocal or equivalently the inverse process of Gγ,µ_t . It must be noticed thateγ_µ =g_µ−γbecause G−_tγ,µ =1/Gγ,µ_t . Furthermore, we introduce the most important tool we deal with in this paper, the Mellin convolutionsg_µγ,⋆_¯ n(see formula (3.14)) ande_µ⋆_¯n(see formula (3.13)) wheree_µ⋆_¯nstands fore_µ1,⋆_¯ n. In Section 4 we draw some useful transforms of the distributionhνof the stable subordinator ˜τνt and the distributionlν of the inverse process Lν_t. Similar calculations can be found in the paper by Schneider and Wyss[29]. The inverse (or hitting time) process is defined once again from the fact that(Lν_t <x) = (τ˜ν

x>t)

(see also [1; 4]). In Section 5 we present our main contribution. We show that the following representations hold true:

hν(x,t) =e⋆µ¯n(x,ϕn+1(t)), x>0, t>0,ν=1/(n+1),n∈N and

lν(x,t) =g (n+1),⋆n

µ (x,ψn+1(t)), x>0, t>0, ν=1/(n+1), n∈N.

where ¯µ= (µ₁, . . . ,µ_n),µ_j = jν, j =1, 2, . . . ,n,ν =1/(n+1), n∈N _{and the time-stetching}

functions are given byϕm(s) = (s/m)mandψm(s) =ms1/m,s∈(0,∞),m∈N,ψ=ϕ−1.

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n_∈N_{∪ {}₀_}_{, the solutions to (1.1) can be written as follows}

uγ,µ_ν (x,t) =

Z∞

g_µγ(x,s1/γ)g_µ1/ν,⋆(1/ν_¯ −1)(s,ψ1/ν(t))ds, x∈(0,∞), t>0 where, forn_∈2N_{∪ {}₀_}_{, we have}

g_µ1/ν,⋆(1/ν_¯ −1)(x,t) = 1

ν1/2ν

π2_t3

1_4ν−νZ∞

0 . . .

Z∞

Q1−ν 2

(x,s1). . .Q1−ν 2

(sn−1,t)ds1. . .dsn−1 and

Q1−ν

2 (x,t) =K 1−ν

(x/t)1/ν_, _x_>_0, _t_>_0.

As a direct consequence of this result we obtain ˜uγ,µ₁ =˜g_µγ, forν=1, where ˜gγ_µ(x,t) =g_µγ(x,t1/γ)

and the governing equation writes ∂

∂tu˜ γ,µ 1 =

1 γ2

∂ ∂xx

2−γ ∂

∂x−(γµ−1) ∂ ∂xx

1−γ

uγ,µ₁ , x>0, t>0. Furthermore, forγ=1, 2 andν_∈(0, 1]we obtain

Dν_t˜u1,µ_ν =

x ∂ 2

∂x2−(µ−2) ∂ ∂x

u1,µ_ν , x>0, t>0, µ >0 (1.2) and

Dν_t˜u2,µ_ν = 1

∂2 ∂x2−

∂ ∂x

(2µ₋1)

u2,µ_ν , x>0, t>0, µ >0. (1.3) Equation (1.3) represents a fractional diffusion around spherical objects and thus, the solutions we deal with obey radial diffusion equations.

2 Preliminaries

The H functions were introduced by Fox[10]in 1996 as a very general class of functions. For our purpose, the Fox’s H functions will be introduced as the class of functions uniquely identified by their Mellin transforms. A function f for which the following Mellin transform exists

M[f(_·)](η) =

Z∞

xηf(x)d x

x , ℜ{η}>0 can be written in terms of H functions by observing that

Z∞

xηHm_p_,,_qn

(ai,αi)i=1,..,p

(bj,βj)j=1,..,q

d x x =M

m,n

p,q (η), ℜ{η} ∈ D (2.1)

where

Mpm,q,n(η) =

j=1Γ(bj+ηβj)

i=1Γ(1−ai−ηαi)

j=m+1Γ(1−bj−ηβj)

i=n+1Γ(ai+ηαi)

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The inverse Mellin transform is defined as f(x) = 1

2πi

Zθ+i∞

θ−i∞

M[f(_·)](η)x−ηdη

at all points xwheref is continuous and for some realθ. Thus, according to a standard notation, the Fox H function is defined as follows

H_pm_,,_qn

(a_i,α_i)_i_=1,..,_p (b_j,β_j)_j_=1,..,_q

= 1

2πi

P(D)

Mpm,q,n(η)x−

η_dη

whereP₍_D₎_{is a suitable path in the complex plane}C_{depending on the fundamental strip (}_D_{) such}

that the integral (2.1) converges. For an extensive discussion on this function see Fox[10]; Mathai and Saxena[20]. The Mellin convolution formula

f1⋆f2(x) =

Z∞

f1(x/s)f2(s) ds

s , x>0 (2.3)

turns out to be very useful later on. Formula (2.3) is a convolution in the sense that

f₁⋆f₂(_·)

(η) =_M

f₁(_·)

(η)_{× M}

f₂(_·)

(η). (2.4)

Throughout the paper we will consider the integral f₁_◦f₂(x,t) =

Z∞

f₁(x,s)f₂(s,t)ds (2.5) (for some well-defined f₁, f₂) which is not, in general, a Mellin convolution. We recall the fol-lowing connections between Mellin transform and both integer and fractional order derivatives. In particular, we consider a rapidly decreasing function f :[0,_∞)_7→[0,_∞), if there existsa_∈R

such that

lim

x→0+x

a−k−1 d

d xkf(x) =0, k=0, 1, . . . ,n−1, n∈N, x∈R+

then we have

dn d xnf(·)

(η) =(₋1)n Γ(η) Γ(η₋n)M

f(_·)

(η₋n) (2.6)

and, for 0< α <1

dα d xαf(·)

(η) = Γ(η) Γ(η₋α)M

f(_·)

(η₋α) (2.7)

(see Kilbas et al.[13]; Samko et al.[28]for details). The fractional derivative appearing in (2.7) must be understood as follows

dα

d xαf(x) = 1

Γ (n−α)

Z x

(x₋s)n−α−1d

n_f

dsn(s)ds, n−1< α <n (2.8)

that is the Dzerbayshan-Caputo sense. We also deal with the Riemann-Liouville fractional deriva-tive

Dα_xf = 1 Γ (n−α)

dn d xn

Z x

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and the fact that

Dα_xf = d

α d xαf −

n−1

k=0 dk d xkf

x=0+

xk−α

Γ(k−α+1), n−1< α <n, (2.10)

see Gorenflo and Mainardi[11]and Kilbas et al.[13]. We refer to Kilbas et al.[13]; Samko et al.

