Explicit solutions to fractional diffusion equations via Generalized Gamma Convolution
aa r X i v : . [ m a t h . P R ] N ov Explicit solutions to fractional diffusion equations viaGeneralized Gamma Convolution
Mirko D’Ovidio ∗ May 8, 2018
Abstract
In this paper we deal with Mellin convolution of generalized Gamma densities whichleads to integrals of modified Bessel functions of the second kind. Such convolutions allow us to ex-plicitly write the solutions of the time-fractional diffusion equations involving the adjoint operators ofa square Bessel process and a Bessel process.
Keywords : Mellin convolution formula, generalized Gamma r.v.’s, Stable subordinators, Fox func-tions, Bessel processes, Modified Bessel functions.
In the last years, the analysis of the compositions of processes and the corresponding governing equa-tions has received the attention of many researchers. Many of them are interested in compositionsinvolving subordinators, in other words, subordinated processes Y ( T ( t )), t > T ( t ), t > Y ( T ( t )) as a fractional equation considering that a fractional time-derivative must betaken into account. In the literature, several authors have studied the solutions to space-time fractionalequations. In the papers by Wyss [30], Schneider and Wyss [29], the authors present solutions of thefractional diffusion equation ∂ λt T = ∂ x T in terms of Fox’s functions (see Section 2). In the works byMainardi et al., see e.g. [17; 18] the authors have shown that the solutions to space-time fractionalequation x D αθ u = t D β ∗ u can be represented by means of Mellin-Barnes integral representations (orFox’s functions) and M-Wright functions (see e.g. Kilbas et al. [13]). The fractional Cauchy problem D αt u = L u has been thoroughly studied by yet other authors and several representations of the solu-tions have been carried out, but an explicit form of the solutions has never been obtained. Nigmatullin[25] gave a physical derivation when L is 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 mathematical approach while Baeumer and Meerschaert [1] established the connec-tions between fractional problem and subordination by means of inverse stable subordinator when L is an infinitely divisible generator on a finite dimensional vector space. In particular, if ∂ t p = Lp is thegoverning equation of X ( t ), then under certain conditions, ∂ βt q = Lq + δ ( x ) t − β / Γ(1 − β ) is the equationgoverning the process X ( V t ) where V t is the inverse or hitting time process to the β -stable subordinator, β ∈ (0 , ∂ νt u = λ ∂ x u only in some particlular cases: ν = (1 / n , n ∈ N and ν = 1 / , / , /
3. Also, they represented thesolutions to the fractional telegraph equations in terms of stable densities, see [3; 26]. In general, the ∗ Department of Statistics, Probability and Applied Statistics. Sapienza University of Rome, P.le Aldo Moro, 5 -00185 Rome (Italy). e-mail: [email protected] L νt ) which is an inverse stable sub-ordinator (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 diffusionequations. In particular, our result allows us to write the distribution of both stable subordinatorand its inverse process whose governing equations are respectively space-fractional or time-fractionalequations. This result turns out to be useful for representing the solutions to the following fractionaldiffusion equation D νt ˜ u γ,µν = G γ,µ ˜ u γ,µν (1.1)where ˜ u γ,µν = ˜ u γ,µν ( x, t ), x > t > D νt is the Riemann-Liouville fractional derivative, ν ∈ (0 , G γ,µ is an operator to be defined below (see formula (3.4)). We present, for ν = 1 / (2 n + 1), n ∈ N ∪ { } , the explicit solutions to (1.1) in terms of integrals of modified Bessel functions of thesecond kind ( K ν ) whereas, for ν ∈ (0 , Q γµ startingfrom which we define the distribution g γµ of the (generalized Gamma) process G γ,µt and the distribution e γµ of the process E γ,µt . The latter can be seen as the reciprocal Gamma process, indeed E γ,µt = 1 /G γ,µt ,or in a more striking interpretation, as the hitting time process for which ( E γ,µt < x ) = ( G γ,µx > t ). Weshall refer to E γ,µt as the reciprocal or equivalently the inverse process of G γ,µt . It must be noticed that e γµ = g − γµ because G − γ,µt = 1 /G γ,µt . Furthermore, we introduce the most important tool we deal within this paper, the Mellin convolutions g γ,⋆n ¯ µ (see formula (3.14)) and e ⋆n ¯ µ (see formula (3.13)) where e ⋆n ¯ µ stands for e ,⋆n ¯ µ . In Section 4 we draw some useful transforms of the distribution h ν of the