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λ,ρϕ(t)= λρϕ(t), ϕ(0) = 1, (1.3) Dt − λ,ρ for λ > 0, t 0 and ρ (0, 1], where t is defined as particular case of the so-called convolution- type derivatives≥ (see definitions∈ (1.15)D and (2.1) below). We prove that it is satisfied by the following relaxation function Γ(ρ; λt) ϕ (t)= , (1.4) λ,ρ Γ(ρ) . + ∞ w ρ 1 where Γ(ρ; x) = x e− w − dw is the so-called upper-incomplete gamma function with parameter ρ (0, 1]. Note that Γ (ρ; x) is well-defined for any ρ R and it is real-valued when x R+. ∈In the specialR case where ρ =1, equation (1.3) reduces∈ to the standard relaxation∈ equation (1.1). The asymptotic behavior of the solution coincides with that of the model in [33], but thanks to the explicit form of the solution, we can obtain here further results: we can prove the completely monotonicity and integrability of ϕλ,ρ and obtain its spectral distribution, which, as we will explain below, can be considered a generalization of stable laws. ηS (t) ηαt We recall that the Laplace transform of the α-stable subordinator, i.e. Φ(η) := Ee− α = e− , t 0, satisfies equation (1.1), with λ = ηα. We then generalize this result and define a new class of stochastic≥ processes by means of the n-times Laplace transform of its density, as follows
n α Γ ρ; Ξ (tk tk 1) k,n − Pn η (t ) k=1 − Ee− k=1 kXα,ρ k = , η , ...η 0,n N (1.5) P Γ (ρ) 1 n ≥ ∈
α n α where Ξk,n := ( i=k ηi) , for 0 < t1 < .... < tn, for α (0, 1) and ρ (0, 1] . In the one-dimensional case (for n = 1), the Laplace transform in (1.5) coincides∈ with (1.4) for∈ λ = ηα, i.e. P α η (t) Γ (ρ; η t) Ee− Xα,ρ = , η> 0. (1.6) Γ (ρ) Moreover, in the special case ρ = 1, we obtain from (1.5) the Laplace transform of the α-stable subordinator. As we will see in the next section, both (1.5) and (1.6) are completely monotone functions (with respect to η1,..., ηn and η, respectively), under the conditions that α, ρ 1 and αρ 1, respectively. Thus we will assume hereafter ρ (0, 1/α] when treating the one-dimensional≤ distribution,≤ whereas we will restrict to the case ρ (0,∈1] in the n-dimensional case. Moment generating functions expressed as power∈ series of incomplete gamma functions can be also found in [32], where they are used to generalize the gamma distribution. The process α,ρ := α,ρ(t),t 0 defined in (1.5) displays a self-similar property of order 1/α, for any ρ. On theX other{X hand, it is≥ a L´evy} process only in the stable case (i.e. for ρ = 1), since its distribution is not, in general, infinitely divisible. The main feature of self-similar processes is the invariance in distribution under suitable scaling of time and space. Self-similarity is usually associated with long-range dependence (see, for example, [28]). The main tools used to describe the dependence structure of a stochastic processes are both the autocorrelation coefficient and the spectral analysis. However, in the case of heavy-tailed distributions, where the moments are not finite, we must resort to other tools, such as the auto-codifference function. In particular, referring to the definition given in [29] and [38], we are able to interpret ρ as a a measure of the local deviation from the temporal dependence structure displayed in the standard α-stable case. In order to derive the differential equation satisfied by the transition function hα,ρ(x, t) of α,ρ, we distinguish the two cases 0 < ρ 1 and 1 < ρ 1/α: indeed, in the first one, we proveX that ≤ ≤
2 hα,ρ(x, t) satisfies an integro-differential equation where a space-dependent time-fractional derivative (of convolution-type) and a Caputo space-fractional derivative appear. In the case where ρ > 1, the differential equation satisfied by the transition density is obtained only in the special case ρ = 2 and it involves the so-called logarithmic differential operator introduced in [2] and defined in (3.9) below. We are able to write the explicit expression of hα,ρ(x, t), in terms of H-functions, which for ρ =1 and t =1, reduces to the stable law formula given in (2.28) of [23]. Finally we present some properties of the generalized stable r.v. α,ρ, such as the tails’ behavior and the moments. In particular, we prove that it possesses finite momentsX of order δ <αρ, which can be written as
δ δ c α Γ(ρ ) E ( )δ = − α . Xα,ρ Γ(ρ) Γ(1 δ) − We now recall some well-known definitions and results which will be used throughout the paper. m,n Let Hp,q denote the Fox’s H-function defined as (see [23] p.13):
m,n (a1, A1) ... (ap, Ap) Hp,q z (1.7) | (b1,B1) ... (bq,Bq) m n s Γ(bj + Bj s) Γ(1 aj Aj s) z− ds . 1 (j=1 )(j=1 − − ) = , 2πi Q q Q p ZL Γ(1 bj Bj s) Γ(aj + Aj s) (j=m+1 − − )(j=n+1 ) Q Q + with z = 0, m,n,p,q N0, for 1 m q, 0 n p, aj ,bj R, Aj ,Bj R , for i = 1, ..., p, j =1, ...,6 q and is a contour∈ such≤ that the≤ following≤ ≤ condition is∈ satisfied ∈ L A (b + α) = B (a k 1), j =1, ..., m, l =1, ..., n, α, k =0, 1, ... (1.8) l j 6 j l − − Let us recall the following particular case of the H-function which we will use later, i.e. the Mittag-Leffler function, defined as
+ ∞ zj (0, 1) E (z) := = H1,1 z , (1.9) α,β Γ(αj + β) 1,2 − | (0, 1) (1 β, α) j=0 X − for α > 0 (see formula (1.136) in [23]). Moreover, we recall the other special case of (1.7) given by the Meijer’s G-function, which is obtained for Ai = 1, for any i = 1, ..., p and for Bj = 1, for any j =1, ..., q: m,n (a1, ...ap) m,n (a1, 1) ... (ap, 1) Gp,q z := Hp,q z . (1.10) | (b1, ..., bq) | (b1, 1) ... (bq, 1) In particular, we will use (1.10) in the case where n = 0 and p = q = m, for which the following result holds (see properties 3 and 9 in [14]):
p,0 (a1, ...ap) Gp,p x =0, x > 1, (1.11) | (b1, ..., bq) | | p while, under the assumption that tbj taj 0, it is j=1 − ≥ p,0 P (a1, ...ap) Gp,p x 0, 0 < x < 1. (1.12) | (b1, ..., bq) ≥ | | More details on the properties of the function (1.10), together with its applications in fractional 1,0 calculus, can be found in [17]. In particular, for p = q = 1, the function G1,1 is proved to be the kernel of the Erdelyi-Kober fractional integral (see also [15]).
