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arXiv:math/0702133v1

[math.P

R]6Feb2007

FRACALMO PRE-PRINT www.fracalmo.org

Fractional Calculus and Applied Analysis, Vol. 6 No 4 (2003), pp. 441-459An International Journal for Theory and Applications ISSN 1311-0454

www.diogenes.bg/fcaa/

Mellin transform and subodination lawsin fractional diffusion processes

Francesco MAINARDI(1), Gianni PAGNINI(2) and Rudolf GORENFLO(3)

(1) Department of Physics, University of Bologna, and INFN,

Via Irnerio 46, I-40126 Bologna, ItalyCorresponding Author; E-mail: mainardi@bo.infn.it

(2)ENEA: Italian Agency for New Technologies, Energy and the Environment

Via Martiri di Monte Sole 4, I-40129 Bologna, Italy

(3) Department of Mathematics and Informatics, Free University of Berlin,

Arnimallee 3, D-14195 Berlin, Germany

This paper is dedicated to Paul Butzer, Professor Emeritus, Rheinisch-Westfalische Technische Hochschule (RWTH), Aachen, Germany, on theoccasion of his 75-th birthday (April 15, 2003)

AbstractThe Mellin transform is usually applied in probability theory to the prod-uct of independent random variables. In recent times the machinery of theMellin transform has been adopted to describe the Levy stable distributions,and more generally the probability distributions governed by generalized dif-fusion equations of fractional order in space and/or in time. In these casesthe related stochastic processes are self-similar and are simply referred to asfractional diffusion processes. In this note, by using the convolution proper-ties of the Mellin transform, we provide some (interesting) integral formulasinvolving the distributions of these processes that can be interpreted in termsof subordination laws.

2000 Mathematics Subject Classification: 26A33, 33C60, 42A38, 44A15,44A35, 60G18, 60G52.

Key Words and Phrases: Random variables, Mellin transform, Mellin-Barnes integrals, stable distributions, subordination, self-similar processes.

http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1http://arxiv.org/abs/math/0702133v1

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1. Introduction

The role of the Mellin transform in probability theory is mainly related tothe product of independent random variables: in fact it is well-known thatthe probability density of the product of two independent random variablesis given by the Mellin convolution of the two corresponding densities. Lessknown is their role with respect to the class of the Levy stable distributions,that was formerly outlined by Zolotarev [38] and Schneider [34], see also [36].A general class of probability distributions (evolving in time), that includesthe Levy strictly stable distributions, is obtained by solving, through themachinery of the Mellin transform, generalized diffusion equations of frac-tional order in space and/or in time, see [18, 26]. In these cases the relatedstochastic processes turn out as self-similar and are referred to as fractionaldiffusion processes.

In this note, after the essential notions and notations concerning theMellin transform, we first show the role of the Mellin convolution betweenprobability densities to establish subordination laws related to self-similarstochastic processes. Then, for the fractional diffusion processes we establisha sort of Mellin convolutions between the related probability densities, thatcan be interpreted as subordination laws. This is carried out starting for therepresentations through Mellin-Barnes integrals of the probability densities.

We point out that our results, being based on simple manipulations, canbe understood by non-specialists of transform methods and special functions;however they could be derived through a more general analysis involving the

class of higher transcendental functions of Fox H type to which the prob-ability densities arising as fundamental solutions of the fractional diffusionequation belong.

2. The Mellin transform

The Mellin transform of a sufficiently well-behaved function f(x), x IR+,is defined by

M{f(x); s} = f(s) =+0

f(x) xs1 dx , s IC , (2.1)

when the integral converges. Here we assume f(x) Lloc(IR+) according tothe most usual approach suitable for applied scientists. The basic proper-ties of the Mellin transform follow immediately from those of the bilateralLaplace transform since the two transforms are intimately connected.

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Recently the theory of the Mellin transform has been the object

of intensive researches by Professor Butzer and his associates, see e.g.[6, 7, 8, 9, 10, 11]; in particular Butzer and Jansche [6, 7] have introduced atheory independent from Laplace or Fourier transforms.

The integral (2.1) defines the Mellin transform in a vertical strip in thes-plane whose boundaries are determined by the analytic structure of f(x)as x 0+ and x +.

