q-Invariant Functions for Some Generalizations of the Ornstein-Uhlenbeck Semigroup
aa r X i v : . [ m a t h . P R ] J a n q -INVARIANT FUNCTIONS FOR SOME GENERALIZATIONS OF THEORNSTEIN-UHLENBECK SEMIGROUP P. PATIE
Abstract.
We show that the multiplication operator associated to a fractional power of aGamma random variable, with parameter q >
0, maps the convex cone of the 1-invariantfunctions for a self-similar semigroup into the convex cone of the q -invariant functions for theassociated Ornstein-Uhlenbeck (for short OU) semigroup. We also describe the harmonic func-tions for some other generalizations of the OU semigroup. Among the various applications, wecharacterize, through their Laplace transforms, the laws of first passage times above and over-shoot for certain two-sided α -stable OU processes and also for spectrally negative semi-stableOU processes. These Laplace transforms are expressed in terms of a new family of power serieswhich includes the generalized Mittag-Leffler functions and generalizes the family of functionsintroduced by Patie [20]. Introduction and main results
Let E = R , R + or [0 , ∞ ) and let X be the realization of ( P t ) t ≥ , a Feller semigroup on Esatisfying, for α >
0, the α -self-similarity property, i.e. for any c > f ∈ B (E), thespace of bounded Borelian functions on E, we have the following identity(1) P c α t f ( cx ) = P t ( d c f ) ( x ) , x ∈ E , where d c is the dilatation operator, i.e. d c f ( x ) = f ( cx ). We denote by ( P x ) x ∈ E the family ofprobability measures of X which act on D (E), the Skorohod space of c`adl`ag functions from[0 , ∞ ) to E, and by ( F Xt ) t ≥ its natural filtration. We also mention that, throughout thepaper, E stands for a reference expectation operator. Moreover, A (resp. D ( A )) stands for itsinfinitesimal generator (resp. its domain). We have in mind the following situations(1) E = R and X is an α -stable L´evy process.(2) E = R + or [0 , ∞ ) and X is a α -semi-stable processes in the terminology of Lamperti.More precisely, let ξ be a L´evy process starting from x ∈ R , Lamperti [16] showed that the timechange process(2) X t = e ξ At , t ≥ , where A t = inf { u ≥ V u := Z u e αξ s ds > t } , I wish to thank an anonymous referee for very helpful comments and careful reading that led to improving thepresentation of the paper. This work was partially carried out while I was visiting the Project OMEGA of INRIAat Sophia-Antipolis and the Department of Mathematics of ETH Z¨urich. I would like to thank the members ofboth groups for their hospitality. s an α -semi-stable positive Markov process starting from e x . Further, we assume that −∞ b > b ≥ ξ is spectrally negative), it is plain that X has infinite lifetimeand we know from Bertoin and Yor [2, Theorem 1] (resp. [3, Proposition 1]), that the familyof probability measures ( P x ) x> converges in the sense of finite-dimensional distributions to aprobability measure, denoted by P , as x → b <
0, then it is plain that ξ drift towards −∞ and X has a finite lifetime which is T X = inf { u ≥ X u − = 0 , X u = 0 } . Inthis case, we assume that there exists a unique θ > E [ e θξ ] = 1 . Then, under the additional condition 0 < θ < α , Rivero [24] showed that the minimal process(
X, T X ) admits an unique recurrent extension that hits and leaves 0 continuously a.s. and whichis a α -semi-stable process on [0 , ∞ ). With a slight abuse of notation, we write ( P x ) x> for thefamily of laws of such a recurrent extension. We gather the different possibilities in the following. H0 . E = [0 , ∞ ) and either b > b < θ < α . Moreover, if ξ isspectrally negative, the case b = 0 is also allowed.We will also use the following hypothesis H1 . E ⊆ [0 , ∞ ). ξ has finite exponential moments of arbitrary positive orders, i.e. ψ ( m ) < ∞ for every m ≥ ψ ( m ) = − Ψ( − im ).If ξ is spectrally negative, excluding the degenerate cases, then H1 holds and for b ∈ R , wewrite θ for the largest root of the equation ψ ( u ) = 0. Then, being continuous and increasingon [ θ , ∞ ), ψ has a well-defined inverse function φ : [0 , ∞ ) → [ θ , ∞ ) which is also continuousand increasing.For r >