3 We will use the definition and properties of completely monotone functions, both in the univariate and multivariate cases; thus we recall that, for Theorem 4.2.1 p.87 in [5], a function f(z1, ..., zn) in an octant 0 for a measure ρ such that ρ(A) 0, for any A, if and only if it is C∞ and ≥ k1+...+kn k1+...+kn ∂ ( 1) f(z1, ..., zn) 0, (1.14) k1 kn − ∂z1 ...∂zn ≥ for all combinations k1 0, ..., kn 0 (the representation being unique). The measure ρ is called spectral distribution of f≥and the function≥ satisfying (1.14) is said to be completely monotone (hereafter CM). By the way, we note that the function (1.13) is linked to the kernel of a particular case of the Obrechkoff transform (see, for example, [7]). We also recall the definition of the convolution-type derivative on the positive half-axes, in the sense of Caputo (see [34], Def.2.4, for b = 0): let ν(ds) be a non-negative L´evy measure (i.e. satisfying + + ∞ ∞ the condition 0 (z 1)ν(dz) < ) and let ν(s) = s ν(dz) be its tail, under the assumption that it is absolutely continuous.∧ Let,∞ moreover, g : R+ R+, be a Bernstein function, i.e. with the R + sx R→ following representation g(x)= a + bx + ∞(1 e− )ν(ds), for a,b 0, then 0 − ≥ Rt d gu(t) := u(t s)ν(s)ds, t> 0, (1.15) Dt ds − Z0 for an absolutely continuous function such that u(t) Meλt, for some M,λ> 0 and for any t. Convolution-type derivatives (or derivatives| defined|≤ as integrals with memory kernels) have been treated recently by many authors: see, among the others, [18], [19], [33], [9], [34]. The Laplace transform of g is given by Dt + ∞ θt g g(θ) e− u(t)dt = g(θ)u(θ) u(0), (θ) >θ , (1.16) Dt − θ R 0 Z0 (see [34], Lemma 2.5). It is easy to check that,e in the trivial case where g(θ) = θ, the convolution- type derivative coincides with the standard first-order derivative. When g(θ)= θα, for α (0, 1), this ∈ Bernstein function coincides with the Laplace exponent of the α-stable subordinator α(t) (distributed, for any t, as a stable r.v. (σ,β,µ), with σ = (cos (πα/2) t)1/α, β = 1 and µ = 0),S since Sα θ (t) α Ee− Sα = exp θ t , t,θ> 0. (1.17) {− } α 1 In this case, the L´evy measure reduces to ν(ds)= αs− − /Γ(1 α) and thus (1.15) coincides with the fractional Caputo derivative of order α, i.e. − t C α 1 d α u(t) := u(s)(t s)− ds, t> 0, α (0, 1). (1.18) Dt Γ(1 α) ds − ∈ − Z0 Finally, we will use the Riesz-Feller (RF) fractional derivative, which is defined, by means of its Fourier transform. Let Lc( ) of functions for which the Riemann improper integral on any open interval absolutely convergesI (see [22]). For any f Lc( ), the RF fractional derivative is defined as I ∈ I α f(x); ξ = ψα(ξ) f(x); ξ , α (0, 2], θ min α, 2 α , (1.19) F Dx,θ − θ F{ } ∈ | |≤ { − } 4 with symbol ψα(ξ) := ξ αei sign(ξ)θπ/2, ξ R, (1.20) θ | | ∈ (see [16], p.131). Since we are interested in the special case where the support of the density is the positive half-line, we can take ξ > 0 and θ = α (which corresponds to the stable law totally skewed to the right). − 2 Tempered relaxation function Under the assumption that ρ 1, we prove that the (normalized) upper-incomplete Gamma function given in (1.4) is the solution of≤ a tempered relaxation equation, which extends the standard one, i.e. (1.1), by means of the tempered fractional derivative, in the Caputo sense. The latter can be obtained from the definition given in (1.15), by choosing the Bernstein function g(x) = (λ + x)ρ λρ, for −λs −ρ−1 − λ, x R+ and the L´evy measure ν(ds)= ρe s ds. Thus the tail L´evy measure is taken equal to ∈ Γ(1 ρ) ρλρΓ( ρ;λs) − ν(ds) = Γ(1−ρ) ds (see [34] and [33], for details). For ρ = 1, we have that g(x)= x and thus the tempered derivative− reduces to the first-order derivative. Lemma 1 λ,ρ R+ R+ Let t be defined, for any absolutely continuous function u : , such that u(t) cekt, forD some c,k > 0 and for any t 0, as → | |≤ ≥ ρλρ t d λ,ρu(t)= u(t s)Γ( ρ; λs)ds, λ> 0, ρ (0, 1], (2.1) Dt Γ(1 ρ) dt − − ∈ − Z0 then the solution of the following tempered relaxation equation λ,ρu(t)= λρu(t), (2.2) Dt − with initial condition u(0) = 1, is given by Γ(ρ; λt) ϕ (t)= . (2.3) λ,ρ Γ(ρ) Proof. The Laplace transform of the l.h.s. of (2.2) can be obtained from (1.16), by choosing g(x)= (λ + x)ρ λρ and considering the initial condition: − + ρ ρ ∞ θt λ,ρ ρ ρ (λ + θ) λ e− u(t)dt = [(λ + θ) λ ] u(θ) − . (2.4) Dt − − θ Z0 Then, by taking into account the r.h.s. of (2.2), we get e (λ + θ)ρ λρ u(θ)= − (2.5) θ(λ + θ)ρ which coincides with the Laplace transforme of (2.3), since + + + 1 ∞ θt 1 ∞ θt ∞ w ρ 1 e− Γ (ρ; λt) dt = e− e− w − dwdt Γ(ρ) Γ(ρ) Z0 Z0 Zλt ρ + w λ ∞ λw ρ 1 θt = e− w − e− dtdw Γ(ρ) Z0 Z0 ρ + λ ∞ λw θw ρ 1 = e− (1 e− )w − dw. θΓ(ρ) − Z0 We now derive some properties of ϕλ,ρ( ), which are usually required to a relaxation function, and obtain its spectral distribution in terms of· H-function. The latter will be proved to coincide with the density of the process defined in the next section (in the special case α = 1). Xα,ρ 5 Theorem 2 The tempered relaxation function ϕλ,ρ( ), defined in (2.3) is ∞, completely monotone and locally integrable. The spectral distribution of ϕ · ( ) is given by C λ,ρ · λ1z>λ λ 2 K (z)= G1,0 , ρ (0, 1), (2.6) λ Γ(ρ) 1,1 z 1+ ρ ∈ where Gm,n[ ] is the Meijer’s G-function defined in (1.10). p,q · Proof. It is well-known that the upper incomplete gamma function is ∞ and thus the same holds C true for ϕλ,ρ( ). Moreover, we can derive complete monotonicity by repeatedly differentiating (2.3) · k and showing that ( 1)k d ϕ (t) 0, for any k N and t 0: − dtk λ,ρ ≥ ∈ ≥ d ρ λt ρ 1 ϕ (t) = λ e− t − 0 −dt λ,ρ ≥ 2 d ρ+1 λt ρ 1 ρ λt ρ 2 ϕ (t) = λ e− t − (ρ 1)λ e− t − 0 dt2 λ,ρ − − ≥ 3 d ρ+2 λt ρ 1 ρ+1 λt ρ 2 ϕ (t) = λ e− t − 2(ρ 1)λ e− t − + −dt3 λ,ρ − − ρ λt ρ 3 + (ρ 1)(ρ 2)λ e− t − 0 − − ≥ and so on. The sign of the k-derivative depends only on ρ j +1, for j =2, ..., k, which is non-positive, for any k N and for ρ 1. As far as the integrability,− we first derive an alternative expression of (2.3) in terms∈ of Mittag-Leffler≤ function: recalling formula (3.7) p.316 in [25] and formula (4.2.8) in [12], we can write λt w ρ 1 0 e− w − dw ϕλ,ρ(t) = 1 (2.7) − R Γ (ρ) λt ρ λt ρ = 1 e− λ (ρ, λ)=1 e− (λt) E (λt), − Et − 1,ρ+1 eλx x ν 1 w where (ν, λ)= w − e− dw is the so-called Miller-Ross function. Ex Γ(ν) 0 By applying formula (6.5.31) in [36], we easily obtain the following limiting behavior of (2.3) R e λtλρ 1tρ 1 ϕ (t) − − − , t + . (2.8) λ,ρ ≃ Γ(ρ) → ∞ Since ϕλ,ρ(0) = 1 and the function ϕλ,ρ( ). is bounded by one on any compact set, (2.8) is enough to prove its integrability. Finally, we check that· + ∞ tz ϕλ,ρ(t)= e− Kλ(z)dz, (2.9) Z0 for K(z) given in (2.6). Indeed we can write (2.6), by (1.10), in terms of a H-functions, as λ λ (2, 1) K (z) = H1,0 λ Γ(ρ) 1,1 z (1 + ρ, 1) λ 0,1 z ( ρ, 1) = [ by(1.60) in [23]] = H − , Γ(ρ) 1,1 λ ( 1, 1) − 6 so that we can apply formula (2.19) in [23] and get + 1 ∞ tz 0,1 z ( ρ, 1) e− H1,1 − dz λΓ(ρ) 0 λ ( 1, 1) Z − 1 0,2 1 (0, 1)( ρ, 1) = H − Γ(ρ)λt 2,1 λt ( 1, 1) − 1 (1, 1) = [by (1.58) and (1.60) in [23]] = H2,0 λt Γ(ρ) 1,2 | (0, 1)(ρ, 1) 1 1 s Γ(s)Γ(ρ + s) = (λt)− ds = ϕ (t), Γ(ρ) 2πi Γ(1 + s) λ,ρ ZL by means of the Mellin-Barnes integral form of the incomplete gamma function: 1 Γ(s + ρ)x s Γ(ρ; x)= − ds, (2.10) 2πi Γ(ρ)s ZL where the contour avoids all the poles of the gamma functions (see [27] for more details). Note that the indicator functionL in (2.6) comes from (1.11). Remark 3 We note that, as t + , the function ϕλ,ρ( ) decays exponentially fast and thus faster than in the fractional case. On the→ other∞ hand, for small valu· es of t, the tempered relaxation function exhibits the same fast decay of the fractional relaxation (see [21]): indeed we can obtain the behavior of ϕ ( ), for t 0+, by considering (2.7) together with λ,ρ · → 1 z E (z) + , z 0+ α,β ≃ Γ(β) Γ(β + α) → (see [12], p.64): thus we have that e λtλρtρ tρ ϕ (t) 1 − 1 , t 0+. (2.11) λ,ρ ≃ − Γ(ρ + 1) ≃ − Γ(ρ + 1) → We thus obtain asymptotic behaviors similar to those of the relaxation function introduced in [33] (by time-changing with inverse tempered subordinators), but, in this case, we have also a closed expression of the solution, in terms of well-known