If we suppose that

f(x) =

O (x1) as x 0+ ,

O (x2) as x + ,(2.2)

for every (small) > 0 and 1 < 2, the the integral (2.1) converges abso-lutely and defines an analytic function in the strip 1 < s < 2 . This stripis known as the strip of analyticity ofM{f(x); s} = f(s)

The inversion formula for (2.1) follows directly from the correspondinginversion formula for the bilateral Laplace transform. We have

M1{f(s); x} = f(x) = 12i

+ii

f(s) xs ds , 1 < < 2 , (2.3)

at all points x 0 where f(x) is continuous.Let us now consider the most relevant Operational Rules. Denoting by

M

the juxtaposition of a function f(x) on x > 0 with its Mellin transform

f(s) , we have

xa f(x)M f(s + a) , a IC ,

f(xb)M 1|b|f

(s/b) , b IC , b = 0 ,

f(cx)M cs f(s) , c IR , c > 0 ,

from which

xa f(cxb)M 1|b| c

(s+a)/b f

s + a

b

. (2.4)

Furthermore we have

h(x) =0

fx

g() d

M f(s) g(s) = h(s) , (2.5)

which is known as the Mellin convolution formula.

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3. Subordination in stochastic processes via Mellin

convolution

In recent years a number of papers have appeared where explicitly or im-plicitly subordinated stochastic processes have been treated in view of theirrelevance in physical and financial applications, see e.g. [1, 2, 3, 30, 31, 33,35, 36, 37] and references therein. Historically, the notion of subordinationwas originated by Bochner, see [4, 5]. In brief, according to Feller [17], asubordinated process X(t) = Y(T(t)) is obtained by randomizing the timeclock of a stochastic process Y(t) using a new clock T(t), where T(t) is arandom process with non-negative independent increments. The resultingprocess Y(T(t)) is said to be subordinated to Y(t), called the parent process,

and is directed by T(t) called the directing process. The directing processis often referred to as the operational time1. In particular, assuming Y()to be a Markov process with a spatial probability density function (pdf) ofx, evolving in time , q(x) q(x; ), and T() to be a process with non-negative independent increments with pdf of depending on a parameter t,ut() u(; t), then the subordinated process X(t) = Y(T(t)) is governedby the spatial pdf ofx evolving with t, pt(x) p(x; t), given by the integralrepresentation

pt(x) =

0q(x) ut() d . (3.1)

If the parent process Y() is self-similar of the kind that its pdf q(x) issuch that

q(x) q(x; ) = q x , > 0 , (3.2)

then Eq. (3.1) reads,

pt(x) =

0q

x

ut()

d

. (3.3)

Herewith we show how to interpret Eq. (3.3) in terms of a special convolutionintegral in the framework of the theory of the Mellin transform. Later, inthe next sections, we shall show how to use the tools of the Mellin-Barnesintegral and Mellin transform to treat the subordination for the class of

1ADDED NOTE (2007). In a recent paper by R. Gorenflo, F. Mainardi and A.Vivoli: Continuous time random walk and parametric subordination in fractional dif-fusion (http://arXiv.org/abs/cond-mat/0701126), to be published in Chaos, Solitons& Fractals (2007), the Authors have given an interesting alternative interpretation ofsubordination, that they call parametric subordination.

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self-similar stochastic process, which are governed by fractional diffusion

equations.Let X1 and X2 be two real independent random variables with pdfs

p1(x1) and p2(x2) respectively, with x1 IR and x2 IR+0 . The jointprobability is

p(x1, x2) = p1(x1)p2(x2) . (3.4)

Denoting by X the random variable obtained by the product ofX1 and X2 ,

i.e. x = x1 x2 , and carrying out the transformation

x1 = x/ ,

x2 = ,(3.5)

we get the identity

p(x, ) dxd = p1(x/)p2() J dxd , (3.6)where

J =

x1x

x1x2

x2x

x2x2

=

1

x

+10 1

(3.7)is the Jacobian of the transformation (3.5). Noting that J = 1/ andintegrating (3.6) in d we finally get the pdf of X,

p(x) =

IR+

p1

x

p2()

d

, x IR . (3.8)

For = 1, by comparing with Eq. (2.5), we recover the well known resultthat the probability density of the product of two independent random vari-ables is given by the Mellin convolution of the two corresponding densities.