0, we write P rt = e − rt P t . We say that a non-negative function, I r , is r -excessive(resp. r -invariant) for P t if for any t > P rt I r ( x ) ≤ I r ( x ) , (resp. if we have = in place of ≤ ) and lim t ↓ P rt I r ( x ) = I r ( x ) pointwise. Taking r = 1 andwriting simply I = I , we have P t I ( x ) = I ( x ). The self-similarity property (1) then yields(4) P rt (cid:16) d r α I (cid:17) ( r − α x ) = I ( x ) , which entails the identity I r ( x ) = I ( r α x ) for all x ∈ E and r >
0. We denote the convex coneof 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 t ) t ≥ , is defined, for f ∈ B (E), by(5) Q t f ( x ) = P e χ ( t ) (cid:16) d e ′ λ ( − t ) f (cid:17) ( x ) , x ∈ E , t ≥ , where e λ ( t ) = e λt − λ , χ = αλ and we write v λ ( . ) for the continuous increasing inverse function of e λ ( . ). We mention that such a deterministic transformation of self-similar processes traces back o 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 t ) t ≥ is a Feller semigroupwith infinitesimal generator, for f ∈ D ( A ), given by U f ( x ) = A f ( x ) − λxf ′ ( x ) . Let U be the realization of the Feller semigroup ( Q t ) t ≥ . It follows from (5) that(6) U t = e ′ λ ( − t ) X e χ ( t ) , t ≥ . We deduce, with obvious notation, that T U = v χ ( T X ) a.s.. If E = R (resp. otherwise), we call U a self-similar (resp. semi-stable) OU process. We denote by ( Q x ) x> the family of probabilitymeasures of a semi-stable OU process. We deduce, from the Lamperti mapping (2) and thediscussions above the following.
Proposition 1.1.
For any x > , there exists a one to one mapping between the law of a L´evyprocess starting from log( x ) and the law of a semi-stable OU process starting from x . Moreprecisely, we have (7) U t = e − λt e ξ △ t , t < T U , where △ t = R t U − αs ds . Note that for b > , the previous identity holds for any t ≥ .Moreover, if (3) holds with < θ < α , then the minimal process ( U, T U ) admits a recurrentextension which hits and leaves continuously a.s. which is the OU process associated to ( X, P x ) .We write its family of laws by ( Q x ) x> . Under the condition H1 , for any γ >
0, we denote by Q ( γ ) x the law of the semi-stable OUprocess, starting at x ∈ R + , associated to the L´evy process having Laplace exponent ψ γ ( u ) = ψ ( u + γ ) − ψ ( γ ) , u ≥ m is invariant for U if it satisfies, for any f ∈ B (E), Z E Q t f ( x ) m ( dx ) = Z E f ( x ) m ( dx ) . Finally, let G q be a gamma random variable independent of X , with parameter q >
0, whoselaw is given by γ ( dr ) = e − r r q − Γ( q ) dr . We are now ready to state the following. Theorem 1.2.
If E = R or H0 holds then the Feller process U is positively recurrent and itsunique invariant measure is χ P ( X ∈ dx ) .Next, assume that I ∈ L ( γ ( dr )) . For any q > , we introduce the function I ( q ; x ) defined by (8) I ( q ; x ) = χ qχ E (cid:20) I (cid:18)(cid:16) χG qχ (cid:17) α x (cid:19)(cid:21) , x ∈ E . Then, if
I ∈ I ( X ) ( resp. E ( X )) then I ( q ; x ) ∈ I q ( U ) ( resp. E q ( U )) .Consequently, if I ∈ I ( X ) , we have, for any q > , (9) (1 + χt ) − qχ P t (cid:16) d (1+ χt ) − α I (cid:17) ( q ; x ) = I ( q ; x ) , x ∈ E . Remark . (1) We call the multiplication operator (8) associated to a fractional power ofa Gamma random variable, the Γ -transform .