stablesubordinator ˜ τ νt and the distribution l ν of the inverse process L νt . Similar calculations can be foundin the paper by Schneider and Wyss [29]. The inverse (or hitting time) process is defined once againfrom the fact that ( L νt < x ) = (˜ τ νx > t ) (see also [1; 4]). In Section 5 we present our main contribution.We show that the following representations hold true: h ν ( x, t ) = e ⋆n ¯ µ ( x, ϕ n +1 ( t )) , x > , t > , ν = 1 / ( n + 1) , n ∈ N and l ν ( x, t ) = g ( n +1) ,⋆n ¯ µ ( x, ψ n +1 ( t )) , x > , t > , ν = 1 / ( n + 1) , n ∈ N . where ¯ µ = ( µ , . . . , µ n ), µ j = j ν , j = 1 , , . . . , n , ν = 1 / ( n + 1), n ∈ N and the time-stetching functionsare given by ϕ m ( s ) = ( s/m ) m and ψ m ( s ) = ms /m , s ∈ (0 , ∞ ), m ∈ N , ψ = ϕ − .The discussion made so far allows us to introduce the result stated in Theorem 1. For ν = 1 / ( n + 1), n ∈ N ∪ { } , the solutions to (1.1) can be written as follows˜ u γ,µν ( x, t ) = Z ∞ g γµ ( x, s /γ ) g /ν,⋆ (1 /ν − µ ( s, ψ /ν ( t )) ds, x ∈ (0 , ∞ ) , t > n ∈ N ∪ { } , we have g /ν,⋆ (1 /ν − µ ( x, t ) = 1 ν / ν (cid:16) xπ t (cid:17) − ν ν Z ∞ . . . Z ∞ Q − ν ( x, s ) . . . Q − ν ( s n − , t ) ds . . . ds n − and Q − ν ( x, t ) = K − ν (cid:18) q ( x/t ) /ν (cid:19) , x > , t > . As a direct consequence of this result we obtain ˜ u γ,µ = ˜ g γµ , for ν = 1, where ˜ g γµ ( x, t ) = g γµ ( x, t /γ ) andthe governing equation writes ∂∂t ˜ u γ,µ = 1 γ (cid:18) ∂∂x x − γ ∂∂x − ( γµ − ∂∂x x − γ (cid:19) ˜ u γ,µ , x > , t > . γ = 1 , ν ∈ (0 ,
1] we obtain D νt ˜ u ,µν = (cid:18) x ∂ ∂x − ( µ − ∂∂x (cid:19) ˜ u ,µν , x > , t > , µ > D νt ˜ u ,µν = 12 (cid:18) ∂ ∂x − ∂∂x (2 µ − x (cid:19) ˜ u ,µν , x > , t > , µ > . (1.3)Equation (1.3) represents a fractional diffusion around spherical objects and thus, the solutions wedeal with obey radial diffusion equations. The H functions were introduced by Fox [10] in 1996 as a very general class of functions. For ourpurpose, the Fox’s H functions will be introduced as the class of functions uniquely identified by theirMellin transforms. A function f for which the following Mellin transform exists M [ f ( · )]( η ) = Z ∞ x η f ( x ) dxx , ℜ{ η } > Z ∞ x η H m,np,q (cid:20) x (cid:12)(cid:12)(cid:12)(cid:12) ( a i , α i ) i =1 ,..,p ( b j , β j ) j =1 ,..,q (cid:21) dxx = M m,np,q ( η ) , ℜ{ η } ∈ D (2.1)where M m,np,q ( η ) = Q mj =1 Γ( b j + ηβ j ) Q ni =1 Γ(1 − a i − ηα i ) Q qj = m +1 Γ(1 − b j − ηβ j ) Q pi = n +1 Γ( a i + ηα i ) . (2.2)The inverse Mellin transform is defined as f ( x ) = 12 πi Z θ + i ∞ θ − i ∞ M [ f ( · )]( η ) x − η dη at all points x where f is continuous and for some real θ . Thus, according to a standard notation, theFox H function is defined as follows H m,np,q (cid:20) x (cid:12)(cid:12)(cid:12)(cid:12) ( a i , α i ) i =1 ,..,p ( b j , β j ) j =1 ,..,q (cid:21) = 12 πi Z P ( D ) M m,np,q ( η ) x − η dη where P ( D ) is a suitable path in the complex plane C depending on the fundamental strip ( D ) suchthat the integral (2.1) converges. For an extensive discussion on this function see Fox [10]; Mathai andSaxena [20]. The Mellin convolution formula f ⋆ f ( x ) = Z ∞ f ( x/s ) f ( s ) dss , x > M [ f ⋆ f ( · )] ( η ) = M [ f ( · )] ( η ) × M [ f ( · )] ( η ) . (2.4)Throughout the paper we will consider the integral f ◦ f ( x, t ) = Z ∞ f ( x, s ) f ( s, t ) ds (2.5)3for some well-defined f , f ) which is not, in general, a Mellin convolution. We recall the followingconnections between Mellin transform and both integer and fractional order derivatives. In particular,we consider a rapidly decreasing function f : [0 , ∞ ) [0 , ∞ ), if there exists a ∈ R such thatlim x → + x a − k − d k dx k f ( x ) = 0 , k = 0 , , . . . , n − , n ∈ N , x ∈ R + then we have M (cid:20) d n dx n f ( · ) (cid:21) ( η ) =( − n Γ( η )Γ( η − n ) M [ f ( · )] ( η − n ) (2.6)and, for 0 < α < M (cid:20) d α dx α f ( · ) (cid:21) ( 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) mustbe understood as follows d α dx α f ( x ) = 1Γ ( n − α ) Z x ( x − s ) n − α − d n fds n ( s ) ds, n − < α < n (2.8)that is the Dzerbayshan-Caputo sense. We also deal with the Riemann-Liouville fractional derivative D αx f = 1Γ ( n − α ) d n dx n Z x ( x − s ) n − α − f ( s ) ds, n − < α < n (2.9)and the fact that D αx f = d α dx α f − n − X k =0 d k dx k f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 + x k − α Γ( k − α + 1) , n − < 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). In this section we introduce and study the Mellin convolution of generalized gamma densities. In