We now adapt Eq. (3.8) to our subordination formula (3.3) by identifyingp1, p2 with q and ut, respectively. We can now interpret the subordinationformula (3.3) as follows. The pdf of the subordinated process X, pt(x), turnsout to be the pdf of the product of the independent random variables Xqand Xu

distributed according to q(xq) and ut(xu), respectively.

4. Fractional diffusion equations and probability

distributions

An interesting way to generalize the classical diffusion equation

2

x2u(x, t) =

tu(x, t) , < x < + , t 0 , (4.1)

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is to replace in (4.1) the partial derivatives in space and time by suitable

linear integro-differential operators, to be intended as derivatives of noninteger order, that allow the corresponding Green function (see below) tobe still interpreted as a spatial probability density evolving in time with anappropriate similarity law.

The Space-Time Fractional Diffusion EquationRecalling the approach by Mainardi, Luchko and Pagnini in [26], to

which we refer the interested reader for details, it turns out that this general-ized diffusion equation, that we call space-time fractional diffusion equation,is

xD u(x, t) = tD

u(x, t) , < x < + , t 0 , (4.2)

where the , , are real parameters restricted as follows

0 < 2 , || min{, 2 } , (4.3)

0 < 1 or 1 < 2 . (4.4)Here xD

and tD

are integro-differential operators, the Riesz-Feller space

fractional derivative of order and asymmetry and the Caputo time frac-tional derivative of order , respectively.

The relevant cases of the space-time fractional diffusion equation (4.2)include, in addition to the standard case of normal diffusion{ = 2, = 1},the limiting case of the DAlembert wave equation { = 2, = 2}, thespace fractional diffusion

{0 < < 2, = 1

}, the time fractional diffusion

{ = 2, 0 < < 2}, and the neutral fractional diffusion {0 < = < 2}.When 1 < 2 we speak more properly about the fractional diffusion-waveequation in that the corresponding equation governs intermediate phenom-ena between diffusion and wave processes.

Let us now resume the essential definitions of the fractional derivativesin (4.2)-(4.4) based on their Fourier and Laplace representations.

By denoting the Fourier transform of a sufficiently well-behaved (gener-

alized) function f(x) as f() = F {f(x); } = +

e +ix f(x) dx, IR ,the Riesz-Feller space fractional derivative of order and skewness isdefined as

F { xD f(x); } = () f() ,(4.5)

() = || ei(sign )/2 , 0 < 2 , || min {, 2 } .

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In other words the symbol of the pseudo-differential operator xD is the

logarithm of the characteristic function of the generic strictly stable proba-bility density according to the Feller parameterization [16, 17], as revisitedby Gorenflo and Mainardi [21]. For this density we write

L(x)F L() = exp () , (4.6)

where is just the stability exponent (0 < 2) and is a real parameterrelated to the asymmetry (|| min {, 2}) improperly called skewness.

By denoting the Laplace transform of a sufficiently well-behaved (gen-eralized) function f(t) as f(s) = L{f(t); s} = 0 est f(t) dt , (s) > af ,the Caputo time fractional derivative of order (m 1 < m , m IN)is defined through2

LtD f(t); s

= s f(s) m1k=0

s1k f(k)(0+) , m 1 < m . (4.7)

This leads to define, see e.g. [20, 32],

tD f(t) :=

1

(m )t0

f(m)() d

(t )+1m , m 1 < < m,dm

dtmf(t) , = m .

(4.8)

The Green FunctionWhen the diffusion equations (4.1), (4.2) are equipped by the initial and

boundary conditions

u(x, 0+) = (x) , u(, t) = 0 , (4.9)

their solution reads u(x, t) =+

G(, t) (x ) d , where G(x, t) denotesthe fundamental solution (known as the Green function) corresponding to(x) = (x), the Dirac generalized function. We note that when 1 < 2

2The reader should observe that the Caputo fractional derivative introduced in[12, 13, 14] represents a sort of regularization in the time origin for the classical Riemann-

Liouville fractional derivative see e.g. [20, 32]. We note that the Caputo fractional deriva-tive coincides with that introduced (independently of Caputo) by Djrbashian & Nersesian[15], which has been adopted by Kochubei, see e.g. [23, 24] for treating initial value prob-lems in the presence of fractional derivatives. In [11] the authors have pointed out thatsuch derivative was also considered by Liouville himself, but it should be noted that itwas disregarded by Liouville who did not recognize its role.

