q-INVARIANT FUNCTIONS FOR SOME GENERALIZATIONS OF THE ORNSTEIN-UHLENBECK SEMIGROUP P. PATIE 8 0 Abstract. We show that the multiplication operator associated to a fractional power of a 0 Gamma random variable, with parameter q > 0, maps the convex cone of the 1-invariant 2 functions for a self-similar semigroup into the convex cone of the q-invariant functions for the n associated Ornstein-Uhlenbeck(for short OU)semigroup. Wealso describetheharmonicfunc- a tions for some other generalizations of theOU semigroup. Among thevarious applications, we J characterize, through their Laplace transforms, the laws of first passage times above and over- 4 shoot for certain two-sided α-stable OU processes and also for spectrally negative semi-stable 1 OUprocesses. TheseLaplacetransforms areexpressedintermsofanewfamily ofpowerseries which includes the generalized Mittag-Leffler functions and generalizes the family of functions ] introduced by Patie [20]. R P . h t a m 1. Introduction and main results [ Let E = R,R+ or [0, ) and let X be the realization of (Pt)t≥0, a Feller semigroup on E ∞ 1 satisfying, for α > 0, the α-self-similarity property, i.e. for any c > 0 and every f B(E), the v space of bounded Borelian functions on E, we have the following identity ∈ 1 1 (1) Pcαtf(cx)= Pt(dcf)(x), x E, 1 ∈ 2 where d is the dilatation operator, i.e. d f(x) = f(cx). We denote by (P ) the family of c c x x∈E . 1 probability measures of X which act on D(E), the Skorohod space of c`adla`g functions from 0 [0, ) to E, and by (FX) its natural filtration. We also mention that, throughout the 8 ∞ t t≥0 paper, E stands for a reference expectation operator. Moreover, A (resp. D(A)) stands for its 0 : infinitesimal generator (resp. its domain). We have in mind the following situations v i (1) E = R and X is an α-stable L´evy process. X (2) E = R+ or [0, ) and X is a 1-semi-stable processes in the terminology of Lamperti. r ∞ α a More precisely, let ξ be a L´evy process starting from x R, Lamperti [16] showed that the time ∈ change process (2) Xt = eξAt, t 0, ≥ where u A = inf u 0; V := eαξs ds > t , t u { ≥ } Z0 Iwishtothankananonymousrefereeforveryhelpfulcommentsandcarefulreadingthatledtoimprovingthe presentationofthepaper. ThisworkwaspartiallycarriedoutwhileIwasvisitingtheProjectOMEGAofINRIA at Sophia-Antipolis and the Department of Mathematics of ETH Zu¨rich. I would like to thank the members of both groups for their hospitality. 1 is an 1-semi-stable positive Markov process starting from ex. Further, we assume that < α −∞ b := E[ξ ] < and denote the characteristic exponent of ξ by Ψ. We also suppose that ξ is 1 ∞ not arithmetic (i.e. does not live on a discrete subgroup rZ for some r > 0). Then, on the one hand, for b> 0 (resp. b 0 and ξ is spectrally negative), it is plain that X has infinite lifetime ≥ and we know from Bertoin and Yor [2, Theorem 1] (resp. [3, Proposition 1]), that the family of probability measures (P ) converges in the sense of finite-dimensional distributions to a x x>0 probability measure,denotedbyP , asx 0+. Ontheother hand,ifb <0, thenitisplainthat 0 → ξ drift towards and X has a finite lifetime which is TX = inf u 0; X = 0,X = 0 . In −∞ 0 { ≥ u− u } thiscase, weassumethatthereexistsauniqueθ > 0whichyieldstheso-called Cram´ercondition (3) E[eθξ1]= 1. Then, under the additional condition 0 < θ < α, Rivero [24] showed that the minimal process (X,TX) admits an uniquerecurrentextension that hits and leaves 0 continuously a.s. and which 0 is a 1-semi-stable process on [0, ). With a slight abuse of notation, we write (P ) for the α ∞ x x>0 familyoflawsofsucharecurrentextension. Wegatherthedifferentpossibilitiesinthefollowing. H0. E = [0, ) and either b > 0 or b < 0 and (3) holds with θ < α. Moreover, if ξ is ∞ spectrally negative, the case b = 0 is also allowed. We will also use the following hypothesis H1. E [0, ). ξ has finiteexponential moments of arbitrary positive orders, i.e. ψ(m) < ⊆ ∞ ∞ for every m 0 where ψ(m) = Ψ( im). ≥ − − If ξ is spectrally negative, excluding the degenerate cases, then H1 holds and for b R, we ∈ write θ for the largest root of the equation ψ(u) = 0. Then, being continuous and increasing 0 on [θ , ), ψ has a well-defined inverse function φ : [0, ) [θ , ) which is also continuous 0 0 ∞ ∞ → ∞ and increasing. For r > 0, we write Pr = e−rtP . We say that a non-negative function, , is r-excessive t t Ir (resp. r-invariant) for P if for any t > 0, t Pr (x) (x), t Ir ≤ Ir (resp. if we have = in place of ) and lim Pr (x) = (x) pointwise. Taking r = 1 and ≤ t↓0 t Ir Ir writing simply = , we have P1 (x)= (x). The self-similarity property (1) then yields I I1 t I I (4) Ptr drα1I (r−α1x) = I(x), (cid:16)1 (cid:17) which entails the identity r(x) = (rαx) for all x E and r > 0. We denote the convex cone I I ∈ of 1-excessive (resp. 1-invariant) functions for X by E(X) (resp. I(X)). For any λ > 0, the Ornstein-Uhlenbeck (for short OU) semigroup, (Q ) , is defined, for t t≥0 f B(E), by ∈ (5) Qtf(x)= Peχ(t) de′λ(−t)f (x), x∈ E, t ≥ 0, where e (t) = eλt−1, χ = αλ and we writ(cid:16)e v (.) for(cid:17)the continuous increasing inverse function of λ λ λ e (.). We mention that such a deterministic transformation of self-similar processes traces back λ 2 to Doob [11] who studied the generalized OU processes driven by symmetric stable L´evy pro- cesses. Moreover, Carmona et al. [7, Proposition 5.8] showed that (Q ) is a Feller semigroup t t≥0 with infinitesimal generator, for f D(A), given by ∈ Uf(x)= Af(x) λxf′(x). − Let U be the realization of the Feller semigroup (Q ) . It follows from (5) that t t≥0 (6) U = e′ ( t)X , t 0. t λ − eχ(t) ≥ We deduce, with obvious notation, that TU = v (TX) a.s.. If E = R (resp. otherwise), we call 0 χ 0 U a self-similar (resp. semi-stable) OU process. We denote by (Q ) the family of probability x x>0 measures of a semi-stable OU process. We deduce, from the Lamperti mapping (2) and the discussions above the following. Proposition 1.1. For any x > 0, there exists a one to one mapping between the law of a L´evy process starting from log(x) and the law of a semi-stable OU process starting from x. More