[28]for a close examination of the fractional derivatives (2.8) and (2.9).

3 Mellin convolution of generalized Gamma densities

In this section we introduce and study the Mellin convolution of generalized gamma densities. In the literature, it is well-known that generalized Gamma r.v. possess density law given by

Qγ_µ(z) =γz γµ−1

Γ µexp{−z

}, z>0,γ >0,µ >0. Our discussion here concerns the function

g_µγ(x,t) =sign(γ)1

tQ γ µ

=_|γ_| x γµ−1 tγµ_Γ(_µ₎exp

−x

γ tγ

, x>0,t>0,γ₆=0,µ >0. (3.1) Let us introduce the convolution

gγ1

µ1⋆g

γ2

µ2(x,t) =

Z∞

0 gγ1

µ1(x,s)g

γ2

µ2(s,t)ds=sign(γ1γ2)

1 t

Z∞

0 Qγ1

µ1(x/s)Q

γ2

µ2(s/t)

s (3.2)

for which we have (see formula (2.4))

gγ1

µ1⋆g

γ2

µ2(·,t) i

(η) =_M

gγ1

µ1(·,t

1/2₎i₍_η₎

× M

gγ2

µ2(·,t

1/2₎i₍_η₎ _(3.3) as a straightforward calculation shows. We now introduce the generalized Gamma process (GGP in short). Roughly speaking, the function (3.1) can be viewed as the distribution of a GGP_{Gγ,µ_t ,t> 0_}in the sense that_∀tthe distribution of the r.v. Gγ,µ_t is the generalized Gamma distribution (3.1). Thus, we make some abuse of language by considering a process without its covariance structure. In the literature there are several non-equivalent definitions of the distribution onRn

+of Gamma distributions, see e.g. Kotz et al.[16]for a comprehensive discussion. In Section 5 (Corollary 1) we will show that the distribution (3.1) satisfies the p.d.e.

∂ ∂t g

γ µ=

d(tγ)

d t Gγ,µg γ

µ, x>0,t>0 where

Gγ,µf = 1 γ2

∂ ∂xx

2−γ ∂

∂x −(γµ−1) ∂ ∂xx

1−γ

f, x>0, t>0 (3.4) andγ₆=0, f ∈D(_G_γ,µ). Forγ=1, equation (3.1) becomes the distribution of a 2µ-dimensional squared Bessel process {BESSQ(2µ)_t_/2,t > 0} and, for γ = 2 we obtain the distribution of a 2µ-dimensional Bessel process_{BES_t(2µ)_/2,t >0_}, both starting from zero. Some interesting distribu-tions can be realized through Mellin convolution of distributiong_µγ. Indeed, after some algebra we arrive at

gγ_µ

1⋆g

−γ µ2(x,t) =

γ B(µ1,µ2)

xγµ1−1_tγµ2

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and

g_µ−γ

1 ⋆g

µ2(x,t) =

γ B(µ₁,µ₂)

xγµ2−1_tγµ1

(tγ₊_xγ₎µ1+µ2, x>0,t>0,γ >0 (3.6)

whereB(_·,·)is the Beta function (see e.g. Gradshteyn and Ryzhik[12, formula 8.384]). Moreover, in light of the Mellin convolution formula (2.4), the following holds true

g_µγ

1⋆g

−γ µ2(·,t)

(η) =_M

g_µ−γ

2 ⋆g

γ µ1(·,t)

(η).

A further distribution arising from convolution can be presented. In particular, forγ₆=0, we have g_µγ

1⋆g

µ2(x,t) =

2|γ_|(xγ/tγ₎µ1+µ₂2

xΓ(µ₁)Γ(µ₂) Kµ2−µ1 2 r

xγ tγ

, x>0, t>0 (3.7) which proves to be very useful further on. The functionKνappearing in (3.7) is the modified Bessel function of imaginary argument (see e.g[12, formula 8.432]). For the sake of completeness we have writen the following Mellin transforms:

g_µγ(_·,t)

(η) = Γ

η−1 γ +µ

Γ µ t

η−1_, _t_>_0,

ℜ{η_}>1₋γµ,γ₆=0, and

Mhg_µγ(x,·)i(η) = Γ

µ₋η γ

Γ µ x

η−1_, _x_>_0,_ℜ{_η_}_{> γµ,}_γ₆₌_0. _(3.8) Formula (3.8) suggests that

gγ1

µ1⋆g

γ2

µ2(x,·) i

(η) =_M

gγ1

µ1(x

1/2 ,·)

(η)_{× M}

gγ2

µ2(x

1/2 ,·)

(η).

For the one-dimensional GGP we are able to define the inverse generalized Gamma process{Eγ,µ_t , t>0}(IGGP in short) by means of the following relation

P r{Etγ,µ<x}=P r{Gγ,µx >t}.