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we must add a second initial condition ut(x, 0+) = (x), which implies

two Green functions corresponding to {u(x, 0+

) = (x), ut(x, 0+

) = 0} and{u(x, 0+) = 0, ut(x, 0+) = (x)}. Here we restrict ourselves to consider thefirst Green function because only for this it is legitimate to demand it to bea spatial probability density evolving in time, see below.

It is straightforward to derive from (4.2) the composite Fourier-Laplacetransform of the Green function by taking into account the Fourier transformfor the Riesz-Feller space fractional derivative, see (4.5),, and the Laplacetransform for the Caputo time fractional derivative, see (4.7). We have, see[26]

G,(, s) =s1

s+ (). (4.10)

By using the known scaling rules for the Fourier and Laplace transforms,we infer without inverting the two transforms,

G,(x, t) = tK,(x/t

) , = /, (4.11)

where the one-variable function K, is the reduced Green functionand x/t

is the similarity variable. We note fromG,(0, s) = 1/s G,(0, t) = 1 ,the normalization property+

G,(x, t) dx =

+

K,(x) dx = 1 ,

and, from() =

() = () ,

the symmetry relationK,(x) = K,(x) ,

allowing us to restrict our attention to x 0 .For 1 < 2 we can show, see e.g. [28], that the second Green function

is a primitive (with respect to the variable t) of the first Green function(4.11), so that, being no longer normalized in IR, cannot be interpreted asa spatial probability density.

When = 2 ( = 0) and = 1 the inversion of the Fourier-Laplacetransform in (4.10) is trivial: we recover the Gaussian density, evolving in

time with variance 2 = 2t, typical of the normal diffusion,

G02,1(x, t) =1

2

texpx2/(4t)

, x IR , t > 0 , (4.12)

which exhibits the similarity law (4.11) with = 1/2.

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For the analytical and computational determination of the reduced Green

function we restrict our attention to x > 0 because of the symmetry relation.In this range Mainardi, Luchko and Pagnini [26] have provided the Mellin-Barnes3 integral representation

K,(x) =1

x

1

2i

+ii

( s)(1 s)(1 s)(1 s) ( s)(1 s)

x s ds , =

2 ,

(4.13)where is a suitable real constant.

The Space Fractional Diffusion : {0 < < 2 , = 1}In this case we recover the class L(x) of the strictly stable (non-

Gaussian) densities exhibiting fat tails (with the algebraic decay propor-

tional to |x|(+1)

) and infinite variance,

K,1(x) = L(x) =

1

x

1

2i

+ii

(s/)(1 s)( s)(1 s) x

s ds , =

2 ,

(4.14)where 0 < < min{, 1} .

A stable pdf with extremal value of the skewness parameter is calledextremal. One can prove that all the extremal stable pdfs with 0 < < 1are one-sided, the support being IR+0 if = , and IR0 if = + . Theone-sided stable pdfs with support in IR+0 can be better characterized bytheir (spatial) Laplace transform, which turn out to be

L (s) :=

0 esx

L

(x) dx = es

, (s) > 0 , 0 < < 1 . (4.15)In terms of Mellin-Barnes integral representation we have

L (x) =1

x

1

2i

+ii

(s/)

(s)x s ds , 0 < < < 1 . (4.16)

The Time Fractional Diffusion : { = 2 , 0 < < 2}In this case we recover the class of the Wright type densities exhibiting

stretched exponential tails and finite variance proportional to t ,

K02,(x) =1

2M/2(x) =

1

2x

1

2i +i

i

(1 s)(1

s/2)

x s ds , (4.17)

3The names refer to the two authors, who in the first 1910s developed the theory ofthese integrals using them for a complete integration of the hypergeometric differentialequation. However, these integrals were first used by S. Pincherle in 1888. For a revisitedanalysis of the pioneering work by Pincherle (1853-1936, Professor of Mathematics at theUniversity of Bologna from 1880 to 1928) we refer to Mainardi and Pagnini [27].