theliterature, it is well-known that generalized Gamma r.v. possess density law given by Q γµ ( z ) = γ z γµ − Γ ( µ ) exp {− z γ } , z > , γ > , µ > . Our discussion here concerns the function g γµ ( x, t ) = sign( γ ) 1 t Q γµ (cid:16) xt (cid:17) = | γ | x γµ − t γµ Γ( µ ) exp (cid:26) − x γ t γ (cid:27) , x > , t > , γ = 0 , µ > . (3.1)Let us introduce the convolution g γ µ ⋆ g γ µ ( x, t ) = Z ∞ g γ µ ( x, s ) g γ µ ( s, t ) ds = sign( γ γ ) 1 t Z ∞ Q γ µ ( x/s ) Q γ µ ( s/t ) dss (3.2)for which we have (see formula (2.4)) M (cid:2) g γ µ ⋆ g γ µ ( · , t ) (cid:3) ( η ) = M h g γ µ ( · , t / ) i ( η ) × M h g γ µ ( · , t / ) i ( η ) (3.3)4s a straightforward calculation shows. We now introduce the generalized Gamma process (GGP inshort). Roughly speaking, the function (3.1) can be viewed as the distribution of a GGP { G γ,µt , t > } in the sense that ∀ t the distribution of the r.v. G γ,µt 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 on R n + of Gammadistributions, see e.g. Kotz et al. [16] for a comprehensive discussion. In Section 5 (Corollary 1) wewill show that the distribution (3.1) satisfies the p.d.e. ∂∂t g γµ = d ( t γ ) dt G γ,µ g γµ , x > , t > G γ,µ f = 1 γ (cid:18) ∂∂x x − γ ∂∂x − ( γµ − ∂∂x x − γ (cid:19) f, x > , t > γ = 0, f ∈ D ( G γ,µ ). For γ = 1, equation (3.1) becomes the distribution of a 2 µ -dimensionalsquared Bessel process { BESSQ (2 µ ) t/ , t > } and, for γ = 2 we obtain the distribution of a 2 µ -dimensional Bessel process { BES (2 µ ) t/ , t > } , both starting from zero. Some interesting distributionscan be realized through Mellin convolution of distribution g γµ . Indeed, after some algebra we arrive at g γµ ⋆ g − γµ ( x, t ) = γB ( µ , µ ) x γµ − t γµ ( t γ + x γ ) µ + µ , x > , t > , γ > g − γµ ⋆ g γµ ( x, t ) = γB ( µ , µ ) x γµ − t γµ ( t γ + x γ ) µ + µ , x > , t > , γ > B ( · , · ) is the Beta function (see e.g. Gradshteyn and Ryzhik [12, formula 8.384]). Moreover, inlight of the Mellin convolution formula (2.4), the following holds true M (cid:2) g γµ ⋆ g − γµ ( · , t ) (cid:3) ( η ) = M (cid:2) g − γµ ⋆ g γµ ( · , t ) (cid:3) ( η ) . A further distribution arising from convolution can be presented. In particular, for γ = 0, we have g γµ ⋆ g γµ ( x, t ) = 2 | γ | ( x γ /t γ ) µ µ x Γ( µ )Γ( µ ) K µ − µ r x γ t γ ! , x > , t > K ν appearing in (3.7) is the modified Besselfunction of imaginary argument (see e.g [12, formula 8.432]). For the sake of completeness we havewriten the following Mellin transforms: M (cid:2) g γµ ( · , t ) (cid:3) ( η ) = Γ (cid:16) η − γ + µ (cid:17) Γ ( µ ) t η − , t > , ℜ{ η } > − γµ, γ = 0 , and M (cid:2) g γµ ( x, · ) (cid:3) ( η ) = Γ (cid:16) µ − ηγ (cid:17) Γ ( µ ) x η − , x > , ℜ{ η } > γµ, γ = 0 . (3.8)Formula (3.8) suggests that M (cid:2) g γ µ ⋆ g γ µ ( x, · ) (cid:3) ( η ) = M h g γ µ ( x / , · ) i ( η ) × M h g γ µ ( x / , · ) i ( η ) . For the one-dimensional GGP we are able to define the inverse generalized Gamma process { E γ,µt , t > } (IGGP in short) by means of the following relation P r { E γ,µt < x } = P r { G γ,µx > t } . e γµ = e γµ ( x, t ) of the IGGP can be carried out by observing that e γµ ( x, t ) = P r { E γ,µt ∈ dx } /dx = Z ∞ t ∂∂x g γµ ( s, x ) ds, x > , t > M (cid:2) e γµ ( · , t ) (cid:3) ( η ) = Z ∞ t M (cid:20) ∂∂x g γµ ( s, · ) (cid:21) ( η ) ds, ℜ{ η } <
1= [by (2.6)] = − ( η − Z ∞ t M (cid:2) g γµ ( s, · ) (cid:3) ( η − ds = [by (3.8)] = − ( η − Z ∞ t Γ (cid:16) µ − η − γ (cid:17) Γ ( µ ) s η − ds = Γ (cid:16) µ − η − γ (cid:17) Γ ( µ ) t η − The derivative under the integral sign in (3.9) is allowed from the fact that Ξ ( s ) = ∂∂x g γµ ( s, x ) ∈ L ( R + )as a function of s . From (2.2) and the fact that H m,np,q (cid:20) x (cid:12)(cid:12)(cid:12)(cid:12) ( a i , α i ) i =1 ,..,p ( b j , β j ) j =1 ,..,q (cid:21) = c H m,np,q (cid:20) x c (cid:12)(cid:12)(cid:12)(cid:12) ( a i , cα i ) i =1 ,..,p ( b j , cβ j ) j =1 ,..,q (cid:21) (3.10)for all c > e γµ ( x, t ) = γx H , , " t γ x γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( µ, µ, , x > , t > , γ > . (3.11)By observing that M (cid:2) e γµ ( · , t ) (cid:3) (1) = 1, we immediately verify that (3.11) integrates to unity. Thedensity law g γµ can be expressed in terms of H functions as well, therefore we have g γµ ( x, t ) = γx H , , " x γ t γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( µ, µ, , x > , t > , γ > . (3.12)In view of (3.11) and (3.12) we can argue that E γ,µt law = G − γ,µt law = 1 /G γ,µt , t > , γ > , µ > e γµ ( x, t ) = g − γµ ( x, t ), γ > x > t > Remark 1.