The density laweγ_µ=eγ_µ(x,t)of the IGGP can be carried out by observing that e_µγ(x,t) =P r{Eγ,µt ∈d x}/d x=

Z∞

∂ ∂xg

µ(s,x)ds, x>0,t>0 (3.9) and, making use of the Mellin transform, we obtain

eγ_µ(_·,t)

(η) =

Z∞

∂ ∂xg

γ µ(s,·)

(η)ds, _ℜ{η_}<1

by (2.6)

=₋(η₋1)

Z∞

g_µγ(s,_·)

(η₋1)ds

by (3.8)

=₋(η₋1)

Z∞

Γµ₋η−1 γ

Γ µ s

η−2_ds₌ Γ

µ₋η−1 γ

Γ µ t

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The derivative under the integral sign in (3.9) is allowed from the fact thatΞ1(s) = _∂∂_xgγµ(s,x)∈ L1₍_R

+)as a function ofs. From (2.2) and the fact that Hm,n

p,q

(ai,αi)i=1,..,p

(bj,βj)j=1,..,q

=c Hm,n p,q

(ai,cαi)i=1,..,p

(bj,cβj)j=1,..,q

(3.10) for allc>0 (see Mathai and Saxena[20]), we have that

eγ_µ(x,t) = γ

xH 1,0 1,1





tγ xγ

(µ, 0) (µ, 1)



, x>0,t>0,γ >0. (3.11)

By observing thatMheγ_µ(_·,t)i(1) =1, we immediately verify that (3.11) integrates to unity. The density lawgγ_µcan be expressed in terms of H functions as well, therefore we have

g_µγ(x,t) = γ

xH 1,0 1,1





xγ tγ

(µ, 0) (µ, 1)



, x>0,t>0,γ >0. (3.12)

In view of (3.11) and (3.12) we can argue that

E_tγ,µl aw= G−_tγ,µl aw= 1/G_tγ,µ, t>0,γ >0,µ >0 andeγ_µ(x,t) =g_µ−γ(x,t),γ >0,x>0,t>0.

Remark 1. We notice that the inverse process_{E1,1/2t ,t>0}can be written as

E1,1/2_t =inf_{s;B(s) =p2t_}

whereB is a standard Brownian motion. Thus,E1,1/2can be interpreted as the first-passage time of a standard Brownian motion through the levelp2t.

In what follows we will consider the Mellin convolutione⋆n

µ (x,t) =eµ1⋆. . .⋆eµn(x,t)(see formulae

(2.4) and (3.2)) where ¯µ= (µ₁, . . . ,µ_n),µ_j > 0, j = 1, 2, . . . ,nand, for the sake of simplicity, eµ(x,t) =e1µ(x,t). For the density lawe

⋆n

µ (x,t),x>0,t>0 we have

e_µ⋆_¯n(_·,t)

(η) =

j=1

eµj(·,t

1/n₎i₍_η_{) =}_tη−1

j=1

Γµ_j+1₋η

Γµ_j (3.13) with_ℜ{η_}<1. Furthermore, for the Mellin convolutiong_µ_¯γ,⋆n(x,t) =g_µγ

1⋆, . . . ,⋆g

µn(x,t)we have M

g_µγ,⋆_¯ n(_·,t)

(η) =

j=1

g_µγ

j(·,t

1/n₎i₍_η_{) =}_tη−1

j=1

η−1 γ +µj

Γµ_j (3.14) withℜ{η_}>1−minj{µj}.

Lemma 1. The functions gγ_µand e_µγare commutative under⋆-convolution.

Proof. Consider the Mellin convolution (3.13). Leteµj be the distribution of the processX

σj_{, then}

formula (3.13) means that

E{Xσ1₍_Xσ2₍_{. . .}_Xσn₍_t₎_{. . .}₎₎_}η−1₌_E¦_Xσ1₍_t1/n₎_Xσ2₍_t1/n₎_{· · ·}_Xσn₍_t1/n₎©η−1

for all possible permutations of_{σ_j_},j=1, 2, . . . ,n. The same result can be shown fore_µγ

j. Suppose

now that the processXσj _{possesses distribution}_gγ

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4 Stable subordinators

The ν-stable subordinators _{τ˜(ν)_t , t > 0}, ν _∈ (0, 1), are defined as non-decreasing, (totally) positively skewed, Lévy processes with Laplace transform

Eexp_{−λτ˜(ν_t )_}=exp_{−tλν_}, t>0, λ >0 (4.1) and characteristic function

Eexp_{iξ˜τ(ν)_t _}=exp_{−tΨν(ξ)}, ξ∈R (4.2) where

Ψν(ξ) =

Z∞

(1₋e−iξu) ν Γ (1₋ν)

du uν+1 (see Bertoin[4]; Zolotarev[32]). After some algebra we get

Ψν(ξ) =σ|ξ|ν

1₋isgn(ξ)tan

πν

=_|ξ_|νexp

−iπν 2

|ξ_|

For the density law of theν-stable subordinator_{τ˜(ν_t ),t>0_}, sayhν =hν(x,t), x>0, t>0 we have the t-Mellin transforms

Mˆhν(ξ,·)

(η) =_|ξ_|−ην_exp

iπην 2

|ξ_|

Γ η

(4.3) and

M˜hν(λ,·)

(η) =λ−ηνΓ η

(4.4) where ˆhν(ξ,t) = F

hν(·,t)

(ξ) is the Fourier transform appearing in (4.2) and ˜hν(λ,t) =

hν(·,t)

(λ) is the Laplace transform (4.1). By inverting (4.3) we obtain the Mellin trans-form with respect totof the densityhν which reads

hν(x,·)

(η) = 1

2π

e−iξxMˆh(ξ,_·)(η)dξ (4.5)

=Γ η

Γ 1₋ην

2π

(

eiπην2

(i x)1−ην +

e−iπην2

(₋i x)1−ην

)

=Γ η

Γ 1₋ην

2πx1−ην

exp

−iπ

2+iπην

+exp

iπ

2 −iπην

ªª

=Γ η

Γ 1₋ην

πx1−ην sinπην=

Γ η

Γ ηνx

ην−1_, _x_>_0,_ν

∈(0, 1)

where_ℜ{ην_{} ∈}(0, 1). Formula (4.5) can be also obtained by inverting (4.4). We are also able to evaluate the Mellin transform with respect toxof the density lawhν. From (4.3) and the fact that

Z∞

xη−1e−iξxd x= Γ(η)

(iξ)η, where (±iξ) ν₌

|ξ_|νexp

±iνπ 2

|ξ_|

, ν_∈(0, 1) (4.6) we obtain

h_ν(_·,t)

(η) =Γ η

2π

|ξ_|−ηexp

−iπη 2

|ξ_|−tΨν(ξ)

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=Γ(η)

2π

e−iπη2

Z∞

ξ−ηe−tΦν(ξ)_dξ₊_ei πη

Z∞

ξ−ηe−tΦν(−ξ)_dξ «

=Γ(η)

2πνΓ

1₋η ν

tη−ν1 ¦

eiπ(1−η)₊_e−iπ(1−η)©

=Γ

1₋η ν

tη−ν1

νΓ 1−η, ℜ{η} ∈(0, 1),t>0. (4.7)

The inversion of Fourier and Laplace transforms by making use of Mellin transform has been also treated by Schneider and Wyss[29].