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where 0 < < 1 . As a matter of fact 12M/2(x) turns out to be a symmetric

probability density related to the transcendental function M(z) defined forany (0, 1) and z IC as

M(z) =n=0

(z)nn! [n + (1 )] =

1

n=1

(z)n1(n 1)! (n)sin(n). (4.18)

We note that M(z) is an entire function of order = 1/(1 ) , that turnsout to be a special case of the Wright function4. Restricting our attentionto the positive real axis (r 0) we have: the Laplace transform

L{M(r); s} = E(s) , (s) > 0 , 0 < < 1 , (4.19)where E

is the Mittag-Leffler function, and the asymptotic representation

M(r) A0 Y 1/2 exp (Y) , x ,(4.20)

A0 =1

2 (1 )21 , Y = (1 ) (r)1/(1) ,

a result formerly obtained by Wright himself, and, independently, byMainardi and Tomirotti [29] by using the saddle point method. Becauseof the above exponential decay, any moment of order > 1 for M(r) isfinite and given as

0

r M(r) dr =( + 1)

( + 1)

, >

1 . (4.21)

In particular we get the normalization property in IR+,

0 M(r) dr = 1. Inview of Eqs (4.11) and (4.21) the moments (of even order) of the fundamentalsolution G02,(x, t) turn out to be, for n = 0, 1, 2, . . . and t 0 ,+

x2nG02,(x, t) dx = tn

0x2nM/2(x) dx =

(2n + 1)

(n + 1)tn. (4.22)

4The Wright function is defined by the series representation, valid in the whole complexplane,

,(z) :=

n=0zn

n! (n + ), > 1 , IC , z IC .

Then, M(z) := ,1(z) with 0 < < 1 . The function M(z) provides a generaliza-tion of the Gaussian and of the Airy function in that

M1/2(z) =1

exp z2/4

, M1/3(z) = 32/3 Ai

z/31/3

.

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We agree to call M(r) ( (0, 1), r IR+0 ) the M-Wright function oforder , understanding that its half represents the spatial pdf correspondingto the time fractional diffusion equation of order 2. Relevant propertiesof this function, see e.g. [19, 25, 26], are concerning the limit expressionfor = 1, i.e. M1(r) = (r 1) and its relation with the extremal stabledensities, i.e.

1

c1/L

r

c1/

=

c

r+1M

c

r

, 0 < < 1 , c > 0 , r > 0 . (4.23)

We note that, in both limiting cases of space fractional ( = 2) and timefractional ( = 1) diffusion, we recover the Gaussian density of the normaldiffusion, for which

K02,1(x) =1

2x

1

2i

+ii

(1 s)(1 s/2) x

s ds

= L02(x) =1

2M1/2(x) =

1

2

ex2/4 .

(4.24)

The Neutral Fractional Diffusion : {0 < = < 2}In this case, surprisingly, the corresponding (reduced) Green function

can be expressed (in explicit form) in terms of a (non-negative) simple ele-mentary function, that we denote by N(x) , as it is shown in [26]:

N(x) = 1x1

2i+ii

( s)(1 s)( s)(1 s) x

s ds

=1

x1 sin[2 ( )]1 + 2x cos[2 ( )] + x2

.

(4.25)

The Fractional Diffusion ProcessesThe self-similar stochastic processes generated by the above probability

densities evolving in time can be considered as generalizations of the stan-dard diffusion processes and therefore distinguished from it with the labelfractional. When 0 < < 1 random walk models can be introduced

to generalize the classical Brownian motion of the standard diffusion, as itwas investigated in a number of papers of our group see e.g. [21, 22]. Inthe case of space fractional diffusion we obtain a special class of Marko-vian processes, called stable Levy motions, which exhibit infinite varianceassociated to the possibility of arbitrarily large jumps (Levy flights). In

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the case of time fractional diffusion we obtain a class of stochastic pro-

cesses which are non-Markovian and exhibit a variance consistent with slowanomalous diffusion. For the general genuine space-time fractional diffusion(0 < < 2 , 0 < < 1) we generate a class of densities (symmetric ornon-symmetric according to = 0 or = 0) which exhibit fat tails with analgebraic decay |x|(+1) . Thus they belong to the domain of attraction ofthe Levy stable densities of index and can be referred to as fractional sta-ble densities. The related stochastic processes possess the characteristics ofthe previous two classes; indeed, they are non-Markovian (being 0 < < 1)and exhibit infinite variance associated to the possibility of arbitrarily largejumps (being 0 < < 2).