We notice that the inverse process { E , / t , t > } can be written as E , / t = inf { s ; B ( s ) = √ t } where B is a standard Brownian motion. Thus, E , / can be interpreted as the first-passage time ofa standard Brownian motion through the level √ t .In what follows we will consider the Mellin convolution e ⋆n ¯ µ ( x, t ) = e µ ⋆ . . . ⋆ e µ n ( x, t ) (see formulae(2.4) and (3.2)) where ¯ µ = ( µ , . . . , µ n ), µ j > j = 1 , , . . . , n and, for the sake of simplicity, e µ ( x, t ) = e µ ( x, t ). For the density law e ⋆n ¯ µ ( x, t ), x > t > M (cid:2) e ⋆n ¯ µ ( · , t ) (cid:3) ( η ) = n Y j =1 M h e µ j ( · , t /n ) i ( η ) = t η − n Y j =1 Γ ( µ j + 1 − η )Γ ( µ j ) (3.13)with ℜ{ η } <
1. Furthermore, for the Mellin convolution g γ,⋆n ¯ µ ( x, t ) = g γµ ⋆, . . . , ⋆g γµ n ( x, t ) we have M (cid:2) g γ,⋆n ¯ µ ( · , t ) (cid:3) ( η ) = n Y j =1 M h g γµ j ( · , t /n ) i ( η ) = t η − n Y j =1 Γ (cid:16) η − γ + µ j (cid:17) Γ ( µ j ) (3.14)with ℜ{ η } > − min j { µ j } . 6 emma 1. The functions g γµ and e γµ are commutative under ⋆ -convolution.Proof. Consider the Mellin convolution (3.13). Let e µ j be the distribution of the process X σ j , thenformula (3.13) means that E { X σ ( X σ ( . . . X σ n ( t ) . . . )) } η − = E n X σ ( t /n ) X σ ( t /n ) · · · X σ n ( t /n ) o η − for all possible permutations of { σ j } , j = 1 , , . . . , n . The same result can be shown for e γµ j . Supposenow that the process X σ j possesses distribution g γµ j , from (3.14) we obtain the claimed result. The ν -stable subordinators { ˜ τ ( ν ) t , t > } , ν ∈ (0 , E exp {− λ ˜ τ ( ν ) t } = exp {− tλ ν } , t > , λ > E exp { iξ ˜ τ ( ν ) t } = exp {− t Ψ ν ( ξ ) } , ξ ∈ R (4.2)where Ψ ν ( ξ ) = Z ∞ (1 − e − iξu ) ν Γ (1 − ν ) duu ν +1 (see Bertoin [4]; Zolotarev [32]). After some algebra we getΨ ν ( ξ ) = σ | ξ | ν (cid:16) − i sgn( ξ ) tan (cid:16) πν (cid:17)(cid:17) = | ξ | ν exp (cid:26) − i πν ξ | ξ | (cid:27) . For the density law of the ν -stable subordinator { ˜ τ ( ν ) t , t > } , say h ν = h ν ( x, t ), x > t > t -Mellin transforms M h ˆ h ν ( ξ, · ) i ( η ) = | ξ | − ην exp (cid:26) i πην ξ | ξ | (cid:27) Γ ( η ) (4.3)and M h ˜ h ν ( λ, · ) i ( η ) = λ − ην Γ ( η ) (4.4)where ˆ h ν ( ξ, t ) = F [ h ν ( · , t )] ( ξ ) is the Fourier transform appearing in (4.2) and ˜ h ν ( λ, t ) = L [ h ν ( · , t )] ( λ )is the Laplace transform (4.1). By inverting (4.3) we obtain the Mellin transform with respect to t ofthe density h ν which reads M [ h ν ( x, · )] ( η ) = 12 π Z R e − iξx M h ˆ h ( ξ, · ) i ( η ) dξ (4.5)= Γ ( η ) Γ (1 − ην )2 π (cid:26) e i πην ( ix ) − ην + e − i πην ( − ix ) − ην (cid:27) = Γ ( η ) Γ (1 − ην )2 π x − ην n exp n − i π iπην o + exp n i π − iπην oo = Γ ( η ) Γ (1 − ην ) π x − ην sin πην = Γ ( η )Γ ( ην ) x ην − , x > , ν ∈ (0 , ℜ{ ην } ∈ (0 , x of the density law h ν . From (4.3) and the fact that Z ∞ x η − e − iξx dx = Γ( η )( iξ ) η , where ( ± iξ ) ν = | ξ | ν exp (cid:26) ± i νπ ξ | ξ | (cid:27) , ν ∈ (0 ,
1) (4.6)7e obtain M [ h ν ( · , t )] ( η ) = Γ ( η )2 π Z R | ξ | − η exp (cid:26) − i πη ξ | ξ | − t Ψ ν ( ξ ) (cid:27) dξ = Γ( η )2 π (cid:26) e − i πη Z ∞ ξ − η e − t Φ ν ( ξ ) dξ + e i πη Z ∞ ξ − η e − t Φ ν ( − ξ ) dξ (cid:27) = Γ( η )2 πν Γ (cid:18) − ην (cid:19) t η − ν n e iπ (1 − η ) + e − iπ (1 − η ) o =Γ (cid:18) − ην (cid:19) t η − ν ν Γ (1 − η ) , ℜ{ η } ∈ (0 , , t > . (4.7)The inversion of Fourier and Laplace transforms by making use of Mellin transform has been alsotreated 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 > } and an inverse process { L ( ν ) t , t > } (ISP in short) such that P r { L ( ν ) t < x } = P r { ˜ τ ( ν ) x > t } we have the following relationship between density laws l ν ( x, t ) = P r { L ( ν ) t ∈ dx } /dx = Z ∞ t ∂∂x h ν ( s, x ) ds, x > , t > . (4.8)We observe that ∂∂x h ν ( s, x ) exists and there exists ζ ( s ) ∈ L ( R + ) such that Ξ ( s ) = ∂∂x h ν ( s, x ) = const · D νs h ν ( s, x ) ≤ ζ ( s ). The function h ν is the distribution of a totally skewed stable process,thus h ν ( x ), x ∈ R n + belongs to the space of functions in D (( −△ ) ν/ ), see Samko et al. [28]. Thus, theintegral in (4.8) converges. The density law (4.8) can be written in terms of Fox functions by observingthat M [ l ν ( · , t )] ( η ) = Z ∞ t M (cid:20) ∂∂x h ν ( s, · ) (cid:21) ( η ) ds = [by (2.6)] = − ( η − Z ∞ t M [ h ν ( s, · )] ( η − ds = [by (4.5)] = − Z ∞ t Γ ( η )Γ ( ην − ν ) s ην − ν − ds = Γ ( η )Γ ( ην − ν + 1) t ν ( η − , ℜ{ η } < /ν, t > . (4.9)Thus, by direct inspection of (2.2), we recognize that l ν ( x, t ) = 1 t ν H , , " xt ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 − ν, ν )(0 , , x > , t > , ν (0 , . (4.10)Density (4.10) integrates to unity, indeed M [ l ν ( · , t )] (1) = 1. The t -Laplace transform L [ l ν ( x, · )]( λ ) = λ ν − exp {− xλ ν } , λ > , ν ∈ (0 ,