We investigate the relationship between stable subordinators and their inverse processes. For a ν-stable subordinator_{τ˜(ν)_t ,t>0}and an inverse process{L(ν)_t ,t>0}(ISP in short) such that

P r{L(νt )<x}=P r{τ˜(ν)x >t}

we have the following relationship between density laws lν(x,t) =P r{L

(ν)

t ∈d x}/d x =

Z∞

∂

∂xhν(s,x)ds, x>0,t>0. (4.8) We observe that ∂

∂xhν(s,x)exists and there existsζ(s)∈L

1₍_R

+)such thatΞ2(s) = _∂∂_xhν(s,x) = const _·Dν

shν(s,x)≤ζ(s). The functionhν is the distribution of a totally skewed stable process, thushν(x),x∈Rn+belongs to the space of functions inD((−△)

ν/2₎_{, see Samko et al.}_[₂₈_]_{. Thus,} the integral in (4.8) converges. The density law (4.8) can be written in terms of Fox functions by observing that

lν(·,t)

(η) =

Z∞

∂

∂xhν(s,·)

(η)ds

by (2.6)

=₋(η₋1)

Z∞

hν(s,·)

(η₋1)ds

by (4.5)

=₋

Z∞

Γ η

Γ ην₋νs

ην−ν−1_ds

= Γ η

Γ ην₋ν+1t

ν(η−1)_,

ℜ{η_}<1/ν,t>0. (4.9) Thus, by direct inspection of (2.2), we recognize that

lν(x,t) = 1 tνH

1,0 1,1





x tν

(1₋ν,ν) (0, 1)



, x>0,t>0,ν(0, 1). (4.10)

Density (4.10) integrates to unity, indeed_M

lν(·,t)

(1) =1. Thet-Laplace transform

L[l_ν(x,_·)](λ) =λν−1exp_{−xλν_}, λ >0,ν_∈(0, 1) (4.11) comes directly from the fact that

Z∞

e−λt_M

lν(·,t)

(η)d t= Γ η

λην−ν+1=

Z∞

xη−1_L

lν(x,·)

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From (4.11) we retrieve the well-known fact that_L

lν(·,t)

(λ) =Eν(−λtν)(see also Bondesson et al.[5]) whereEβ is the Mittag-Leffler function which can be also written as

Eν(−λtν) = 1 π

Z∞

exp¦₋λ1/νt x© x

ν−1_sin_πν

1+2xν_cos_πν₊_x2νd x, t>0,λ >0. (4.12) The distribution lν satisfies the fractional equation ∂

∂tνlν = −_∂∂_xlν, x > 0, t > 0 subject to lν(x, 0) =δ(x)where the fractional derivative must be understood in the Dzerbayshan-Caputo sense (formula (2.8)). The governing equation of lν can be also presented by considering the Riemann-Liouville derivative (2.9) and the relation (2.10) (see e.g. Baeumer and Meerschaert

[1]; Meerschaert and Scheffler[21]; Baeumer et al.[2]). It is well-known that the ratio involv-ing two independent stable subordinator_{1τ˜

(ν)

t ,t>0}and{2τ˜ (ν)

t ,t>0}has a distribution,∀t,

given by

r(w) =P r{1τ˜ (ν)

t /2τ˜ (ν)

t ∈d w}/d w=

1 π

wν−1_sin_πν

1+2wν_cos_πν₊_w2ν, w>0,t>0. (4.13) Here we study the ratio of two independent inverse stable processes_{₁L(ν)t ,t>0}and{2L

(ν)

t ,t>

0_}by evaluating its Mellin transform as follows E

1L (ν)

t /2L (ν)

oη−1

=_M

lν(·,t)

(η)_{× M}

lν(·,t)

(2−η) = 1

sinνπ₋ηνπ

sinηπ (4.14) withℜ{η_{} ∈}(0, 1). By inverting (4.14) we obtain

k(x) = 1

νπ

sinνπ

1+2xcosνπ+x2= 1 2πi

Zθ+i∞

θ−i∞

sinνπ₋ηνπ sinηπ x

−η_dη _(4.15) for some realθ_∈(0, 1). From (4.13) and (4.15) we can argue that

1τ˜ (ν)

t /2τ˜ (ν)

l aw

= ₁L(νt )/2L (ν)

t , ∀t>0. (4.16)

We notice that the equivalence in law (4.16) is independent oftas the formulae (4.13) and (4.15) entail. The distributionhν◦lν(x,t)of the process{τ˜

(ν)

L(ν)t

,t>0}has Mellin transform (by making use of the formulae (4.7) and (4.9)) given by

hν◦lν(·,t)

(η) =_M

hν(·, 1)

(η)_{× M}

lν(·,t)

η₋1 ν +1

= 1

sinπη sinπ1−η

tη−1, t>0 withℜ{η_{} ∈}(0, 1). Thus, we can infer that

˜ τ(ν)

L(ν)t

l aw

= t_×1τ˜(νt )/2τ˜(ν)t t>0

andhν ◦lν(x,t) = t−1r(x/t)where r(w)is that in (4.13). For the process{L (ν) ˜ τ(ν)t

, t >0} with distributionlν◦hν(x,t)we obtain (from (4.9) and (4.7))

lν◦hν(·,t)

(η) =_M

lν(·, 1)

(η)_{× M}

hν(·,t)

(ην₋ν+1) = 1

sinπν₋πην sinπη t

η−1

withℜ{η_{} ∈}(0, 1)and thus

L(ν) ˜ τ(ν)t

l aw

= t_×₁L(ν)_t /₂L(ν)_t , t>0. We have thatlν◦hν(x,t) =t−1k(x/t)wherek(x)is that in (4.15).