precisely, we have (7) Ut = e−λteξ△t, t < T0U, where = tU−αds. Note that for b > 0, the previous identity holds for any t 0. △t 0 s ≥ Moreover, if (3) holds with 0 < θ < α, then the minimal process (U,TU) admits a recurrent R 0 extension which hits and leaves 0 continuously a.s. which is the OU process associated to (X,P ). x We write its family of laws by (Q ) . x x>0 Under the condition H1, for any γ > 0, we denote by Q(γ) the law of the semi-stable OU x process, starting at x R+, associated to the L´evy process having Laplace exponent ψ (u) = γ ∈ ψ(u+γ) ψ(γ), u 0. − ≥ We say that a probability measure m is invariant for U if it satisfies, for any f B(E), ∈ Q f(x)m(dx) = f(x)m(dx). t ZE ZE Finally, let G be a gamma random variable independent of X, with parameter q > 0, whose q law is given by γ(dr) = e−rrq−1dr. We are now ready to state the following. Γ(q) Theorem 1.2. If E = R or H0 holds then the Feller process U is positively recurrent and its unique invariant measure is χP (X dx). 0 1 ∈ Next, assume that L1(γ(dr)). For any q > 0, we introduce the function (q;x) defined by I ∈ I 1 (8) (q;x) = χχqE χGq α x , x E. I I χ ∈ (cid:20) (cid:18) (cid:19)(cid:21) (cid:16) (cid:17) Then, if I(X)(resp. E(X)) then (q;x) Iq(U)(resp. Eq(U)). I ∈ I ∈ Consequently, if I(X), we have, for any q > 0, I ∈ −q (9) (1+χt) χPt d −1 (q;x) = (q;x), x E. (1+χt) αI I ∈ (cid:16) (cid:17) Remark 1.3. (1) We call the multiplication operator (8) associated to a fractional power of a Gamma random variable, the Γ-transform. 3 (2) The characterization of time-space invariant functions of the form (9), associated to self-similar processes, has been first identified by Shepp [26] in the case of the Brown- ian motion and by several authors for some specific processes: Yor [27] for the Bessel processes, Novikov [19] and Patie [22] for the one sided-stable processes. Whilst in the mentioned papers, the authors made used of specific properties of the studied processes to derive the time-space martingales, we provide a proof which is based simply on the self-similarity property. We proceed by investigating the process Y, defined, for any x,β R and ξ = 0 a.s., by 0 ∈ t (10) Y = eαξt x+β e−αξsds , t 0. t ≥ (cid:18) Z0 (cid:19) We call Y the L´evy OU process. We mention that this generalization of the OU process is a specific instance of the continuous analogue of random recurrence equations, as shown by de Haan and Karandikar [9]. They have been also well-studied by Carmona et al. [6], Erickson and Maller[12],Bertoinetal.[1]andbyKondoetal.[14]. In[6],itisprovedthatY isahomogeneous Markov process with respect to the filtration generated by ξ. Moreover, they showed, from the stationarity and the independency of the increments of ξ, that, for any fixed t 0, ≥ t Y (=d) xeαξt +β eαξsds. t Z0 Then, if E[ξ ] < 0, they deduced that, as t , ξ (a.s) and Y (d) βV = ∞eαξsds. We 1 → ∞ t → −∞ t → ∞ 0 refertoBertoinandYor[4]forathoroughsurveyontheexponentialfunctionalofL´evyprocesses. R In the spectrally negative case, it is well know that the law of V is self-decomposable, hence ∞ absolutely continuous and unimodal. Moreover, under the additional assumption that θ < α, its law has been computed in term of the Laplace transform by Patie [20]. Now, we introduce the process Z defined, for any x= 0,β R and ξ = 0 a.s., by 0 6 ∈ t −1 (11) Z = eαξt x+β eαξsds , t 0. t ≥ (cid:18) Z0 (cid:19) Before stating the next result, we introduce some notation. Let B be a Borel subset of E and we write TU for the first exit time from B by U. With a slight abuse of terminology, we say B that for any x E, a non-negative function is a (q∆,B)-harmonic function for (U,Q ) if x ∈ H (12) Ex e−q△TBUH(UTBU)I{TBU<T0U} = H(x). h i When ∆ is replaced by t in the previous expression, we simply say that is a (q,B)-harmonic t H function for (U,Q ). We are ready to state the following. x Theorem 1.4. Set β = αλx in (11). Then, to a process Z starting from 1, with x = 0, one x 6 can associate a semi-stable OU process (U,Q ) such that 1 (13) Z = x−1Uα , t < TU, t ▽t 0 where = tZ ds and its inverse is given by = tU−αds. Note that for b> 0, the previous ▽t 0 s △t 0 s identity holds for any t 0. Consequently, with x > 0, Z is a Feller process on (0, ). R ≥ R ∞ Moreover, letq > 0, 0 a< b + and x (a,b). Then, a(q∆,TU )-harmonic function ≤ ≤ ∞ ∈ (ax,bx) for (U,Q ) is a (q,TZ )-harmonic function for the process Z starting from x−α. Similarly, a 1 (aα,bα) 4 (q∆,T(Ux,x))-harmonic for (U,Q1) is a (q,T(Ybaα,bα))-harmonic function for the process Y starting b a from xα, the L´evy OU process associated to the L´evy process ξ = ξ, the dual of ξ with respect − b to the Lebesgue measure. Finally, assume that H1 holds and write p (x) = xq forbx,q > 0. If the function is q H (λφ(q),B)-harmonic functionfor(U,Q(φ(q)))then the functionp is(q∆,B)-harmonic func- x φ(q) H tion for (U,Q ). x 2. Proofs 2.1. Proof of Theorem 1.2. The description of the unique invariant measure is a refinement of [7, Proposition 5.7] where therein the proof is provided for R-valued self-similar processes and can be extended readily for the R+-valued case under the condition H0, which ensures that (X,P ) admits an entrance law at 0. x Next, let us assume that L1(γ(dr)) I(X). We need to show that for any q > 0, I ∈ ∩ e−qtQ (q;x) = (q;x). For x E, we deduce from the definition of (Q ) that t t t≥0 I I ∈ q e−qtQt (q;x) = χχ e−qtEx ∞ (χr)α1Ut e−rrχq−1dr I Γ χq (cid:20)Z0 I(cid:16) (cid:17) (cid:21) (cid:16)q (cid:17) = Γχχχq e−qtEx(cid:20)Z0∞I(cid:16)(χr)α1e′λ(−t)Xeχ(t)(cid:17)e−rrχq−1dr(cid:21). Using the change of variable u =(cid:16) χ(cid:17)e′ ( t)r, Fubini theorem and (4), we get χ − e−qtQtI(q;x) = Γ 1χq Ex(cid:20)Z0∞e−ueχ(t)I(cid:16)uα1Xeχ(t)(cid:17)e−χuuχq−1du(cid:21) (cid:16)1 (cid:17) ∞ 1 −u q−1 = uαx e χuχ du Γ χq Z0 I(cid:16) (cid:17) = ((cid:16)q;x(cid:17)) I where the last line follows after the change of variable u = χr. The case L1(γ(dr)) E(X) I ∈ ∩ is obtained by following the same line of reasoning. The last assertion is deduced from (5) and (8) by performing the change of variable u= v (t), with v (t) = 1 log(1+χt). χ χ χ 2.2. Proof of Theorem 1.4. Setting β = αλx, the Lamperti mapping (2) yields t −1 Z = x−1eαξt 1+αλ eαξsds t (cid:18) Z0 (cid:19) α 1 = x−1 X 1 . (1+αλ.)