1) (4.11)comes directly from the fact that Z ∞ e − λt M [ l ν ( · , t )] ( η ) dt = Γ ( η ) λ ην − ν +1 = Z ∞ x η − L [ l ν ( x, · )] ( λ ) dx. L [ l ν ( · , t )] ( λ ) = E ν ( − λt ν ) (see also Bondesson et al.[5]) where E β is the Mittag-Leffler function which can be also written as E ν ( − λt ν ) = 1 π Z ∞ exp n − λ /ν tx o x ν − sin πν x ν cos πν + x ν dx, t > , λ > . (4.12)The distribution l ν satisfies the fractional equation ∂ ν ∂t ν l ν = − ∂∂x l ν , x > t > l ν ( x,
0) = δ ( x ) where the fractional derivative must be understood in the Dzerbayshan-Caputo sense (formula(2.8)). The governing equation of l ν can be also presented by considering the Riemann-Liouvillederivative (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 involving two independent stable subordinator { ˜ τ ( ν ) t , t > } and { ˜ τ ( ν ) t , t > } has a distribution, ∀ t , given by r ( w ) = P r { ˜ τ ( ν ) t / ˜ τ ( ν ) t ∈ dw } /dw = 1 π w ν − sin πν w ν cos πν + w ν , w > , t > . (4.13)Here we study the ratio of two independent inverse stable processes { L ( ν ) t , t > } and { L ( ν ) t , t > } by evaluating its Mellin transform as follows E n L ( ν ) t / L ( ν ) t o η − = M [ l ν ( · , t )] ( η ) × M [ l ν ( · , t )] (2 − η ) = 1 ν sin νπ − ηνπ sin ηπ (4.14)with ℜ{ η } ∈ (0 , k ( x ) = 1 νπ sin νπ x cos νπ + x = 12 πi Z θ + i ∞ θ − i ∞ sin νπ − ηνπ sin ηπ x − η dη (4.15)for some real θ ∈ (0 , (cid:16) ˜ τ ( ν ) t / ˜ τ ( ν ) t (cid:17) ν law = L ( ν ) t / L ( ν ) t , ∀ t > . (4.16)We notice that the equivalence in law (4.16) is independent of t as the formulae (4.13) and (4.15)entail. The distribution h ν ◦ l ν ( x, t ) of the process { ˜ τ ( ν ) L ( ν ) t , t > } has Mellin transform (by making useof the formulae (4.7) and (4.9)) given by M [ h ν ◦ l ν ( · , t )] ( η ) = M [ h ν ( · , η ) × M [ l ν ( · , t )] (cid:18) η − ν + 1 (cid:19) = 1 ν sin πη sin π − ην t η − , t > ℜ{ η } ∈ (0 , τ ( ν ) L ( ν ) t law = t × ˜ τ ( ν ) t / ˜ τ ( ν ) t t > h ν ◦ l ν ( x, t ) = t − r ( x/t ) where r ( w ) is that in (4.13). For the process { L ( ν )˜ τ ( ν ) t , t > } withdistribution l ν ◦ h ν ( x, t ) we obtain (from (4.9) and (4.7)) M [ l ν ◦ h ν ( · , t )] ( η ) = M [ l ν ( · , η ) × M [ h ν ( · , t )] ( ην − ν + 1) = 1 ν sin πν − πην sin πη t η − with ℜ{ η } ∈ (0 ,
1) and thus L ( ν )˜ τ ( ν ) t law = t × L ( ν ) t / L ( ν ) t , t > . We have that l ν ◦ h ν ( x, t ) = t − k ( x/t ) where k ( x ) is that in (4.15).9 Main results
In this section we consider compositions of processes whose governing equations are (generalized) frac-tional diffusion equations. When we consider compositions involving Markov processes and stablesubordinators we still have Markov processes. Here we study Markov processes with random timewhich is the inverse of a stable subordinator. Such a process does not belong to the family of sta-ble 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, seeMainardi et al. [19]. Hereafter, we exploit the Mellin convolution of generalized Gamma densities inorder to write explicitly the solutions to fractional diffusion equations. We first present a new repre-sentation of the density law h ν by means of the convolution e ⋆n ¯ µ introduced in Section 3. To do thiswe also introduce the time-stretching function ϕ m ( s ) = ( s/m ) m , m ≥ s ∈ (0 , ∞ ). Lemma 2.
The Mellin convolution e ⋆n ¯ µ ( x, ϕ n +1 ( t )) where µ j = j ν , for j = 1 , , . . . , n is the densitylaw of a ν -stable subordinator { ˜ τ ( ν ) t , t > } with ν = 1 / ( n + 1) , n ∈ N . Thus, we have h ν ( x, t ) = e ⋆n ¯ µ ( x, ϕ n +1 ( t )) , x > , t > , ν = 1 / ( n + 1) , n ∈ N . Proof.