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5 Main results

In this section we consider compositions of processes whose governing equations are (generalized) fractional diffusion equations. When we consider compositions involving Markov processes and stable subordinators we still have Markov processes. Here we study Markov processes with random time which is the inverse of a stable subordinator. Such a process does not belong to the family of stable subordinators (see (4.12)) and the resultant composition is not, in general, a Markov process. This somehow explains the effect of the fractional derivative appearing in the governing equation, see Mainardi et al.[19]. Hereafter, we exploit the Mellin convolution of generalized Gamma densities in order to write explicitly the solutions to fractional diffusion equations. We first present a new representation of the density lawhνby means of the convolutioneµ⋆¯nintroduced in Section 3. To do this we also introduce the time-stretching functionϕ_m(s) = (s/m)m_, _m_≥_1,

s_∈(0,_∞).

Lemma 2. The Mellin convolution e⋆n

µ (x,ϕn+1(t))whereµj= jν, for j=1, 2, . . . ,n is the density

law of aν-stable subordinator_{τ˜(ν)_t ,t>0_}withν=1/(n+1), n_∈N_{. Thus, we have}

hν(x,t) =eµ⋆¯n(x,ϕn+1(t)), x>0, t>0,ν=1/(n+1), n∈N.

Proof. From (3.13) we have that

e⋆_µ_¯n(_·,ϕ_n+1(t))

(η) =

j=1Γ

1₋η+µ_j

j=1Γ

µ_j ϕn+1(t)

η−1

. (5.1)

From Gradshteyn and Ryzhik[12, formula 8.335.3]we deduce that

k=1

k n+1

= (2π) n

n+1, n∈

N _(5.2)

and formula (5.1) reduces to

e⋆_µ_¯n(_·,ϕ_n+1(t))

(η) =

j=1Γ

1₋η+µ_j

(2π)n/2p_ν ϕn+1(t)

η−1

. (5.3)

Furthermore, by making use of the (product theorem) relation

Γ(nx) = (2π)1−2nnnx−1/2

n−1

k=0

x+k

(5.4) (see Gradshteyn and Ryzhik[12, formula 3.335]) formula (5.3) becomes

e_µ⋆_¯n(_·,ϕ_n(t))

(η) =Γ

₁₋_η

(2π)n/2₍_n₊₁₎η/ν−n

Γ 1₋η

(2π)n/2 ϕn+1(t)

η−1

= Γ

₁₋_η

νΓ 1₋ηt η−1

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In light of the last result we are able to write explicitly the density law of a stable subordinator. Forν=1/2, Lemma 2 says that

h1/2(x,t) =e⋆1µ¯ (x,ϕ2(t)) =e1/2(x,(t/2)2) =

x−1/2−1e−t

2 4x

t−1p₄_Γ1 2

, x>0,t>0 (5.5)

which is the well-known density law of the 1/2-stable subordinator or the first-passage time of a standard Brownian motion trough the levelt/p2. Forν=1/3, from (3.7), we obtain

h_1/3(x,t) =e⋆2_µ_¯(x,ϕ₃(t)) =e_1/3⋆e_2/3(x,(t/3)3) = 1

3π t3/2 x3/2K13

2 33/2

t3/2

, x>0,t>0. Forν=1/4, by (3.7) (and the commutativity under⋆, see Lemma 1), we have

h1/4(x,t) =eµ⋆3¯ (x,ϕ4(t)) =e1/4⋆e2/4⋆e3/4(x,(t/4)4) =e1/2⋆(e1/4⋆e3/4)(x,(t/4)4) whereK1/2(z) =

π/2zexp_{−z}(see[12, formula 8.469]). We notice that

e_µ⋆3_¯ (_·,ϕ₄(t))

(η) =_Mh1/2◦h1/2(·,t)

(η)

which is in line with the well-known fact that Eexp

−λ₁τ˜(ν1) 2τ˜

(ν2)

=Eexp

−λν1

2τ˜ (ν2)

=exp{−tλν1ν2_}_,

0< νi <1,i=1, 2. Forν=1/5, by exploiting twice (3.7) (and the commutativity under⋆), we

can write

h1/5(x,t) =e⋆4µ¯ (x,(t/5) 5_{) =(}_e

1/5⋆e2/5)⋆(e3/5⋆e4/5)(x,(t/5)5) (5.6)

= t

7/2

53_π2_x3/10+1

Z∞

s−2/5−1K1 5

s x

K1 5

2 55/2

t5/2

ds or equivalently

h_1/5(x,t) =e_µ_¯⋆4(x,(t/5)5) =(e_1/5⋆e_3/5)⋆(e_2/5⋆e_4/5)(x,(t/5)5) (5.7)

= t

55/2_π2_x2/5+1

Z∞

s−1/5−1K2 5

s x

K2 5

2 55/2

t5/2

ds. Forν=1/(2n+1),n_∈N_{, by using repeatedly (3.7) we arrive at}

hν(x,t) =

xν/2t1/ν−3/2 ν2−1/ν_π1/2ν−1/2K

◦n

x,(νt)1/ν, x>0,t>0 where

K◦n

ν (x,t) =

Z∞

0 . . .

Z∞

Kν(x,s1). . .Kν(sn−1,t)ds1. . .dsn−1

is the integral (2.5) (as the symbol "_◦n" denote) wherenfunctions are involved and_Kν(x,t) = x−2ν−1Kν

2pt/x

, x > 0, t > 0. We state a similar result for the density law lν and the convolution g_µγ,⋆_¯ n (see Section 3). Let us consider the time-stretching functionψm(s) = m s1/m,

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Lemma 3. The Mellin convolution g_µ(_¯n+1),⋆n(x,ψ_n+1(t)) where µj = jν, j = 1, 2, . . . ,n and ν =

1/(n+1), n∈N_{, is the density law of a}_{ν-inverse process}_{_L(ν)

t ,t>0}. Thus, we have

lν(x,t) =gµ(¯n+1),⋆n(x,ψn+1(t)), x>0, t>0,ν=1/(n+1),n∈N.