α ! Vt = x−1 U α vχ(.) Vt 5 (cid:0) (cid:1) wherethelastidentityfollowsfrom(5). Theproofoftheassertion(13)iscompletedbyobserving that t −1 (v (V ))′ = eαξt 1+αλ eαξsds . χ t (cid:18) Z0 (cid:19) Moreover, since the mapping x xα is a homeomorphism of R+, the Feller property follows 7→ from its invariance by ”nice” time change of Feller processes, see Lamperti [15, Theorem 1]. We also obtain the following identities TZ = inf u 0; Z / (a,b) (a,b) { ≥ u ∈ } = inf u 0; Uα / (ax,bx) ≥ ▽u ∈ 1 1 = (cid:8)inf u 0; Uu / (ax)(cid:9)α,(bx)α . △ ≥ ∈ (cid:16) n (cid:16) (cid:17)o(cid:17) The characterization of the harmonic functions of Z follows. The characterization of the har- monic functions of Yˆ are readily deduced from the ones of Z and the identity 1 Yˆ = , t 0. t Z ≥ t The proof of Theorem is then completed by using the following Lemma together with an appli- cation of the optional stopping theorem. Lemma 2.1. Assume that H1 holds, then for γ,δ 0 and x > 0, we have ≥ γ−δ U (1) dQ(γ) = t eλ(γ−δ)t−(ψ(γ)−ψ(δ))△tdQ(δ), on FU t < TU . x x x t ∩{ 0 } (cid:18) (cid:19) Note that for b> 0 the condition ” on t < TU ” can be omitted. For the particular case γ = θ { 0 } and δ = 0, the absolute continuity relationship (1) reduces to θ U dQ(θ) = t eλθtdQ , on FU t < TU . x x x t ∩{ 0 } (cid:18) (cid:19) Proof. We start by recalling that in [20], the following power Girsanov transform has been derived, under H1, for γ,δ 0 and x > 0, with obvious notation, ≥ γ−δ X dP(γ) = t e−(ψ(γ)−ψ(δ))AtdP(δ), on FX t < TX . x x x t ∩{ 0 } (cid:18) (cid:19) The assertion (1) follows readily by time change and recalling that t A = U−αdu. eχ(t) u Z0 We complete the proof by recalling that for b> 0, U does not reach 0 a.s.. (cid:3) 3. Applications In this section, we illustrate our results to some new interesting examples. 6 3.1. First passage times and overshoot of stable OU processes. Let X be an α-stable L´evy process whose characteristic exponent satisfy, for u R, ∈ απ Ψ(iu) = cuα 1 iβsgn(u)tan − | | − 2 (cid:16) (cid:16) (cid:17)(cid:17) where 1 < α < 2 and for convenience we take c = 1+β2tan2 απ −1/2. Then, we introduce 2 the constant ρ= P(X > 0) which was evaluated by Zolotarev [28] as follows 1 (cid:0) (cid:0) (cid:1)(cid:1) 1 1 απ ρ= + tan−1 βtan . 2 πα 2 (cid:16) (cid:16) (cid:17)(cid:17) Following Doney [10], we introduce, for any integers k,l, the class C of stable processes such k,l that ρ+k = lα˜ where α˜ = 1. For m N, x R and z C, introduce the function α ∈ ∈ ∈ m f (x,z) = z+eix(m−2i)π . m Yi=0(cid:16) (cid:17) Next, we recall from the Wiener-Hopf factorization of L´evy processes due to Rogozin [25], that the law of the first passage times τX and the over(under)shoot of X at the level 0 is described 0 by the following identities, for δ,r > 0 and p 0, ≥ Z0∞e−δxE−x(cid:20)e−rτ0X−pXτ0X−(cid:21)dx = δ−1 p 1− ΨΨ++((−−rrαα11pδ))! Z0∞e−δxEx(cid:20)e−rτ0X−pXτ0X−(cid:21)dx = δ−1 p 1− ΨΨ−−((rrαα11pδ))! where(1 Ψ(δ))−1 = Ψ−(δ)Ψ+(δ). HereΨ+(δ)(resp.