From (3.13) we have that M (cid:2) e ⋆n ¯ µ ( · , ϕ n +1 ( t )) (cid:3) ( η ) = Q nj =1 Γ (1 − η + µ j ) Q nj =1 Γ ( µ j ) ( ϕ n +1 ( t )) η − . (5.1)From Gradshteyn and Ryzhik [12, formula 8.335.3] we deduce that n Y k =1 Γ (cid:18) kn + 1 (cid:19) = (2 π ) n √ n + 1 , n ∈ N (5.2)and formula (5.1) reduces to M (cid:2) e ⋆n ¯ µ ( · , ϕ n +1 ( t )) (cid:3) ( η ) = Q nj =1 Γ (1 − η + µ j )(2 π ) n/ √ ν ( ϕ n +1 ( t )) η − . (5.3)Furthermore, by making use of the (product theorem) relationΓ( nx ) = (2 π ) − n n nx − / n − Y k =0 Γ (cid:18) x + kn (cid:19) (5.4)(see Gradshteyn and Ryzhik [12, formula 3.335]) formula (5.3) becomes M (cid:2) e ⋆n ¯ µ ( · , ϕ n ( t )) (cid:3) ( η ) = Γ (cid:0) − ην (cid:1) (2 π ) n/ ( n + 1) η/ν − n Γ (1 − η ) (2 π ) n/ ( ϕ n +1 ( t )) η − = Γ (cid:0) − ην (cid:1) ν Γ (1 − η ) t η − ν (with ℜ{ η } ∈ (0 , ν = 1 /
2, Lemma 2 says that h / ( x, t ) = e ⋆ µ ( x, ϕ ( t )) = e / ( x, ( t/ ) = x − / − e − t x t − √ (cid:0) (cid:1) , x > , t > / t/ √
2. For ν = 1 /
3, from (3.7), we obtain h / ( x, t ) = e ⋆ µ ( x, ϕ ( t )) = e / ⋆ e / ( x, ( t/ ) = 13 π t / x / K (cid:18) / t / √ x (cid:19) , x > , t > . For ν = 1 /
4, by (3.7) (and the commutativity under ⋆ , see Lemma 1), we have h / ( x, t ) = e ⋆ µ ( x, ϕ ( t )) = e / ⋆ e / ⋆ e / ( x, ( t/ ) = e / ⋆ ( e / ⋆ e / )( x, ( t/ )where K / ( z ) = p π/ z exp {− z } (see [12, formula 8.469]). We notice that M (cid:2) e ⋆ µ ( · , ϕ ( t )) (cid:3) ( η ) = M (cid:2) h / ◦ h / ( · , t ) (cid:3) ( η )which is in line with the well-known fact that E exp (cid:26) − λ ˜ τ ( ν ) ˜ τ ( ν t (cid:27) = E exp n − λ ν ˜ τ ( ν ) t o = exp {− tλ ν ν } , < ν i < , i = 1 ,
2. For ν = 1 /
5, by exploiting twice (3.7) (and the commutativity under ⋆ ), we canwrite h / ( x, t ) = e ⋆ µ ( x, ( t/ ) =( e / ⋆ e / ) ⋆ ( e / ⋆ e / )( x, ( t/ ) (5.6)= t / π x / Z ∞ s − / − K (cid:18) r sx (cid:19) K (cid:18) / t / √ s (cid:19) ds or equivalently h / ( x, t ) = e ⋆ µ ( x, ( t/ ) =( e / ⋆ e / ) ⋆ ( e / ⋆ e / )( x, ( t/ ) (5.7)= t / π x / Z ∞ s − / − K (cid:18) r sx (cid:19) K (cid:18) / t / √ s (cid:19) ds. For ν = 1 / (2 n + 1), n ∈ N , by using repeatedly (3.7) we arrive at h ν ( x, t ) = x ν/ t /ν − / ν − /ν π / ν − / K ◦ nν (cid:16) x, ( νt ) /ν (cid:17) , x > , t > K ◦ nν ( x, t ) = Z ∞ . . . Z ∞ K ν ( x, s ) . . . K ν ( s n − , t ) ds . . . ds n − is the integral (2.5) (as the symbol ” ◦ n ” denote) where n functions are involved and K ν ( x, t ) = x − ν − K ν (cid:16) p t/x (cid:17) , x > t >
0. We state a similar result for the density law l ν and the convolution g γ,⋆n ¯ µ (see Section 3). Let us consider the time-stretching function ψ m ( s ) = m s /m , s ∈ (0 , ∞ ), m ∈ N ,( ψ = ϕ − where ϕ has been introduced in the previous Lemma). Lemma 3.
The Mellin convolution g ( n +1) ,⋆n ¯ µ ( x, ψ n +1 ( t )) where µ j = j ν , j = 1 , , . . . , n and ν =1 / ( n + 1) , n ∈ N , is the density law of a ν -inverse process { L ( ν ) t , t > } . Thus, we have l ν ( x, t ) = g ( n +1) ,⋆n ¯ µ ( x, ψ n +1 ( t )) , x > , t > , ν = 1 / ( n + 1) , n ∈ N . Proof.
The proof can be carried out as the proof of Lemma 2.11e obtain that l / ( x, t ) = g / ( x, t / ) = e − x t / √ πt , x > t >
0. Moreover, by making use of(3.7) and (5.2), we have that l / ( x, t ) = g / ⋆ g / ( x, t / ) = 1 π r xt K (cid:18) / x / √ t (cid:19) , x > , t > l / ( x, t ) = g / ⋆ g / ⋆ g / ( x, t / ) follows (thank to the commutativity under ⋆ ) from g / ⋆ g / ⋆ g / ( x, t ) = g / ⋆ ( g / ⋆ g / )( x, t ) = 2 / π xt Z ∞ exp (cid:26) − ( sx ) − st ) (cid:27) ds where g / ⋆ g / ( x, t ) is given by (3.7) and K / ( z ) = p π/ z exp {− z } (see [12, formula 8.469]). In amore general setting, by making use of (3.7) we can write down g /ν,⋆ (1 /ν − µ ( x, t ) = 1 ν / ν (cid:16) xπ t (cid:17) − ν ν Q ◦ n − ν ( x, t ) , ν = 1 / (2 n + 1) , n ∈ N (5.8)where the symbol ” ◦ n ” stands for the integral (2.5) where n functions Q − ν are involved and Q − ν ( x, t ) = K − ν (cid:18) q ( x/t ) /ν (cid:19) , x > , t > . (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 diffusionequations involving the adjoint operators of both Bessel and squared Bessel processes. Let us introducethe distribution ˜ u γ,µν = ˜ g γµ ◦ l ν where ˜ g γµ ( x, t ) = g γµ ( x, t /γ ) and the Mellin transform of ˜ u γ,µν whichreads M [˜ u γ,µν ( · , t )] ( η ) = Γ (cid:16) η − γ + µ (cid:17) Γ (cid:16) η − γ + 1 (cid:17) Γ( µ )Γ (cid:16) η − γ ν + 1 (cid:17) t η − γ ν , − γµ < ℜ{ η } < γ/ν − γ. (5.10)We state the following result. Theorem 1.