Proof. The proof can be carried out as the proof of Lemma 2. We obtain thatl1/2(x,t) =g1/22 (x, 2t

1/2_{) =}_e−x2

4t/pπt, x>0,t>0. Moreover, by making use of

(3.7) and (5.2), we have that l1/3(x,t) =g2/33 ⋆g

3 1/3(x, 3t

1/3_{) =} 1 π

x tK13

2 33/2

x3/2

, x>0, t>0 andl1/4(x,t) =g_3/44 ⋆g_2/44 ⋆g4_1/4(x, 4t1/4)follows (thank to the commutativity under⋆) from

g_3/44 ⋆g4_2/4⋆g4_1/4(x,t) =g_1/24 ⋆(g4_3/4⋆g4_1/4)(x,t) =2

3/2

π x t

Z∞

0 exp

−(s x)4₋ 2 (st)2

ds whereg_3/44 ⋆g_1/44 (x,t)is given by (3.7) andK1/2(z) =

π/2zexp_{−z_}(see[12, formula 8.469]). In a more general setting, by making use of (3.7) we can write down

g_µ1/ν_¯ ,⋆(1/ν−1)(x,t) = 1

ν1/2ν

π2_t3

1−ν 4ν

Q◦1n−ν 2

(x,t), ν=1/(2n+1),n_∈N _(5.8)

where the symbol ”_◦n” stands for the integral (2.5) wherenfunctions_Q1−ν

2 are involved and

Q1−ν

2 (x,t) =K 1−ν

(x/t)1/ν_, _x_>_0, _t_>_0. _(5.9) Now, we present the main result of this paper concerning the explicit solutions to (generalized) fractional diffusion equations. We study a generalized problem which leads to fractional diffusion equations involving the adjoint operators of both Bessel and squared Bessel processes. Let us introduce the distribution ˜uγ,µ

ν = ˜gµγ◦lν where ˜gµγ(x,t) =gµγ(x,t1/γ)and the Mellin transform of ˜

uγ,µ_ν which reads

Mu˜γ,µ_ν (_·,t)(η) = Γ

η−1 γ +µ

η−1 γ +1

Γ(µ)Γ

η−1 γ ν+1

η−1

γ ν_, ₁₋_{γµ <}_ℜ{_η_}_<₁₊_γ/ν₋_γ. _(5.10)

We state the following result.

Theorem 1. Let the previous setting prevail. Forν=1/(2n+1), n_∈N_{∪ {}₀_}_{, the solutions to}

Dν_tu˜γ,µ_ν =_Gγ,µu˜γ,µν , x>0, t>0 (5.11) can be represented in terms of generalized Gamma convolution as

uγ,µ_ν (x,t) =γx

2µ−1_ν1−4ν3ν

(π2_t3ν₎14ν−ν

Z∞

0 s4ν1−

1 4−µe−x

γ_/_s

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where_Gγ,µis the operator appearing in(3.4), vν(s,t) =

Z∞

0 . . .

Z∞

Q1−ν

2 (s,s1). . .Q 1−ν

2 (sn−1,t

ν_/ν₎

ds1. . .dsn−1 and_Q1−ν

2 is that in(5.9). Moreover, forν∈(0, 1], we have

uγ,µ_ν (x,t) = γ

x tν/γH 2,0 2,2   xγ tν

(1,ν); (µ, 0) (1, 1); (µ, 1)



 (5.13)

in terms of H Fox functions.

Proof. By exploiting the property (2.6) of the Mellin transform and the fact that

Z∞

xη−1xθf(x)d x=_M

f(_·)

(η+θ), for the operator (3.4) we have that

MGγ,µu˜γ,µν (·,t)

(η) =₋ 1

γ2(η−1)M

∂ ∂x˜u

γ,µ ν (·,t)

(η₋γ+1) + 1

γ2(γµ−1)(η−1)M

uγ,µ_ν (_·,t)(η₋γ) =1

γ2(η−1)(η−γ)M

uγ,µ_ν (_·,t)(η₋γ) + 1

γ2(γµ−1)(η−1)M

uγ,µ_ν (_·,t)(η₋γ) =1

γ2(η−1)(η−1+γµ−γ)M

uγ,µ_ν (_·,t)(η₋γ) (5.14) whereMu˜γ,µ_ν (_·,t)(η)is that in (5.10). We obtain

MGγ,µ˜uγ,µν (·,t)

(η) =1

γ2(η−1)(η−1+γµ−γ)

η−γ−1 γ +µ

η−γ−1 γ +1

Γ(µ)Γ

η−γ−1 γ ν+1

η−γ−1

γ ν

γ(η−1)

η−1 γ +µ

η−1 γ

Γ(µ)Γ

η−1

γ ν−ν+1

t η−1

γ ν−ν

= Γ

η−1 γ +µ

η−1 γ +1

Γ(µ)Γ

η−1

γ ν−ν+1

η−1

γ ν−ν₌_Dν

uγ,µ_ν (_·,t)(η)

and ˜uγ,µ_ν (x,t)solves (5.11) forν_∈(0, 1). In view of Lemma 3 we can write ˜

uγ,µ_ν (x,t) =

Z∞

0 ˜

g_µγ(x,s)g_µ1/ν,⋆(1/ν_¯ −1)(s,ψ_1/ν(t))ds

and by means of (5.8) result (5.12) appears. Formula (5.13) follows directly from (2.2) by con-sidering formula (3.10) and the fact that

Hm_p_,,_qn

(a_i,α_i)_i=1,..,p

(b_j,β_j)_j=1,..,q

= 1

xc H m,n p,q

(a_i+cα_i,α_i)_i=1,..,p

(b_j+cβ_j,β_j)_j=1,..,q

(5.15) for allc∈R_{(see Mathai and Saxena}[20]).

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We specialize the previous result by keeping in mind formula (2.10) and the operator (3.4). Corollary 1. Forν=1, Theorem 1 says that

∂ ∂tu˜

γ,µ 1 =

1 γ2

∂ ∂xx

2−γ ∂

∂x −(γµ−1) ∂ ∂xx

1−γ

uγ,µ₁ , x>0, t>0, γ₆=0 whereu˜γ,µ₁ =˜gγ

µis the distribution of the GGP.