functions. Remark 4 As far as the spectral distribution Kλ(z), z 0, given in (2.6), we can check, by (1.12), that it is a proper probability distribution: indeed it is no≥n-negative, since t1+ρ t2 0, for ρ < 1 + − ≥ and t (0, 1). Furthermore, ∞ K (z)dz =1, by using (2.9) and considering that ϕ (0) = 1. From ∈ 0 λ λ,ρ (2.6) we can draw the conclusion that the process governed by ϕλ,ρ(t) can be expressed in terms of a continuous distribution ofR standard (exponential) relaxation processes with frequencies on the range (λ, + ) instead of the whole real positive semi-axes. In the special case ρ =1/2, it is well-known that ∞ 1 0,1 1x>1 G x − 2 = , 1,1 | 1 √πx√x 1 − − (see (1.143) in [23]), so that we get 5/2 λ 1z>λ Kλ(z)= . (2.12) πz√z λ − Remark 5 Let Sλ,ρ(t),t 0, be the tempered subordinator, for λ 0 and ρ (0, 1], with transition density h (z,t). Let moreover≥ L (t),t 0, be its inverse, i.e. ≥L (t) :=∈ inf s 0, S (s) > t λ,ρ λ,ρ ≥ λ,ρ { ≥ λ,ρ } 7 (see [1], [10], [24] and [37], for details). As a consequence of Lemma 1 we can derive the following relationship between the tempered relaxation and the density of L , i.e. l (x, t) := P L (t) dx : λ,ρ λ,ρ { λ,ρ ∈ } + Γ (ρ; λt) ∞ λρz ϕ (t)= = e− l (z,t)dz. (2.13) λ,ρ Γ(ρ) λ,ρ Z0 The previous integral expression of the incomplete gamma function can be obtained, by applying (19) of [36]. Let us denote by hρ(u,z) the density of the ρ-stable subordinator (i.e. for λ =0), then + ∞ λρz e− lλ,ρ(z,t)dz 0 Z + t ∞ λρz ∂ = e− hλ,ρ(u,z)dudz − 0 ∂z 0 Z t + Z t ρ λu ∞ λρz+λρz = λ e− e− hρ(u,z)dzdu + hλ,ρ(u, 0)du − 0 0 0 Z t Z + Z ρ λu ∞ = λ e− h (u,z)dzdu +1, − ρ Z0 Z0 + ∞ ρ 1 for t> 0, since hλ,ρ(u, 0) = δ(u). We now notice that 0 hρ(u,z)dz = u − /Γ (ρ) (as can be checked by the Laplace transform) and the result (2.13) easily follows. The relationship (2.13) can be also checked by consideringR that both sides satisfy (2.2): by taking the tempered derivative of the r.h.s. of (2.13): + + λ,ρ ∞ λρz ∞ λρz λ,ρ Γ (ρ) e− l (z,t)dz = Γ (ρ) e− l (z,t)dz Dt λ,ρ Dt λ,ρ Z0 Z0 = [by (6.13) and (2.33) in [34]] + + ∞ λρz ∂ ∞ λρz = Γ (ρ) e− l (z,t)dz Γ (ρ) ν(t) e− l (z, 0)dz − ∂z λ,ρ − λ,ρ Z0 Z0 + + λρz ∞ ρ ∞ λρz = Γ (ρ) e− lλ,ρ(z,t) Γ (ρ) ν(t) Γ (ρ) λ e− lλ,ρ(z,t)dz − z=0 − − 0 h + i Z ρ ∞ λρz = Γ (ρ) λ e− l (z,t)dz, − λ,ρ Z0 ρλρΓ( ρ,λt) where we have considered the initial conditions lλ,ρ(0,t)= Γ(1−ρ) and lλ,ρ(x, 0) = δ(x) (see [34], for details). − 3 Generalized stable process ηS (t) ηαt We recall that the Laplace transform of the α-stable subordinator, i.e. Φ(η) := Ee− α = e− , t 0, satisfies the standard relaxation equation (1.1), with λ = ηα. We then apply the previous results,≥ in order to define a new stochastic process, which generalizes the α-stable subordinator. We first prove that, under some assumptions, the function n α Γ ρ; Ξk,n (tk tk 1) − − Φ(η , ...η ; t , ...t )= k=1 , η , ...η 0,n N, (3.1) 1 n 1 n P Γ (ρ) 1 n ≥ ∈ α n α where Ξk,n := ( i=k ηi) , can be uniquely represented as a Laplace transform of a probability measure in Rn. P 8 Lemma 1 Let 0 < t < .... < t , α (0, 1) and ρ (0, 1]. The function Φ(η , ...η ; t , ...t ) in (3.1) 1 n ∈ ∈ 1 n 1 n is C∞, is such that Φ(0, .., 0; t1, ...tn)=1 and is completely monotone (CM) with respect to η1, ...ηn, for any choice of t1 < ... < tn. Thus the following integral (unique) representation holds + + ∞ ∞ η1x1... ηnxn Φ(η1, ...ηn; t1, ...tn)= ... e− − π(dx1, ..., dxn; t1, ...tn), Z0 Z0 for a probability measure π( , ..., ; t , ...t ). · · 1 n η (t) Proof. We first check that the one-dimensional function Φ(η,t) = Ee− Xα,ρ given in (3.3) is C∞, since the incomplete gamma function is analytic w.r.t. both its arguments. Moreover, it is such k that Φ(0,t) = 1 and is a CM function, i.e is absolutely differentiable and ( 1)k d Φ(η,t) 0, for − dηk ≥ k =0, 1, .... Indeed, ηα is a Bernstein function and we have proved in Theorem 2 that Γ(ρ; ) is CM. Then it is well-known that the composition of a CM and a Bernstein function is again CM.