Let the previous setting prevail. For ν = 1 / (2 n + 1) , n ∈ N ∪ { } , the solutions to D νt ˜ u γ,µν = G γ,µ ˜ u γ,µν , x > , t > can be represented in terms of generalized Gamma convolution as ˜ u γ,µν ( x, t ) = γ x µ − ν − ν ν ( π t ν ) − ν ν Z ∞ s ν − − µ e − x γ /s v ν ( s, t ) ds, x ≥ , t > where G γ,µ is the operator appearing in (3.4) , v ν ( s, t ) = Z ∞ . . . Z ∞ Q − ν ( s, s ) . . . Q − ν ( s n − , t ν /ν ) ds . . . ds n − and Q − ν is that in (5.9) . Moreover, for ν ∈ (0 , , we have ˜ u γ,µν ( x, t ) = γxt ν/γ H , , " x γ t ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 , ν ); ( µ, , µ, (5.13) in terms of H Fox functions. roof. By exploiting the property (2.6) of the Mellin transform and the fact that Z ∞ x η − x θ f ( x ) dx = M [ f ( · )] ( η + θ ) , for the operator (3.4) we have that M [ G γ,µ ˜ u γ,µν ( · , t )] ( η )= − γ ( η − M (cid:20) ∂∂x ˜ u γ,µν ( · , t ) (cid:21) ( η − γ + 1) + 1 γ ( γµ − η − M [˜ u γ,µν ( · , t )] ( η − γ )= 1 γ ( η − η − γ ) M [˜ u γ,µν ( · , t )] ( η − γ ) + 1 γ ( γµ − η − M [˜ u γ,µν ( · , t )] ( η − γ )= 1 γ ( η − η − γµ − γ ) M [˜ u γ,µν ( · , t )] ( η − γ ) (5.14)where M [˜ u γ,µν ( · , t )] ( η ) is that in (5.10). We obtain M [ G γ,µ ˜ u γ,µν ( · , t )] ( η ) = 1 γ ( η − η − γµ − γ ) Γ (cid:16) η − γ − γ + µ (cid:17) Γ (cid:16) η − γ − γ + 1 (cid:17) Γ( µ )Γ (cid:16) η − γ − γ ν + 1 (cid:17) t η − γ − γ ν = 1 γ ( η −
1) Γ (cid:16) η − γ + µ (cid:17) Γ (cid:16) η − γ (cid:17) Γ( µ )Γ (cid:16) η − γ ν − ν + 1 (cid:17) t η − γ ν − ν = Γ (cid:16) η − γ + µ (cid:17) Γ (cid:16) η − γ + 1 (cid:17) Γ( µ )Γ (cid:16) η − γ ν − ν + 1 (cid:17) t η − γ ν − ν = D νt M [˜ u γ,µν ( · , t )] ( η )and ˜ u γ,µν ( x, t ) solves (5.11) for ν ∈ (0 , u γ,µν ( x, t ) = Z ∞ ˜ g γµ ( x, s ) g /ν,⋆ (1 /ν − µ ( s, ψ /ν ( t )) ds and by means of (5.8) result (5.12) appears. Formula (5.13) follows directly from (2.2) by consideringformula (3.10) and the fact that H m,np,q (cid:20) x (cid:12)(cid:12)(cid:12)(cid:12) ( a i , α i ) i =1 ,..,p ( b j , β j ) j =1 ,..,q (cid:21) = 1 x c H m,np,q (cid:20) x (cid:12)(cid:12)(cid:12)(cid:12) ( a i + cα i , α i ) i =1 ,..,p ( b j + cβ j , β j ) j =1 ,..,q (cid:21) (5.15)for all c ∈ R (see Mathai and Saxena [20]).We specialize the previous result by keeping in mind formula (2.10) and the operator (3.4). Corollary 1.
For ν = 1 , Theorem 1 says that ∂∂t ˜ u γ,µ = 1 γ (cid:18) ∂∂x x − γ ∂∂x − ( γµ − ∂∂x x − γ (cid:19) ˜ u γ,µ , x > , t > , γ = 0 where ˜ u γ,µ = ˜ g γµ is the distribution of the GGP. This is because L t a.s. = t . Indeed, for ν = 1, L νt is the elementary subordinator (see [4]). Proof. If ν = 1, then the equation (5.10) takes the formΨ t ( η ) = M [˜ g γµ ( · , t )]( η ) = Γ (cid:18) η − γ + µ (cid:19) t η − γ Γ( µ ) , ℜ{ η } > − γµ (5.16)13here ˜ g γµ ( x, t ) = g γµ ( x, t /γ ). For γ >
0, we perform the time derivative of (5.16) and obtain ∂∂t Ψ t ( η ) = η − γ Γ (cid:18) η − γ + µ (cid:19) t η − γ − γ = η − γ (cid:18) η − γ − γµγ (cid:19) Γ (cid:18) η − γ − γ + µ (cid:19) t η − γ − γ = 1 γ ( η − η − γ − γµ )Ψ t ( η − γ )which coincides with (5.14) and ˜ u γ,µ ( x, t ) = ˜ g γµ ( x, t ), γ >
0. Similar calculation must be done for γ <
Corollary 2.
Let us write ˜ u µν ( x, t ) = ˜ u ,µν ( x, t ) . The distribution ˜ u µν ( x, t ) , x > , t > µ > , ν ∈ (0 , , solves the following fractional equation ∂ ν ∂t ν u µν = (cid:18) x ∂ ∂x − ( µ − ∂∂x (cid:19) u µν . (5.17)In particular, for ν = 1 /
2, we have˜ u µ / ( x, t ) = x µ − √ πt Γ( µ ) Z ∞ s − µ exp (cid:26) − xs − s t (cid:27) ds, x > , t > , µ > { G ,µ | B (2 t ) | , t > } where B is a standard Brownianmotion run at twice its usual speed and G γ,µt is a GGP. We notice that the process G ,µt is a squaredBessel process starting from zero. Corollary 3.