2) The characterization of time-space invariant functions of the form (9), associated toself-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 Besselprocesses, Novikov [19] and Patie [22] for the one sided-stable processes. Whilst in thementioned papers, the authors made used of specific properties of the studied processesto derive the time-space martingales, we provide a proof which is based simply on theself-similarity property.We proceed by investigating the process Y , defined, for any x, β ∈ R and ξ = 0 a.s., by(10) Y t = e αξ t (cid:18) x + β Z t e − αξ s ds (cid:19) , t ≥ . We call Y the L´evy OU process. We mention that this generalization of the OU process is aspecific instance of the continuous analogue of random recurrence equations, as shown by deHaan and Karandikar [9]. They have been also well-studied by Carmona et al. [6], Erickson andMaller [12], Bertoin et al. [1] and by Kondo et al. [14]. In [6], it is proved that Y is a homogeneousMarkov process with respect to the filtration generated by ξ . Moreover, they showed, from thestationarity and the independency of the increments of ξ , that, for any fixed t ≥ Y t ( d ) = xe αξ t + β Z t e αξ s ds. Then, if E [ ξ ] <
0, they deduced that, as t → ∞ , ξ t ( a.s ) → −∞ and Y t ( d ) → βV ∞ = R ∞ e αξ s ds . Werefer to Bertoin and Yor [4] for a thorough survey on the exponential functional of L´evy processes.In the spectrally negative case, it is well know that the law of V ∞ is self-decomposable, henceabsolutely 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 introducethe process Z defined, for any x = 0 , β ∈ R and ξ = 0 a.s., by(11) Z t = e αξ t (cid:18) x + β Z t e αξ s ds (cid:19) − , t ≥ . Before stating the next result, we introduce some notation. Let B be a Borel subset of E andwe write T UB for the first exit time from B by U . With a slight abuse of terminology, we saythat for any x ∈ E, a non-negative function H is a ( q ∆ , B )-harmonic function for ( U, Q x ) if(12) E x h e − q △ TUB H ( U T UB ) I { T UB Set β = αλx in (11) . Then, to a process Z starting from x , with x = 0 , onecan associate a semi-stable OU process ( U, Q ) such that (13) Z t = x − U α ▽ t , t < T U , where ▽ t = R t Z s ds and its inverse is given by △ t = R t U − αs ds . Note that for b > , the previousidentity holds for any t ≥ . Consequently, with x > , Z is a Feller process on (0 , ∞ ) .Moreover, let q > , ≤ a < b ≤ + ∞ and x ∈ ( a, b ) . Then, a ( q ∆ , T U ( ax,bx ) ) -harmonic functionfor ( U, Q ) is a ( q, T Z ( a α ,b α ) ) -harmonic function for the process Z starting from x − α . Similarly, a q ∆ , T U ( xb , xa ) ) -harmonic for ( U, Q ) is a ( q, T b Y ( a α ,b α ) ) -harmonic function for the process b Y startingfrom x α , the L´evy OU process associated to the L´evy process b ξ = − ξ , the dual of ξ with respectto the Lebesgue measure.Finally, assume that H1 holds and write p q ( x ) = x q for x, q > . If the function H is ( λφ ( q ) , B ) -harmonic function for ( U, Q ( φ ( q )) x ) then the function p φ ( q ) H is ( q ∆ , B ) -harmonic func-tion for ( U, Q x ) . Proofs Proof of Theorem 1.2. The description of the unique invariant measure is a refinementof [7, Proposition 5.7] where therein the proof is provided for R -valued self-similar processesand can be extended readily for the R + -valued case under the condition H0 , which ensures that( X, P x ) admits an entrance law at 0.Next, let us assume that I ∈ L ( γ ( dr )) ∩ I ( X ). We need to show that for any q > e − qt Q t I ( q ; x ) = I ( q ; x ). For x ∈ E, we deduce from the definition of ( Q t ) t ≥ that e − qt Q t I ( q ; x ) = χ qχ Γ (cid:16) qχ (cid:17) e − qt E x (cid:20)Z ∞ I (cid:16) ( χr ) α U t (cid:17) e − r r qχ − dr (cid:21) = χ qχ Γ (cid:16) qχ (cid:17) e − qt E x (cid:20)Z ∞ I (cid:16) ( χr ) α e ′ λ ( − t ) X e χ ( t ) (cid:17) e − r r qχ − dr (cid:21) . Using the change of variable u = χe ′ χ ( − t ) r , Fubini theorem and (4), we get e − qt Q t I ( q ; x ) = 1Γ (cid:16) qχ (cid:17) E x (cid:20)Z ∞ e − ue χ ( t ) I (cid:16) u α X e χ ( t ) (cid:17) e − uχ u qχ − du (cid:21) = 1Γ (cid:16) qχ (cid:17) Z ∞ I (cid:16) u α x (cid:17) e − uχ u qχ − du = I ( q ; x )where the last line follows after the change of variable u = χr . The case I ∈ L ( γ ( dr )) ∩ E ( X )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 χ ( t ), with v χ ( t ) = χ log(1 + χt ).2.2. Proof of Theorem 1.4. Setting β = αλx , the Lamperti mapping (2) yields Z t = x − e αξ t (cid:18) αλ Z t e αξ s ds (cid:19) − = x − αλ. ) α X . ! αV t = x − (cid:0) U v χ ( . ) (cid:1) αV t here the last identity follows from (5). The proof of the assertion (13) is completed by observingthat ( v χ ( V t )) ′ = e αξ t (cid:18) αλ Z t e αξ s ds (cid:19) − . Moreover, since the mapping x x α is a homeomorphism of R + , the Feller property followsfrom its invariance by ”nice” time change of Feller processes, see Lamperti [15, Theorem 1]. Wealso obtain the following identities T Z ( a,b ) = inf { u ≥ Z u / ∈ ( a, b ) } = inf (cid:8) u ≥ U α ▽ u / ∈ ( ax, bx ) (cid:9) = △ (cid:16) inf n u ≥ U u / ∈ (cid:16) ( ax ) α , ( bx ) α (cid:17)o(cid:17) . The characterization of the harmonic functions of Z follows. The characterization of the har-monic functions of ˆ Y are readily deduced from the ones of Z and the identityˆ Y t = 1 Z t , 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 ≥ and x > , we have (1) d Q ( γ ) x = (cid:18) U t x (cid:19) γ − δ e λ ( γ − δ ) t − ( ψ ( γ ) − ψ ( δ )) △ t d Q ( δ ) x , on F Ut ∩ { t < T U } . Note that for b > the condition ” on { t < T U } ” can be omitted. For the particular case γ = θ and δ = 0 , the absolute continuity relationship (1) reduces to d Q ( θ ) x = (cid:18) U t x (cid:19) θ e λθt d Q x , on F Ut ∩ { t < T U } . Proof. We start by recalling that in [20], the following power Girsanov transform has beenderived, under H1 , for γ, δ ≥ x > 0, with obvious notation, d P ( γ ) x = (cid:18) X t x (cid:19) γ − δ e − ( ψ ( γ ) − ψ ( δ )) A t d P ( δ ) x , on F Xt ∩ { t < T X } . The assertion (1) follows readily by time change and recalling that A e χ ( t ) = Z t U − αu du. We complete the proof by recalling that for b > U does not reach 0 a.s.. (cid:3) Applications In this section, we illustrate our results to some new interesting examples. .1. First passage times and overshoot of stable OU processes. Let X be an α -stableL´evy process whose characteristic exponent satisfy, for u ∈ R ,Ψ( iu ) = − c | u | α (cid:16) − iβ sgn( u ) tan (cid:16) απ (cid:17)(cid:17) where 1 < α < c = (cid:0) β tan (cid:0) απ (cid:1)(cid:1) − / . Then, we introducethe constant ρ = P ( X > 0) which was evaluated by Zolotarev [28] as follows ρ = 12 + 1 πα tan − (cid:16) β tan (cid:16) απ (cid:17)(cid:17) . Following Doney [10], we introduce, for any integers k, l , the class C k,l of stable processes suchthat ρ + k = l ˜ α where ˜ α = α . For m ∈ N , x ∈ R and z ∈ C , introduce the function f m ( x, z ) = m Y i =0 (cid:16) z + e ix ( m − i ) π (cid:17) . Next, we recall from the Wiener-Hopf factorization