· Thus the statement is proved for n =1. Moreover, we can check directly that this is true if and only if αρ < 1: d ρ αρ 1 ηαt Φ(η,t)= αt η − e− 0 dη − ≤ 2 d ρ αρ 2 ηαt 2 ρ+1 αρ+α 2 ηαt Φ(η,t)= α(αρ 1)t η − e− + α t η − e− 0 dη2 − − ≥ 3 d ρ αρ 3 ηαt 2 ρ+1 αρ+α 3 ηαt Φ(η,t)= α(αρ 1) (αρ 2) t η − e− + (αρ 1)α t η − e− + dη3 − − − − 2 ρ+1 αρ+α 3 ηαt 3 ρ+2 αρ+2α 3 ηαt +α t (αρ + α 2)η − e− α t η − e− 0 − − ≤ and so on. The sign of the k-th derivatives depends only on αρ j +1, for j =2, ...k, and αρ + (j 2)α j +1, for j =3, ...k, which are all non-positive under the− unique condition αρ < 1. − For− n> 1, we start by proving that the condition (1.14) holds, for n =2, i.e. that ∂k1+k2 ( 1)k1+k2 Φ(η , η ; t ,t ) k1 k2 1 2 1 2 − ∂η1 ∂η2 ∂k1+k2 Γ (ρ; (η + η )αt + ηα(t t )) = ( 1)k1+k2 1 2 1 2 2 − 1 0 k1 k2 − ∂η1 ∂η2 Γ (ρ) ≥ 9 . Let us denote A = (η + η )αt + ηα(t t ); thus we can write 1,2 1 2 1 2 2 − 1 ∂k1+k2 ( 1)k1+k2 Φ(η , η ; t ,t ) k1 k2 1 2 1 2 − ∂η1 ∂η2 k1+k2 k2 k1 1 ( 1) ∂ ∂ − ∂ = − Γ (ρ; A ) k2 k1 1 1,2 Γ (ρ) ∂η2 ∂η1 − ∂η1 k1+k2+1 k2 k1 1 ( 1) ∂ ∂ − A1,2 ρ 1 ∂ = − e− A − A1,2 Γ (ρ) k2 k1 1 1,2 ∂η ∂η2 ∂η1 − 1 = [by the general Leibniz formula] k2 k2 j1 ( 1) ∂ k1 1 j1 ∂ A1 2 = − [ − ( 1) e− , k j1 Γ (ρ) ∂η 2 j1, j2, j3 − ∂η · 2 j1+j2+j3=k1 1 1 X − j2 j3+1 j ∂ ρ 1 j +2 ∂ ( 1) 2 A − ( 1) 3 A ] j2 1,2 j3+1 1,2 · − ∂η1 − ∂η1 l1+j1 1 k2 k1 1 l1+j1 ∂ A1 2 = − ( 1) e− , l j1 Γ (ρ) l1,l2,l3 j1, j2, j3 − ∂η 1 ∂η · l1+l2+l3=k2 j1+j2+j3=k1 1 " 2 1 # X X − l2+j2 l3+j3+1 l +j ∂ ρ 1 l +j ∂ ( 1) 2 2 A − ( 1) 3 3 A . l2 j2 1,2 l3 j3+1 1,2 · " − ∂η2 ∂η1 # " − ∂η2 ∂η1 # Under the assumptions that α (0, 1) and ρ (0, 1] and recalling that t1,t2 t1 0, it is easy to ∈ ∈ ρ 1− ≥ A1,2 show that A1,2 is a Bernstein function w.r.t. η1 and η2. On the other hand, A1,−2 and e− are both CM, being the composition of CM and Bernstein functions (see Theorem 3.7 in [30], p.27). Moreover it can be checked that, in the last step, the sign of the partial derivatives depends only on the order, regardless of the variables involved; thus the functions inside square brackets are all non-negative. This procedure can be successively applied in order to prove that (1.14) is true for any integer n 3. ≥ Definition 2 We define the process := (t),t 0 , under the assumption that, for t =0, Xα,ρ {Xα,ρ ≥ } 0 α,ρ(t0)=0, almost surely, by the n-times Laplace transform of its finite dimensional distributions Xgiven in (3.1), i.e. n α Γ ρ; Ξk,n (tk tk 1) Pn η (t ) k=1 − − Ee− k=1 kXα,ρ k := , η , ...η 0,n N, P Γ (ρ) 1 n ≥ ∈ for Ξα := ( n η )α and α (0, 1), ρ (0, 1]. k,n i=k i ∈ ∈ Remark 3 ItP is easy to check that α,ρ displays the self-similar property of order 1/α, for any ρ: X 1 d indeed we have the following equality of all the finite-dimensional distributions a α (t),t 0 = Xα,ρ ≥ α,ρ(at),t 0 , for any a 0. n o {X On the other≥ } hand, it is≥ a L´evy process only for ρ = 1, when its distribution becomes infinitely divisible. Moreover, only in this special case the process has stationary and independent increments, since formula (3.1) can be rewritten as n n α α Ξk,n(tk tk−1) Φ(η1, ...ηn; t1, ...tn) = exp Ξk,n (tk tk 1) = e− − − − (k=1 ) k=1 n X n Y α α Ξ 1(t t −1) Ξ [ 1(t ) 1(t −1)] = Ee− k,nXα, k − k = Ee− k,n Xα, k −Xα, k . kY=1 kY=1 10 It is evident that, for ρ = 1, the increments have an α-stable distribution (see [29], p.113) and thus reduces to the stable process. Xα,1 We now consider the one-dimensional distribution of the process and we prove that it can be written in terms of the Fox’s H-function defined in (1.7). Theorem 4 The transition density hρ (x, t) of , for any ρ (0, 1/α], is given by α Xα,ρ ∈ α α 0,1 x (1 ρ, 1) hρ (x, t)= H − , x> 0, t 0. (3.2) α xΓ(ρ) 1,1 t (0, α) ≥ Proof. We invert the one-dimensional Laplace transform of the process , i.e. Xα,ρ α η (t) Γ (ρ; η t) Ee− Xα,ρ = , η> 0, (3.3) Γ (ρ) by using (2.10), as follows γ+i α s γ+i α s ρ 1 ∞ Γ(s + ρ)(tx− )− α ∞ Γ(s + ρ)(tx− )− hα(x, t) = ds = ds 2πi γ i xΓ(ρ)Γ(αs)s 2πi γ i xΓ(ρ)Γ(αs + 1) Z − ∞ Z − ∞ α α 1,0 α (1, α) α 0,1 x (1 ρ, 1) = H tx− = H − , (3.4) xΓ(ρ) 1,1 (ρ, 1) xΓ(ρ) 1,1 t (0, α) which exists for any x = 0, since α 1 < 0 (see Theorem 1.1 in [23]). 