The distribution ˜ u ,µν = ˜ u ,µν ( x, t ) , x > , t > , µ > , ν ∈ (0 , solves the followingfractional equation ∂ ν ∂t ν ˜ u ,µν = 12 (cid:18) ∂ ∂x − ∂∂x (2 µ − x (cid:19) ˜ u ,µν . In particular, for ν = 1 /
3, we have˜ u ,µ / ( x, t ) = 2 x µ − π Γ( µ ) √ t Z ∞ e − x s s µ − / K (cid:18) / s / √ t (cid:19) ds, x > , t > , µ > µ = 1 / u , / / ( x, t ) = 2 π / √ t Z ∞ e − x s K (cid:18) / s / √ t (cid:19) ds, x > , t > | B ( L / t ) | where | B ( t ) | is a folded Brownian motion with variance t/ Acknowledgement
The author is grateful to the anonymous referee for careful checks and com-ments.
References [1] B. Baeumer and M. Meerschaert. Stochastic solutions for fractional cauchy problems.
Fract. Calc.Appl. Anal. , 4(4):481 – 500, 2001.[2] B. Baeumer, M. Meerschaert, and E. Nane. Space-time duality for fractional diffusion.
J. Appl.Probab. , 46:1100 – 1115, 2009. 143] L. Beghin and E. Orsingher. The telegraph process stopped at stable-distributed times and itsconnection with the fractional telegraph equation.
Fract. Calc. Appl. Anal. , 6:187 – 204, 2003.[4] J. Bertoin.
L´evy Processes . Cambridge University Press, 1996.[5] L. Bondesson, G. K. Kristiansen, and F. W. Steutel. Infinite divisibility of random variables andtheir integer parts.
Stat. Prob. Lett , 28:271 – 278, 1996.[6] R. D. DeBlassie. Higher order PDEs and symmetric stable process.
Probab. Theory Rel. Fields ,129:495 – 536, 2004.[7] M. D’Ovidio and E. Orsingher. Composition of processes and related partial differential equations.
J. Theor. Probab. (published on line) , 2010.[8] M. D’Ovidio and E. Orsingher. Bessel processes and hyperbolic Brownian motions stoppedat different random times.
Under revision for Stochastic Processes and their Applications,(arXiv:1003.6085v1) , 2010.[9] W. Feller.
An introduction to probability theory and its applications , volume 2. 2 edition, 1971.[10] C. Fox. The G and H functions as symmetrical Fourier kernels.
Trans. Amer. Math. Soc. , 98:395– 429, 1961.[11] R. Gorenflo and F. Mainardi. Fractional calculus: integral and differential equations of frationalorder, in A. Carpinteri and F. Mainardi (Editors).
Fractals and Fractional Calculus in ContinuumMechanics , pages 223 – 276, 1997. Wien and New York, Springher Verlag.[12] I. S. Gradshteyn and I. M. Ryzhik.
Table of integrals, series and products . Academic Press, 2007.Seventh edition.[13] A. Kilbas, H. Srivastava, and J. Trujillo.
Theory and applications of fractional differential equa-tions (North-Holland Mathematics Studies) , volume 204. Elsevier, Amsterdam, 2006.[14] A. N. Kochubei. The Cauchy problem for evolution equations of fractional order.
DifferentialEquations , 25:967 – 974, 1989.[15] A. N. Kochubei. Diffusion of fractional order.
Lecture Notes in Physics , 26:485 – 492, 1990.[16] S. Kotz, N. Balakrishnan, and N. L. Johnson.
Continuous multivariate distributions , volume 1.New York, Wiley, 2nd edition, 2000.[17] F. Mainardi, Y. Luchko, and G. Pagnini. The fundamental solution of the space-time fractionaldiffusion equation.
Fract. Calc. Appl. Anal. , 4(2):153 – 192, 2001.[18] F. Mainardi, G. Pagnini, and R. Gorenflo. Mellin transform and subordination laws in fractionaldiffusion processes.
Fract. Calc. Appl. Anal. , 6(4):441 – 459, 2003.[19] F. Mainardi, G. Pagnini, and R. Gorenflo. Some aspects of fractional diffusion equations of singleand distributed order.
Applied Mathematics and Computing , 187:295 – 305, 2007.[20] A. Mathai and R. Saxena.
Generalized Hypergeometric functions with applications in statisticsand physical sciences . Lecture Notes in Mathematics, n. 348, 1973.[21] M. Meerschaert and H. P. Scheffler. Triangular array limits for continuous time random walks.
Stoch. Proc. Appl. , 118:1606 – 1633, 2008.[22] M. Meerschaert, E. Nane, and P. Vellaisamy. Fractional cauchy problems on bounded domains.
Ann. of Probab. , 37(3):979 – 1007, 2009. 1523] E. Nane. Higher order PDE’s and iterated processes .
Trans. Amer. Math. Soc. , (5):681–692,2008.[24] E. Nane. Fractional cauchy problems on bounded domains: survey of recent results. arXiv:1004.1577v1 , 2010.[25] R. Nigmatullin. The realization of the generalized transfer in a medium with fractal geometry.
Phys. Status Solidi B , 133:425 – 430, 1986.[26] E. Orsingher and L. Beghin. Time-fractional telegraph equations and telegraph processes withBrownian time.
Probab. Theory Rel. Fields , 128(1):141 – 160, 2004.[27] E. Orsingher and L. Beghin. Fractional diffusion equations and processes with randomly varyingtime.
Ann. Probab. , 37:206 – 249, 2009.[28] S. Samko, A. A. Kilbas, and O. I. Marichev.
Fractional Integrals and Derivatives: Theory andApplications . Gordon and Breach, Newark, N. J., 1993.[29] W. Schneider and W. Wyss. Fractional diffusion and wave equations.
J. Math. Phys. , 30:134 –144, 1989.[30] W. Wyss. The fractional diffusion equations.
J. Math. Phys. , 27:2782 – 2785, 1986.[31] G. Zaslavsky. Fractional kinetic equation for Hamiltonian chaos.
Phys. D , 76:110 – 122, 1994.[32] V. M. Zolotarev.