This is becauseL1_t a=.s.t. Indeed, forν=1,Lν_t is the elementary subordinator (see[4]). Proof. Ifν=1, then the equation (5.10) takes the form

Ψ_t(η) =_M[g˜γ_µ(_·,t)](η) = Γ

η₋1 γ +µ

η−1 γ

Γ(µ), ℜ{η}>1−γµ (5.16)

where ˜g_µγ(x,t) =g_µγ(x,t1/γ). Forγ >0, we perform the time derivative of (5.16) and obtain ∂

∂tΨt(η) = η₋1

γ Γ

η₋1 γ +µ

η−γ−1 γ

=η−1

η₋γ₋1+γµ γ

η₋γ₋1 γ +µ

η−γ−1 γ

= 1

γ2(η−1)(η−γ−1+γµ)Ψt(η−γ)

which coincides with (5.14) and ˜uγ,µ₁ (x,t) =g˜_µγ(x,t),γ >0. Similar calculation must be done for γ <0 and the proof is completed.

Corollary 2. Let us writeu˜µ_ν(x,t) =u˜1,µ_ν (x,t). The distributionu˜µ_ν(x,t), x > 0, t > 0 µ > 0, ν_∈(0, 1], solves the following fractional equation

∂ν ∂tνu

µ ν =

x ∂ 2

∂x2−(µ−2) ∂ ∂x

uµ_ν. (5.17)

In particular, forν=1/2, we have ˜

uµ_1/2(x,t) = x

µ−1

πtΓ(µ)

Z∞

s−µexp

−x

s − s2 4t

ds, x>0,t>0,µ >0 which can be seen as the distribution of the process _{G1,µ

|B(2t)|, t > 0} where B is a standard

Brownian motion run at twice its usual speed andGγ,µ_t is a GGP. We notice that the processG1,µ_t is a squared Bessel process starting from zero.

Corollary 3. The distributionu˜2,µ_ν =˜u2,µ_ν (x,t), x>0, t>0,µ >0,ν_∈(0, 1]solves the following fractional equation

∂ν ∂tνu˜

2,µ ν =

1 22

∂2 ∂x2−

∂ ∂x

(2µ₋1)

˜ u2,µ_ν .

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In particular, forν=1/3, we have

u2,µ_1/3(x,t) = 2x

2µ−1 πΓ(µ)pt

Z_∞

0 e−x

sµ−1/2K13

2 33/2

s3/2

ds, x>0, t>0, µ >0 and forµ=1/2 we obtain

u2,1/2_1/3 (x,t) = 2

π3/2p_t

Z∞

0 e−x

s K1 3

2 33/2

s3/2

ds, x>0, t>0

which is the distribution of_|B(L1/3_t )_|where_|B(t)_|is a folded Brownian motion with variancet/2. AcknowledgementThe author is grateful to the anonymous referee for careful checks and com-ments.

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Lemma 3. The Mellin convolution g_µ(_¯n+1),⋆n(x,ψ_n+1(t)) where µj = jν, j = 1, 2, . . . ,n and ν =

1/(n+1), n_∈N_{, is the density law of a}_ν_{-inverse process}_{_L(ν)

t ,t>0}. Thus, we have

lν(x,t) =gµ¯(n+1),⋆n(x,ψn+1(t)), x>0, t>0,ν=1/(n+1),n∈N.

Proof. The proof can be carried out as the proof of Lemma 2. We obtain thatl1/2(x,t) =g12/2(x, 2t

1/2_{) =}_e−x2

4t/pπt, x>0,t>0. Moreover, by making use of

(3.7) and (5.2), we have that l1/3(x,t) =g23/3⋆g

3 1/3(x, 3t

1/3_{) =} 1 π

x tK13

2 33/2

x3/2 p

, x>0, t>0 andl₁/4(x,t) =g34/4⋆g

4 2/4⋆g

4 1/4(x, 4t

1/4₎_{follows (thank to the commutativity under}_⋆_{) from}

g₃4_/₄⋆g4₂_/₄⋆g4₁_/₄(x,t) =g₁4_/₂⋆(g4₃_/₄⋆g4₁_/₄)(x,t) =2

3/2

x t

Z∞

exp

−(s x)4₋ 2 (st)2

ds whereg₃4_/₄⋆g₁4_/₄(x,t)is given by (3.7) andK1/2(z) =

π/2zexp_{−z_}(see[12, formula 8.469]). In a more general setting, by making use of (3.7) we can write down

g_µ1_¯/ν,⋆(1/ν−1)(x,t) = 1

ν1/2ν

π2_t3

1−ν

4ν Q◦1n−ν

(x,t), ν=1/(2n+1),n_∈N _(5.8)

where the symbol ”_◦n” stands for the integral (2.5) wherenfunctions_Q1−ν

2 are involved and

Q1−ν

2 (x,t) =K 1−ν

(x/t)1/ν_, _x_>_0, _t_>_0. _(5.9) Now, we present the main result of this paper concerning the explicit solutions to (generalized) fractional diffusion equations. We study a generalized problem which leads to fractional diffusion equations involving the adjoint operators of both Bessel and squared Bessel processes. Let us introduce the distribution ˜uγ,µ

ν = ˜gµγ◦lν where ˜gµγ(x,t) =gµγ(x,t1/γ)and the Mellin transform of ˜

uγ,µ

ν which reads Mu˜γ_ν,µ(_·,t)(η) =

η−1

γ +µ

η−1

γ +1

Γ(µ)Γ

η−1

γ ν+1

η−1

γ ν_, ₁₋_{γµ <}_ℜ{_η_}_<₁₊_γ/ν₋_γ_. _(5.10) We state the following result.

Theorem 1. Let the previous setting prevail. Forν=1/(2n+1), n_∈N_{∪ {}₀_}_{, the solutions to}

Dν_tu˜γ_ν,µ=_Gγ_,µu˜γν,µ, x>0, t>0 (5.11) can be represented in terms of generalized Gamma convolution as

uγ_ν,µ(x,t) =γx

2µ−1_ν1−₄3νν (π2_t3ν₎14−νν

Z∞

s41ν−

1 4−µe−x

γ_/_s

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where_Gγ,µis the operator appearing in(3.4), vν(s,t) =

Z∞

. . .