6 − Remark 5 We can check that (3.2) is a proper density function, for any t, by applying the Mellin transform and recalling formula (2.8) in [23]: + + α ∞ α ∞ 1 0,1 x (1 ρ, 1) hρ (x, t)dx = H − dx α Γ(ρ) x 1,1 t (0, α) Z0 Z0 + 1 ∞ 1 0,1 y (1 ρ, 1) 1 = H1,1 − dy = Γ(ρ)=1. Γ(ρ) 0 y t (0, α) Γ(ρ) Z The H-function distribution, which includes, among others, the gamma, beta, Weibull, chi-square, exponential and the half-normal distributions as particular cases, has been introduced by [6] and studied in many other papers (see, for example, [31], [35]). For ρ =1, formula (3.2) reduces to α 1 1 0,1 x (0, 1) h (x, t) = αx− H = [by (1.59) in [23]] α 1,1 t (0, α) 1 1 0,1 x (0, ) = x− H α = [by (1.60) in [23], for σ = 1] 1,1 t1/α (0, 1) − s 1 1 1 s x − 1 0,1 x ( , ) 1 1 Γ(1 + α α ) t1/α ds = H − α α = − t1/α 1,1 t1/α ( 1, 1) t1/α 2πi Γ(2 s) − ZL 1 1 − 1 1 0,1 x (1 , ) = H − α α = [by (1.58) in [23]] t1/α α 1,1 t1/α (0, 1) 1 1 t1/α (1, 1) = H1,0 , t1/α α 1,1 x ( 1 , 1 ) α α which coincides, for t = 1, with the expression of the stable law given in (2.28) of [23], in terms of H-functions. 11 On the other hand, for α = 1, formula (3.2) reduces to (2.6) and thus Theorem 4 shows that the spectral density K( ) of the fractional relaxation function ϕ ( ) coincides with the density of . · 1,ρ · X1,ρ We now study the dependence structure of the process α,ρ, by means of the auto-codifference function: adapting the definition given in [29] and [38] to theX Laplace transform (since our process is non-negative valued), we can write, from (3.1), CD( (t), (s)) (3.5) Xα,ρ Xα,ρ η1 (t) η2 (s) η1 (t) η2 (s) : = ln Ee− Xα,ρ − Xα,ρ ln Ee− Xα,ρ ln Ee− Xα,ρ α α − − Γ(ρ; (η1 + η2) s)Γ(ρ; η2 (t s)) = ln α α − . Γ(ρ; η1 s)Γ(ρ; η2 s) By considering the asymptotic behavior (2.8) of the incomplete gamma function, we easily get CD( (t), (s)) (ηα ηα) t, (3.6) Xα,ρ Xα,ρ ≃ 1 − 2 for t + and for fixed s. The limiting expression (3.6) coincides with the auto-codifference of the α-stable→ subordinator.∞ Therefore, we can consider the parameter ρ as a measure of the local deviation from the temporal dependence structure displayed in the standard α-stable case. In order to obtain the Cauchy problem satisfied by the one-dimensional density of α,ρ given in (3.2), we distinguish the two cases: 0 <ρ 1 and ρ (0, 1/α]. X ≤ ∈ 3.1 Case 0 < ρ 1 ≤ When we restrict to the case ρ 1, we can obtain the integro-differential equation satisfied by the density (3.2), in terms of a convolution-type≤ derivative with space-dependent tail L´evy measure. Theorem 6 Let p ( ) be the law of the α-stable r.v. S (1), for α (0, 1), then, for ρ (0, 1), α · α ∈ ∈ 1. 1 ρ 1 1/α ρw α p (z/w ) M (dw)= − − − α dw, w> 0, (3.7) z Γ(1 ρ) − is a L´evy measure, for any z > 0. ρ 2. If we define the convolution-type derivative (with space-dependent tail L´evy measure) z t := t + D d ∞ 0 dt u(t s)Mz(ds), where Mz(s) = s M z(dw), for any z > 0, then the transition density hρ (x, t) satisfies− the following integro-differential equation: R α R x ρh(x z,t)dz = C αρh(x, t), (3.8) zDt − − Dx Z0 with the following conditions: h(x, 0) = δ(x) limx 0+ h(x, t)=0, ↓ for any x 0 and t 0,respectively. ≥ ≥ Proof. 1. We apply the Property 1.2.15 on the tail’s behavior of the stable law, given in [29]: α 1+β α limx + x P (X > x) = Cα σ , for Cα given in (1.2.9), β = 1 (since the α-stable considered → ∞ 2 here is totally skewed to the right) and for σ = (cos(πα/2)w)1/α. Thus, by taking the first derivative, α 1 we get that p (x) = O C′ wx− − , for x + and a constant C′ . Thus we can conclude that α α → ∞ α 1/α 1 +2 p (z/w ) = O Kw α , for w 0, for a positive constant K. We now verify that the following α → finiteness condition is satisfied by M z: + 1 + ∞ 1 ρ 1/α ∞ 1 ρ 1 1/α (z 1)M (dz)= w− α − p (z/w )dw + w− α − − p (z/w )dw < . ∧ z α α ∞ Z0 Z0 Z1 12 The second integral is finite, since the stable law is finite in the origin, while for the first integral 1 ρ 1/α 2 ρ we can consider that the integrand function w− α − pα(z/w )= O Kw − , for w 0. − 1 −→ρ−1 1 + ρw α p (z/w /α) ∞ α 2. We notice that the tail measure Mz(s) is finite, for any s,z > 0, since s Γ(1 ρ) dw < − − 1 −ρ−1 + ρw α ∞ α s Γ(1 ρ) dw < Then, by applying Lemma 1, with λ = η , we haveR that − ∞ R αρ t ρη d α αρ ϕ α (t s)Γ( ρ; η s)ds = η ϕ α (t), Γ(1 ρ) dt η ,ρ − − − η ,ρ − Z0 α ρΓ( ρ;η s) αρ with ϕηα,ρ(0) = 1, is satisfied by (3.3). Let now µη(s) := Γ(1− ρ) = η− ν(s), then it is the tail of a αρ − new L´evy measure denoted by µη(s)= η− ν(s) (since it differs only by a constant from the original one). Then we get t t ρ d α d ϕηα,ρ(t s)Γ( ρ; η s)ds = ϕηα,ρ(t s)µη(s)ds Γ(1 ρ) 0 dt − − 0 dt − − t Z Z αρ d = η− ϕ α (t s)ν (s)ds = ϕ α (t), dt η ,ρ − η − η ,ρ Z0 which moreover gives, by considering the expression of νη(s), t + ρ d ∞ ηαw ρ 1 αρ ϕ α (t s) e− w− − dwds = η ϕ α (t). Γ(1 ρ) dt η ,ρ − − η ,ρ − Z0 Zs ηαw We can now invert the Laplace transform (in the variable η) in the l.h.s. since ϕηα,ρ(t)e− = hρ ( ,t) p ( , w) = hρ ( ,t) p ( , w) , where denotes the convolution operator: L{ α · }L{ α · } L{ α · ∗ α · } ∗ t x + ρ d ρ ∞ ρ 1 dz h (x z,t s) p (z, w)w− − dwds Γ(1 ρ) dt α − − α − Z0 Z0 Zs = [by the self-similarity property] x t + ρ d ρ ∞ 1/α 1 ρ 1 = dz h (x z,t s) p (z/w )w− α − − dwds, Γ(1 ρ) dt α − − α − Z0 Z0 Zs which coincides with the l.h.s. of (3.8). The r.h.s. follows by taking into account the Laplace transform of the Caputo derivative (1.18), of order αρ < 1, since α,ρ < 1, and the second initial condition, which ρ will be proved below to hold for hα(x, t), together with the first one. ρ Finally, hα(x, 0) = δ(x), by the assumption that α,ρ(0) = 0, almost surely. On the other hand, the limit for x 0+ follows from the following considerations:X the density (3.2) can be rewritten (by (1.60) in [23], with→ σ = 1/α) as − α 1 α 0,1 x (1 ρ , 1) hρ (x, t)= H − − α . α t1/αΓ(ρ) 1,1 t ( 1, α) − The limit for x 0+ is then obtained by considering the asymptotic behavior of the previous → 1 expression (see Theorem 1.3.(ii) in [23]) and since, in this case, µ = α 1 < 0, δ = ρ + α 2 and since A B =1 α> 0: thus we get that − − 1 − 1 − αρ+1− 3α ρ 2 1 α + h (x, t)= O x− 1−α exp (1 α)α 1−α x− 1−α , x 0 . α − − → n o Remark 7 We recall that L´evy measures expressed in terms of α-stable densities have been already treated in [2], in connection with stable processes subordinated by independent Poisson processes. More- over, pseudo-differential operators defined by means of time-dependent L´evy measures are used in [26], [3] and [4], in connection with time-inhomogeneous subordinators. 13 3.2 Case 1 < ρ 1/α ≤ For ρ greater than 1, we derive the fractional differential equation satisfied by the transition density, at least in the special case where ρ =2. To this aim we recall the definition of the logarithmic differential α operator introduced in [2]: let f denote the Fourier transform, then x,c is the operator with the following symbol P \ ( α f)(ξb)= ln(1 + cψα(ξ))f(ξ), c> 0, ξ < 1/c, (3.9) Px,c − θ | | for any f c. ∈ L b Theorem 8 2 Let α (0, 1) and let hα(x, t) denote the transition density of the process defined in (3.2), for ρ =2, then∈ α α 0,1 x ( 1, 1) h2 (x, t)= H − (3.10) α x 1,1 t (0, α) satisfies the following equation ∂ α ∂ α α ∂ u(x, t)= x, αu(x, t) x,tu(x, t)+ x,t u(x, t) , (3.11) ∂t D − − ∂tP P ∂t with initial condition u(x, 0) = δ(x). Proof. By taking the Fourier transform of (3.11) and considering (1.19)-(1.20) (for θ = α) and (3.9), we get − ∂ u(ξ,t) (3.12) ∂t α ∂ α α ∂ = ψb α(ξ)u(ξ,t)+ ln(1 + tψ α(ξ))u(ξ,t) ln(1 + tψ α(ξ)) u(ξ,t) − − ∂t − − − ∂t α α ψ α(ξ) α ∂ = ψ α(ξ)ub(ξ,t)+ − α u(ξ,t) +b ln(1 + tψ α(ξ)) u(ξ,t) b − − 1+ tψ α(ξ) − ∂t − α ∂ ln(1 + tψb α(ξ)) u(ξ,t) b b − − ∂t α α ψ α(ξ) = ψ α(ξ)u(ξ,t)+ b− α u(ξ,t) − − 1+ tψ α(ξ) − On the other hand, we evaluateb the Fourier transformb of (3.2) as follows (by considering formulae 14 (2.49) and (2.59) of the sine and cosine transform of the H-function): + α ∞ α 0,1 x (1 ρ, 1) eiξx H − dx (3.13) x 1,1 t (0, α) Z−∞ + α ∞ 1 0,1 x (1 ρ, 1) = α cos(ξx) H − dx x 1,1 t (0, α) Z−∞ + α ∞ 1 0,1 x (1 ρ, 1) +iα sin(ξx) H − dx x 1,1 t (0, α) Z−∞ α α 1 α α√π 0,2 2 (1, ) (1 ρ, 1) ( , ) = H 2 − 2 2 2 3,1 tξα (0, α) α 1 α α iα√π 0,2 2 ( , ) (1 ρ, 1) (1, ) + H 2 2 − 2 2 3,1 tξ α (0, α) s αs α√π 1 2α − Γ( )Γ(ρ s) = − 2 − ds 2 2πi tξα Γ(1 αs)Γ( 1 + αs ) ZL − 2 2 α s 1 αs iα√π 1 2 − Γ( 2 2 )Γ(ρ s) + α − −αs ds 2 2πi tξ Γ(1 αs)Γ(1 + 2 ) ZL − = [by the duplication formula] s αs αs α 1 1 − Γ( )Γ(ρ s)Γ( ) = − 2 − 2 ds 4 2πi tξα Γ(1 αs)Γ(αs) ZL s − 1 1 − Γ( αs)Γ(ρ s) +iαπ α αs − − αs ds 2πi tξ Γ( 2 )Γ(1 αs)Γ(1 + 2 ) ZL − − = [by the reflection formula] s 1 1 1 − sin(παs)Γ(ρ s) = − ds −2 2πi tξα sin(παs/2)s ZL s 1 1 − sin(παs/2)Γ(ρ s) +i − ds 2πi tξα s ZL 1 α s cos(παs/2)Γ(ρ + s) = (tξ )− ds 2πi s ZL i α s sin(παs/2)Γ(ρ + s) + (tξ )− ds 2πi s ZL s 1 iπα α − Γ(ρ + s) = e− 2 tξ ds. 2πi s ZL In this special case ρ = 2, we can write (3.13) as follows + α s ∞ iξx α 0,1 x (1 ρ, 1) 1 iπα α − e H − dx = e− 2 tξ (1 + s)Γ(s)ds x 1,1 t (0, α) 2πi Z−∞ ZL s s 1 iπα α − 1 iπα α − = e− 2 tξ Γ(s)ds + e− 2 tξ Γ(s + 1)ds (3.14) 2πi 2πi ZL ZL = [by formula (1.38) in [23]] − iπα α 2 iπα α tξ e α 2 ψ−α(ξ)t α = e− 1+ tξ e− = e− (1 + tψ α(ξ)). − It is easy to check that (3.14) satisfies (3.12) as well as the initial condition. 15 4 Properties of the generalized stable law Let now consider a fix time argument and define as the r.v. with Laplace transform Xα,ρ + ∞ e wwρ 1dw α η cηα − − Γ (ρ; cη ) Ee− Xα,ρ = = , η> 0, (4.1) R Γ (ρ) Γ (ρ) for α (0, 1], ρ 1/α and scale parameter σ = c1/α R+. Then an interesting particular case can be obtained∈ for ρ≤= n N : by successive integrating by∈ parts of (4.1), it can be checked that ∈ + w n 1 n 1 α j α∞ e w dw − η cη − − cηα (cη ) Ee− Xα,ρ = = e− , (4.2) Γ (n) j! R j=0 X α thus coinciding with the cumulative distribution function of a Poisson r.v. Zcηα with parameter cη (i.e. P (Z α α s α x − α 0,1 x ( 1, 1) α 1 Γ(2 s) h2 (x) = H − = − c ds α x 1,1 c (0, α) x 2πi Γ(1 αs) ZL − α z 2 x − 2 α 1 Γ( z) c αc c = dz = W α,1 2α( ). x 2πi Γ(1 + αz 2α) x2α+1 − − −xα ZL − Moreover, when ρ = 2 and α =1/2, it reduces to s c − 2 c 1 Γ(s) √x h2 (x) = ds 1/2 2x2 2πi Γ(s/2) ZL = by the duplication formula of the gamma function] s c2 1 s 1 c − = Γ( + ) ds 4x2√π 2πi 2 2 2√x ZL c2 c = H1,0 = [by (1.125) in [23]] 4x2√π 0,1 2√x ( 1 , 1 ) 2 2 3 2 c c = e− 4x , 4x√πx3 2 which is linked to a L´evy distribution Ya with scale parameter a = c /2 by the following relationship: a h2 (x)= P (Y dx) . (4.4) 1/2 x a ∈ 2 which is linked to a L´evy distribution Ya with scale parameter a = c /2 by the following relationship: a h2 (x)= P (Y dx) . (4.5) 1/2 x a ∈ The Laplace transform can be checked to coincide with (4.1), indeed + ∞ w c√η e− wdw = e− (1 + c√η) Zc√η 16 and + 2 d ∞ w d c√η c c√η e− wdw = e− (1 + c√η)= e− dη dη − 2 Zc√η On the other hand, + + d ∞ ηx a ∞ ηx √2aη e− P (Y dx)= a e− P (Y dx)= ae− . dη x a ∈ − a ∈ − Z0 Z0 Moreover, from (4.5) it is evident that the first moment is finite and reads c2 E 1 = . X 2 ,2 2 The tail probabilities of the r.v. α,ρ can be obtained, by means of the Tauberian theorem (see e.g.[8], Theorem XIII-5-4, p.446), as follows:X + E u α,ρ ∞ ux 1 e− X e− P ( α,ρ > x)dx = − 0 X u Z α cu w ρ 1 e− w − dw = 0 = [for u 0] R uΓ (ρ) → uαρ 1 − , ≃ Γ (ρ + 1) by considering the well-known asymptotics of the (lower) incomplete gamma function. Under the condition ρ< 1/α we get, from (5.17) in [8], for a constant L( ), · αρ 1 lim x P ( α,ρ > x)= . (4.6) x + X Γ(ρ + 1)Γ(1 αρ) → ∞ − For ρ = 1 (α-stable case), formula (4.6) reduces with the well-known result (see formula (1.2.10) in [29]). + E δ ∞ δ By taking into account (4.6) and the fact that X = 0 P (X > x)dx, for any positive r.v., we can give the following condition for the finiteness of the δ-th moments: R E ( )δ < + , iff 0 < δ < αρ. Xα,ρ ∞ Therefore, in the case where ρ < 1, the r.v. α,ρ has finite moments only of fractional order less than 1, as happens for the stable law of order αX (0, 1). For any ρ, we can evaluate the moment of order δ (0, αρ), for δ = n N, by applying the∈ Mellin transform formula for the H-function given in (2.8)∈ of [23]: 6 ∈ + α δ α ∞ δ 1 0,1 x (1 ρ, 1) E ( α,ρ) = x − H − dx (4.7) X Γ(ρ) 1,1 c (0, α) Z0 + 1 ∞ δ 1 0,1 z (1 ρ, 1) = z α − H − dz Γ(ρ) 1,1 c (0, α) Z0 δ δ c α Γ(ρ ) = − α . 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