of L´evy processes due to Rogozin [25], thatthe law of the first passage times τ X and the over(under)shoot of X at the level 0 is describedby the following identities, for δ, r > p ≥ Z ∞ e − δx E − x (cid:20) e − rτ X − pX τX − (cid:21) dx = 1 δ − p − Ψ + ( − r α δ )Ψ + ( − r α p ) !Z ∞ e − δx E x (cid:20) e − rτ X − pX τX − (cid:21) dx = 1 δ − p − Ψ − ( r α δ )Ψ − ( r α p ) ! where (1 − Ψ( δ )) − = Ψ − ( δ )Ψ + ( δ ). Here Ψ + ( δ ) (resp. Ψ − ( δ )) is analytic in ℜ ( δ ) < ℜ ( δ ) > 0) continuous and nonvanishing on ℜ ( δ ) ≤ ℜ ( δ ) ≥ C k,l as followsΨ + ( z ) = f k − ( α, ( − l ( − z ) α ) f l − ( ˜ α, ( − k +1 z ) , Arg ( z ) = 0 , Ψ − ( z ) = f l − ( ˜ α, ( − k +1 z ) f k ( α, ( − l z α ) , Arg ( z ) = − π, where z β stands for σ β e iβφ when z = σe iφ with σ > − π < φ ≤ π . Observe also thatΨ + ( − x α ) ∼ x − ρ for large real x . Moreover, using the fact that the function E x (cid:20) e − rτ X − pX τX (cid:21) , x ∈ R , is r -excessive for the semigroup of X , we deduce from the Γ-transform the following. orollary 3.1. For any q, δ > , p ≥ , and for any integers k, l such that X ∈ C k,l , we have Z −∞ e δx E x (cid:20) e − qτ U − p U τU − (cid:21) dx = 1 δ − p χ qχ − qχ ) Z ∞ Ψ + ( − r α δ )Ψ + ( − r α p ) e − rχ r qχ − dr !Z ∞ e − δx E x (cid:20) e − qτ U − p U τU − (cid:21) dx = 1 δ − p χ qχ − qχ ) Z ∞ Ψ − ( r α δ )Ψ − ( r α p ) e − rχ r qχ − dr ! . First passage times of one-sided semi-stable- and L´evy-OU processes. We nowfix ( P t ) t ≥ to be the semigroup of a spectrally negative α -semi-stable process X . X is thenassociated via the Lamperti mapping (2) to a spectrally negative L´evy process, ξ , which weassume to have a finite mean b . Its characteristic exponent ψ has the well known L´evy-Khintchinerepresentation ψ ( u ) = bu + σ u + Z −∞ ( e ur − − ur ) ν ( dr ) , u ≥ , (1)where σ ≥ ν satisfies the integrability condition R −∞ ( r ∧ r ) ν ( dr ) < + ∞ .Patie [20] computes the Laplace transform of the first passage times above of X as follows. Forany r ≥ ≤ x ≤ a , we have E x h e − rT Xa i = I α,ψ ( rx α ) I α,ψ ( ra α )(2)where the entire function, I α,ψ , is given, for γ ≥ α > 0, by I α,ψ ( z ) = ∞ X n =0 a n ( ψ ; α ) z n , z ∈ C and a n ( ψ ; α ) − = n Y k =1 ψ ( αk ) , a = 1 . Using the Γ-transform, we introduce the following power series(3) I α,ψ ( q ; z ) = ∞ X n =0 a n ( ψ ; α )( q ) n z n where ( q ) n = Γ( q + n )Γ( q ) is the Pochhammer symbol and we have used the integral representationof the gamma function Γ( q ) = R ∞ e − r r q − dr, ℜ ( q ) > 0. By means of the following asymptoticformula of ratio of gamma functions, see e.g. Lebedev [17, p.15], for δ > z + n ) δ = z δ (cid:20) δ (2 n + δ − z + O ( z − ) (cid:21) , | arg z | < π − ǫ, ǫ > , (4)we deduce that I α,ψ ( q ; z ) is an entire function in z and is analytic on the domain { q ∈ C ; ℜ ( q ) > − } . For b < 0, we recall that there exists θ > ψ ( θ ) = 0 and thus ψ θ ( u ) = ψ ( θ + u ).In this case, by setting θ α = θα , it is shown in [20] that there exists a positive constant C θ α suchthat I α,ψ ( x α ) ∼ C θ α x θ I α,ψ θ ( x α ) as x → ∞ . e also introduce the function N α,ψ,θ ( q ; x α ) defined by(5) N α,ψ,θ ( q ; x α ) = I α,ψ ( q ; x α ) − C θ α x θ Γ( q + θ α )Γ( q ) I α,ψ θ ( q + θ α ; x α ) , ℜ ( x ) ≥ . Moreover, if we assume that there exists β ∈ [0 , 1] and a constant a β > u →∞ ψ ( u ) /u β = a β , then C θ α is characterized by C θ α = Γ(1 − θ α ) α ( θ α − Q θα − k =1 ψ ( αk ) , if θ α is a positive integer , Γ(1 − θ α ) α a − θ α β e E γ βθ α Q ∞ k =1 e − βθαk ( k + θ α ) ψ ( αk ) kψ ( αk + θ α ) , otherwise , where E γ stands for the Euler-Mascheroni constant. We recall, also from [20], that, for r, x ≥ E x h e − rT X i = I α,ψ ( rx α ) − C θ α ( r α x ) θ I α,ψ θ ( rx α ) . We deduce from Theorems 1.2 and 1.4 the following. Corollary 3.2. Let q ≥ and < x ≤ a . Then, E x h e − qT Ua i = I α,ψ (cid:16) qχ ; χx α (cid:17) I α,ψ (cid:16) qχ ; χa α (cid:17) and E x (cid:20)(cid:16) χT X ( α ) (cid:17) − qχ (cid:21) = I α,ψ (cid:16) qχ ; χx α (cid:17) I α,ψ (cid:16) qχ ; χa α (cid:17) where T X ( α ) = inf { u ≥ X u = a (1 + χu ) α } . We also deduce that E x h e − q △ TUa I { T Ua We first consider a Brownian motion with drift − ν , i.e. ψ ( u ) = u − νu . Setting α = 2, we have θ = 2 ν and therefore we assume ν < ν and thusthe associated Ornstein-Uhlenbeck process is, in the case n = 2 ν + 1 ∈ N , the radial norm of n -dimensional Ornstein-Uhlenbeck process. We get I ,ψ ( x ) = ( x/ ν/ Γ( − ν + 1)I − ν (cid:16) √ x (cid:17) where I ν ( x ) = P ∞ n =0 ( x/ ν +2 n n !Γ( ν + n +1) stands for the modified Bessel function of index ν , see e.g. [17,5.], and I ,ψ ( q ; x ) = Φ (cid:18) q, − ν, x (cid:19) I ,ψ ν ( q ; x ) = Φ (cid:18) q + ν, ν + 1 , x (cid:19) where Φ( q, ν, x ) = P ∞ n =0 ( q ) n ( ν ) n n ! x n stands for the confluent hypergeometric function of the firstkind, see e.g. [17, 9.9]. Using the asymptotic behavior of the Bessel functionI ν ( x ) ∼ e x √ πx as x → ∞ , we deduce that C ν = − Γ( − ν )Γ( ν ) . Hence, N α,ψ ν ( q ; x ) = (cid:18) Φ (cid:18) q, − ν, x (cid:19) + x ν Γ( − ν )Γ( q + ν )Γ( ν )Γ( q ) Φ (cid:18) q, − ν, x (cid:19)(cid:19) = Γ( q )Γ( q + ν )Γ( ν ) Λ (cid:18) q, ν + 1 , x (cid:19) where Λ( q, ν + 1 , x ) is the confluent hypergeometric of the second kind. We mention that, inthis 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. Some generalization of the Mittag-Leffler function. Patie [21] introduced a new parametricfamily of one-sided L´evy processes which are characterized by the following Laplace exponents,for any 1 < α < 2, and γ > − α ,(6) ψ γ ( u ) = 1 α (( u + γ − α − ( γ − α ) . ts characteristic triplet are σ = 0, ν ( dy ) = α ( α − − α ) e ( α + γ − y (1 − e y ) α +1 dy, y < , and b γ = ( γ ) α (Υ( γ − α ) − Υ( γ − λ ) = Γ ′ ( λ )Γ( λ ) is the digamma function. In particular, if γ denotes the zeroof the function γ → b γ , then for γ ≥ γ ∈ (1 − α, b ≥ The case γ = 0 . (6) reduces to ψ ( u ) = α ( u − α . Observe that θ = 1, ψ ′ (1) = Γ( α ) α and a n ( ψ ; α ) − = Γ( α ( n + 1))Γ( α ) , a = 1 . The series (3) can be written as follows I ,ψ ( q ; x ) = Γ( α ) M qα,α ( αx ) I ,ψ ( q ; x ) = Γ( α − M qα,α − ( αx )where M qα,β ( z ) = ∞ X n =0 ( q ) n z n Γ( αn + β ) , z ∈ C , stands for the Mittag-Leffler function of parameter α, β, q > , which was introduced by Prab-hakar [23]. Moreover, we have, see e.g. [20], ∞ X n =0 z αn Γ( αn + β ) ∼ α e x x − β l ( x α ) as x → ∞ , with l a slowly varying function at infinity. Thus, C α = αα − and N α,ψ ( q ; x α ) = M qα,α − ( x α ) − αxα − q + α )Γ( q ) M qα,α ( x α ) . As concluding remarks, we first mention that in the diffusion case, i.e. when ( U, Q x ) is theOrnstein-Uhlenbeck process associated to a Bessel process, see 3.2.1, the law of the first pas-sage time above can be expressed as an infinite convolution of exponential distributions withparameters given by the sequence of positive zeros of the confluent hypergeometric function,see Kent [13] for more details. 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