Z∞

Q1−ν

2 (s,s1). . .Q 1−ν

2 (sn−1,t

ν_/ν_)ds

1. . .dsn−1

and_Q1−ν

2 is that in(5.9). Moreover, forν∈(0, 1], we have

uγ_ν,µ(x,t) = γ x tν/γH

2,0 2,2   xγ tν

(1,ν); (µ, 0) (1, 1); (µ, 1)



 (5.13)

in terms of H Fox functions.

Proof. By exploiting the property (2.6) of the Mellin transform and the fact that

Z∞

xη−1xθf(x)d x=_M

f(_·)

(η+θ), for the operator (3.4) we have that

MGγ,µu˜γν,µ(·,t)

(η) =₋ 1

γ2(η−1)M

∂ ∂x˜u

γ,µ ν (·,t)

(η₋γ+1) + 1

γ2(γµ−1)(η−1)M

uγ_ν,µ(_·,t)(η₋γ) =1

γ2(η−1)(η−γ)M

uγ_ν,µ(_·,t)(η₋γ) + 1

γ2(γµ−1)(η−1)M

uγ_ν,µ(_·,t)(η₋γ) =1

γ2(η−1)(η−1+γµ−γ)M

uγ_ν,µ(_·,t)(η₋γ) (5.14) whereMu˜γ_ν,µ(_·,t)(η)is that in (5.10). We obtain

MGγ,µ˜uγν,µ(·,t)

(η) =1

γ2(η−1)(η−1+γµ−γ)

η−γ−1

γ +µ

η−γ−1

γ +1

Γ(µ)Γ

η−γ−1

γ ν+1

η−γ−1

γ ν

γ(η−1)

η−1

γ +µ

η−1

Γ(µ)Γ

η−1

γ ν−ν+1

η−1

γ ν−ν

= Γ

η−1

γ +µ

η−1

γ +1

Γ(µ)Γ

η−1

γ ν−ν+1

η−1

γ ν−ν₌_Dν

uγ_ν,µ(_·,t)(η) and ˜uγ_ν,µ(x,t)solves (5.11) forν_∈(0, 1). In view of Lemma 3 we can write

uγ_ν,µ(x,t) =

Z∞

g_µγ(x,s)g_µ_¯1/ν,⋆(1/ν−1)(s,ψ₁_/ν(t))ds

and by means of (5.8) result (5.12) appears. Formula (5.13) follows directly from (2.2) by con-sidering formula (3.10) and the fact that

Hm,n_p,q

(a_i,α_i)_i=1,..,p

(b_j,β_j)_j=1,..,q

= 1 xc H

m,n p,q x

(a_i+cα_i,α_i)_i=1,..,p

(b_j+cβ_j,β_j)_j=1,..,q

(5.15) for allc∈R_{(see Mathai and Saxena}[20]).

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We specialize the previous result by keeping in mind formula (2.10) and the operator (3.4). Corollary 1. Forν=1, Theorem 1 says that

∂ ∂tu˜

γ,µ

1 =

γ2

∂ ∂xx

2−γ ∂

∂x −(γµ−1)

∂ ∂xx

1−γ

uγ₁,µ, x>0, t>0, γ₆=0 whereu˜γ₁,µ=˜gγ

µis the distribution of the GGP.

This is becauseL1_t a.s.= t. Indeed, forν=1,Lν_t is the elementary subordinator (see[4]). Proof. Ifν=1, then the equation (5.10) takes the form

Ψ_t(η) =_M[g˜γ_µ(_·,t)](η) = Γ

η₋1

γ +µ

t η−1

Γ(µ), ℜ{η}>1−γµ (5.16) where ˜g_µγ(x,t) =g_µγ(x,t1/γ). Forγ >0, we perform the time derivative of (5.16) and obtain

∂

∂tΨt(η) =

η₋1

γ Γ

η₋1

γ +µ

t η−γ−1

=η−1

η₋γ₋1+γµ γ

η₋γ₋1

γ +µ

t η−γ−1

= 1

γ2(η−1)(η−γ−1+γµ)Ψt(η−γ)

which coincides with (5.14) and ˜uγ₁,µ(x,t) =˜g_µγ(x,t),γ >0. Similar calculation must be done for

γ <0 and the proof is completed.

Corollary 2. Let us writeu˜µ_ν(x,t) =u˜1,_νµ(x,t). The distributionu˜µ_ν(x,t), x > 0, t > 0 µ > 0,

ν_∈(0, 1], solves the following fractional equation

∂ν ∂tνu

µ ν =

x ∂

∂x2−(µ−2) ∂ ∂x

uµ_ν. (5.17)

In particular, forν=1/2, we have ˜

uµ₁_/₂(x,t) = x µ−1

πtΓ(µ)

Z∞

s−µexp

−x s −

s2 4t

ds, x>0,t>0,µ >0 which can be seen as the distribution of the process _{G1,µ

|B(2t)|, t > 0} where B is a standard

Brownian motion run at twice its usual speed andGγ_t,µis a GGP. We notice that the processG1,_t µis a squared Bessel process starting from zero.

Corollary 3. The distributionu˜2,_νµ =˜u_ν2,µ(x,t), x>0, t>0,µ >0,ν_∈(0, 1]solves the following fractional equation

∂ν ∂tνu˜

2,µ ν =

1 22

∂2 ∂x2−

∂ ∂x

(2µ₋1) x

˜ u2,_νµ.

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In particular, forν=1/3, we have

u2,₁_/µ₃(x,t) = 2x

2µ−1 πΓ(µ)pt

Z_∞

e−x

sµ−1/2K13

2 33/2

s3/2 p

ds, x>0, t>0, µ >0 and forµ=1/2 we obtain

u2,1₁_/₃/2(x,t) = 2

π3/2p_t

Z∞

e−x

s K1 3

2 33/2

s3/2

p t

ds, x>0, t>0

which is the distribution of_|B(L1_t/3)_|where_|B(t)_|is a folded Brownian motion with variancet/2. AcknowledgementThe author is grateful to the anonymous referee for careful checks and com-ments.