Ψ−(δ))isanalyticin (δ) < 0(resp. (δ) > − ℜ ℜ 0) continuous and nonvanishingon (δ) 0 (resp. (δ) 0). Doney [10] computes the Wiener- ℜ ≤ ℜ ≥ Hopf factors for stable processes in C as follows k,l f (α,( 1)l( z)α) Ψ+(z) = k−1 − − , Arg(z) = 0, f (α˜,( 1)k+1z) 6 l−1 − f (α˜,( 1)k+1z) Ψ−(z) = l−1 − , Arg(z) = π, f (α,( 1)lzα) 6 − k − where zβ stands for σβeiβφ when z = σeiφ with σ > 0 and π < φ π. Observe also that − ≤ Ψ+( xα1) x−ρforlargerealx. Moreover,usingthefactthatthefunctionEx e−rτ0X−pXτ0X ,x − ∼ ∈ (cid:20) (cid:21) R, is r-excessive for the semigroup of X, we deduce from the Γ-transform the following. 7 Corollary 3.1. For any q,δ > 0, p 0, and for any integers k,l such that X C , we have k,l ≥ ∈ Z−0∞eδxEx(cid:20)e−qτ0U−pUτ0U−(cid:21)dx = δ−1 p χχq − Γ(1χq) Z0∞ ΨΨ++((−−rrαα11pδ))e−χrrχq−1dr! Z0∞e−δxEx(cid:20)e−qτ0U−pUτ0U−(cid:21)dx = δ−1 p χχq − Γ(1χq) Z0∞ ΨΨ−−((rrαα11pδ))e−χrrχq−1dr!. 3.2. First passage times of one-sided semi-stable- and L´evy-OU processes. We now fix (P ) to be the semigroup of a spectrally negative 1-semi-stable process X. X is then t t≥0 α associated via the Lamperti mapping (2) to a spectrally negative L´evy process, ξ, which we assumetohaveafinitemeanb. ItscharacteristicexponentψhasthewellknownL´evy-Khintchine representation σ 0 (1) ψ(u) = bu+ u2+ (eur 1 ur)ν(dr), u 0, 2 − − ≥ Z−∞ where σ 0 and the measure ν satisfies the integrability condition 0 (r r2)ν(dr) < + . ≥ −∞ ∧ ∞ Patie [20] computes the Laplace transform of the first passage times above of X as follows. For R any r 0 and 0 x a, we have ≥ ≤ ≤ (rxα) (2) Ex e−rTaX = Iα,ψ(raα) α,ψ h i I where the entire function, , is given, for γ 0 and α> 0, by α,ψ I ≥ ∞ (z) = a (ψ;α)zn, z C α,ψ n I ∈ n=0 X and n a (ψ;α)−1 = ψ(αk), a = 1. n 0 k=1 Y Using the Γ-transform, we introduce the following power series ∞ (3) (q;z) = a (ψ;α)(q) zn α,ψ n n I n=0 X Γ(q+n) where (q) = is the Pochhammer symbol and we have used the integral representation n Γ(q) of the gamma function Γ(q) = ∞e−rrq−1dr, (q) > 0. By means of the following asymptotic 0 ℜ formula of ratio of gamma functions, see e.g. Lebedev [17, p.15], for δ > 0, R δ(2n+δ 1) (4) (z+n) = zδ 1+ − +O(z−2) , arg z <π ǫ, ǫ > 0, δ 2z | | − (cid:20) (cid:21) we deducethat (q;z) is an entire function in z and is analytic on the domain q C; (q) > α,ψ I { ∈ ℜ 1 . For b< 0, we recall that there exists θ > 0 such that ψ(θ) = 0 and thus ψ (u) = ψ(θ+u). θ − } In this case, by setting θ = θ, it is shown in [20] that there exists a positive constant C such α α θα that (xα) C xθ (xα) as x . Iα,ψ ∼ θα Iα,ψθ → ∞ 8 We also introduce the function (q;xα) defined by α,ψ,θ N Γ(q+θ ) (5) (q;xα) = (q;xα) C xθ α (q+θ ;xα), (x) 0. Nα,ψ,θ Iα,ψ − θα Γ(q) Iα,ψθ α ℜ ≥ Moreover,ifweassumethatthereexistsβ [0,1]andaconstanta > 0suchthatlim ψ(u)/u1+β = β u→∞ ∈ a , then C is characterized by β θα Γ(1−θα) (θα−1)! , if θ is a positive integer, α Qθα−1ψ(αk) α k=1 C =  θα  Γ(1−θα)a−θαeEγβθα ∞ e−βθkα (k+θα)ψ(αk), otherwise, α β k=1 kψ(αk+θα) where E stands for the Euler-MascQheroni constant. We recall, also from [20], that, for r,x 0, γ  ≥ Ex e−rT0X = Iα,ψ(rxα)−Cθα(rα1x)θIα,ψθ(rxα). h i We deduce from Theorems 1.2 and 1.4 the following. Corollary 3.2. Let q 0 and 0 < x a. Then, ≥ ≤ q;χxα Ex e−qTaU = Iα,ψ(cid:16)χq;χaα(cid:17) h i Iα,ψ χ (cid:16) (cid:17) and E 1+χTX −χq = Iα,ψ χq;χxα x(cid:20)(cid:16) (α)(cid:17) (cid:21) Iα,ψ(cid:16)χq;χaα(cid:17) (cid:16) (cid:17) where T(Xα) = inf{u ≥ 0; Xu = a(1+χu)α1}. We also deduce that Exhe−q△TaUI{TaU<T0U}i = (cid:16)xb(cid:17)γ IIαα,,ψψγγ(cid:0)ααγγ;;χχxaαα(cid:1). Moreover, assume b> 0 and set β = αλx and γ = φ(q). Then(cid:0), (cid:1) 1 γ γ;χ 1 Ex1 he−qTaZi =(cid:18)bx(cid:19) Iα,Iψαγ,ψγαγ;(cid:0)χα(ax(cid:1))α , 0 < x ≤ a, (cid:0) (cid:1) Ea e−qTxYb = xa αγ Iα,ψγ (cid:16)γ;αγ;χχaα1(cid:17)α1 , 0< x≤ a. h i (cid:16) (cid:17) Iα,ψγ α x (cid:16) (cid:17) Finally, if b< 0 and 0< θ < α, we have (cid:0) (cid:1) q;χxα Ex e−qT0U = Nα,ψ,θ(cid:16)χq;χxα(cid:17). h i Nα,ψ,θ χ (cid:16) (cid:17) 9 Remark 3.3. From the strong Markov property and the absence of positive jumps, we easily get thatfirstpassagetimesabovefortheprocessesU andZ areinfinitelydivisiblerandomvariables. Hence, we obtain fromCorollary 3.2, that thefunctions(3)and (5)areLaplace transforms, with respect to the parameter q, of infinitely divisible distributions concentrated on the positive real line. We endupby investigating somespecial cases which allow to make some connections between the power series introduced and some well-known or new special functions. 3.2.1. The confluent hypergeometric functions. We first consider a Brownian motion with drift ν, i.e. ψ(u) = 1u2 νu. Setting α = 2, we have θ = 2ν and therefore we assume ν < 1. − 2 − Its associated semi-stable process is well known to be a Bessel process of index ν and thus the associated Ornstein-Uhlenbeck process is, in the case n = 2ν +1 N, the radial norm of ∈ n-dimensional Ornstein-Uhlenbeck process. We get (x) = (x/2)ν/2Γ( ν +1)I √2x 2,ψ −ν I − (cid:16) (cid:17) where I (x) = ∞ (x/2)ν+2n stands for the modified Bessel function of index ν, see e.g. [17, ν n=0 n!Γ(ν+n+1) 5.], and P x2 (q;x2) = Φ q,1 ν, 2,ψ I − 2 (cid:18) (cid:19) x2 (q;x2) = Φ q+ν,ν +1, I2,ψ2ν 2 (cid:18) (cid:19) where Φ(q,ν,x) = ∞ (q)n xn stands for the confluent hypergeometric function of the first n=0 (ν)nn! kind, see e.g. [17, 9.9]. Using the asymptotic behavior of the Bessel function P ex I (x) as x , ν ∼ √2πx → ∞ Γ(−ν) we deduce that C = . Hence, 2ν − Γ(ν) x2 Γ( ν)Γ(q+ν) x2 (q;x2) = Φ q,1 ν, +x2ν − Φ q,1 ν, Nα,ψ2ν − 2 Γ(ν)Γ(q) − 2 (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) Γ(q)Γ(q+ν) x2 = Λ q,ν +1, Γ(ν) 2 (cid:18) (cid:19) where Λ(q,ν +1, x2) is the confluent hypergeometric of the second kind. We mention that, in 2 this case, the results of Corollary 3.2 are well-known and can be found in Matsumoto and Yor [18] and in Borodin and Salminen [5, II.8.2]. 3.2.2. SomegeneralizationoftheMittag-Lefflerfunction. Patie[21]introducedanewparametric family of one-sided L´evy processes which are characterized by the following Laplace exponents, for any 1 < α< 2, and γ > 1 α, − 1 (6) ψ (u) = ((u+γ 1) (γ 1) ). γ α α α − − − 10