5. Subordination for space fractional diffusion pro-cesses

In the book by Feller, see [17], at p. 176 we read: LetX and Y be indepen-dent strictly stable variables, with characteristic exponent and respec-tively. Assume Y to be a positive variable (whence < 1). The productX Y1/ has a stable distribution with exponent .

In other words, this statement means that any strictly stable process (ofexponent = ) is subordinated to a parent strictly stable process (ofexponent ) and directed by an extremal strictly stable process (of exponent0 < < 1). Fellers proof is vague being, as a matter of fact, limited

to symmetric subordinated and parent stable distributions. Furthermore,the proof, scattered in several sections, is essentially based on the use ofFourier and Laplace transforms. Here we would like to make more precisethe previous statement by Feller by considering the possibility of asymmetrycharacterized by the index as previously explained and making use of theMellin machinery outlined in Sections 2 and 3, and of the Mellin-Barnesintegral representation (4.14). So, in virtue of the Mellin inversion formula(2.3) we can write for the generic strictly stable pdf

L(x)M 1

s1

[1 + (s 1)][1 + (s 1)][(s 1)] , =

2

. (5.1)

Let us now consider the evolution in time according to the space fractionaldiffusion equation by writing

L(x; t) := G,1(x, t) = t

1/ L

x

t1/

. (5.2)

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Then we prove the following

Theorem. Let Lpp(x; t), L

qq(x; t) and L

(x; t) be strictly stable densities

with exponents p , q , and asymmetry parameters p , q , , respectively,such that

0 < p 2 , |p| min{p, 2 p} ,0 < q 2 , |q| min{q, 2 q} ,

0 < 1 , = ,then the following subordination formula holds true for 0 < x < ,

Lpp(x; t) =

0Lqq(x; ) L

(; t) d , with p = q , p = q . (5.3)

Because of the scaling property (5.2) of the stable pdfs, we can alternativelystate

t1/pLpp

x

t1/p

=

01/qLqq

x

1/q

t1/L

t1/

d . (5.4)

The proof of Eq. (5.4) is a (straightforward) consequence of the previousconsiderations. By recalling the Mellin pairs for the involved stable densities(that can be easily obtained from Eq. (5.1)-(5.2) by adopting the correctparameters) and the scaling properties of the Mellin transform, see (2.4), wehave

t1/

pLpp xt1/p M t1/p 1t1/p

s 1

p

s1p [1 + (s 1)][1 + p(s 1)][p(s 1)]

(5.5)and

b c xa L (cxb)

M c1 s+ab 1

s+ab + 1

1 +s+ab 1

1 + s+ab 1

s+ab 1 . (5.6)

After some algebra we recognize

M{t1/p Lpp

x

t1/p

; s} = M{bxa c L (cxb); s}M{Lqq (x); s} (5.7)

provided that= , a = q 1 , b = q , c = t1/ , (5.8)

andp = q , p = q . (5.9)

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Recalling the Mellin convolution formula (2.5) we obtain from Eqs. (5.7)-

(5.9) the integral representation

t1/p Lpp

x

t1/p

=

0qq1 Lqq

x

t1/L

q

t1/

d

, (5.10)

and, by replacing 1/q , we finally get Eq. (5.4).Taking into account the relationships in Eq. (5.9) we can point out some

interesting subordination laws. In particular we observe that any symmetricstable distribution with exponent p = (0, 2) (p = = 0) is subordi-nated to the Gaussian distribution (q = 2 , q = 0), see L

02(x) in (4.24),

through an extremal stable density of exponent = /2, that is

L0(x; t) = 0

L02(x; ) L/2/2 (; t) d , 0 < < 2 . (5.11)

Furthermore, by recalling the generalized Cauchy densityof skewness (||