The area under a spectrally positive stable excursion and other related processes

We study the distribution of the area under the normalized excursion of a spectrally positive stable L{\'e}vy process L, as well as the area under its meander, and under L conditioned to stay positive. Our results involve a special case of Wright's function, which may be seen as a generalization of the classic Airy function appearing in similar Brownian's areas.

It is well-known, see for instance [23,Prop. 3.4.1], that the distribution of the area under L also follows a stable distribution : The purpose of this note is to study the distribution of the area under three related processes : the normalized excursion of L, the meander of L, and L conditioned to stay positive. These distributions have already been extensively studied in the Brownian case, see in particular the survey by Janson [8] or the paper by Perman & Wellner [21]. For all these Brownian areas, one observes the occurrence of the classic Airy function Ai. We shall prove that for spectrally positive stable Lévy processes, the role of Ai is played by the following M-Wright's function Φ α (x) = 1 π +∞ n=0 (−1) n n! Γ 1 + n 1 + α sin π 1 + n α + 1 (1 + α) The function Φ α is known to be related to a time-fractional diffusion equation, see [19] and the references within. In particular, it admits the integral representation : 1+α cos cos πα 2 z 1+α 1 + α − zx dz from which we immediately see that Φ 2 (x) = Ai(x). Another function of interest will be its cosine counterpart : (−1) n n! Γ 1 + n 1 + α cos π 1 + n α + 1 (1 + α) We now state the main results of this paper.
1.1. The area under a normalized excursion. Let L (ex) denote a normalized excursion of L on the segment [0, 1] and set Theorem 1. The double Laplace transform of A ex is given by : In the Brownian case, i.e. when α = 2, the distribution of A ex is nowadays designated in the literature as the Airy distribution. We refer the reader to Louchard [16,17] for a study of the Brownian excursion area via the Feynman-Kac formula, to Takács [24] for an approach via random walks and to Flajolet & Louchard [6] for a study of Airy distribution via its moments.
In the physics literature, this distribution has also appeared in the study of fluctuating interfaces, see [18]. We finally mention that more recently, some authors have investigated the area under a normalized Bessel excursion and shown its relation with the cooling of atoms [1,12].
As is the case for the Airy distribution, we may deduce from Theorem 1 a recurrence relation for the moments of A ex . To this end, let us define a sequence (B n,k , 1 ≤ n, 1 ≤ k ≤ n) by 1 B n,1 = (2 − α) n−1 (n + 1)(n + 2) , n ≥ 1, and for k ≥ 1, This sequence corresponds to the values of the exponential partial Bell polynomials taken on the sequence (B n,1 , n ≥ 1), see Comtet [5,Section 3.3]. We then set c (α) 0 = 1 and for p ≥ 1, The coefficients (c (α) p , p ≥ 0) are the ones appearing in the asymptotic expansion of the function Φ α , see Proposition 10 in the Appendix. Corollary 2. Let us set, for n ≥ 1, Then, the sequence (Ω n ) follows the recurrence relation : As consequence, we have the asymptotics 2 : The first moments are given by : Note that in the Brownian case, since B n,1 = 0 as soon as n ≥ 2, the coefficients c (2) n simplify to 2n Γ 3n + 1 2 and we recover a classic recurrence formula for the moments of Airy distribution, see [17]. In this case, a complete asymptotic expansion for P(A ex > x) was computed by Janson & Louchard in [9].
1.2. The area under a stable meander. Let L (me) denotes the meander of L on the segment [0, 1] and set Theorem 3. The double Laplace transform of A me is given by : . (1.5) In particular, and there is the asymptotics, for 1 < α < 2 : 2 We write a p ≍ b p as p → +∞ to state that their exist two constants 0 < κ 1 ≤ κ 2 < +∞ such that κ 1 a p ≤ b p ≤ κ 2 a p for p large enough.
In the Brownian case, this distribution was studied for instance by Takács [25], see also Perman & Wellner [21]. Note that here, since the process is not pinned down at t = 1, the presence of positive jumps yields a polynomial decay of the tail of A me , which is different from the case α = 2, see Janson & Louchard [9]. Remark 4. Again, when α = 2, Theorem 3 simplifies. Indeed, observe that in this case, Ψ 2 is a solution of the following differential equation π hence the derivative of the numerator on the right-hand side of (1.5) equals By integration, this implies that which agrees for instance with [21, Corollary 3.2].
1.3. The area under L conditioned to stay positive. Let L ↑ be the process L started from 0 and conditioned to stay positive. We set Theorem 5. The double Laplace transform of A ↑ is given by : In particular, there is the asymptotics, for 1 < α < 2 : In the Brownian case, A ↑ corresponds, up to a √ 2 factor, to the integral on [0, 1] of a threedimensional Bessel process started from 0. As for the meander, Formula (1.6) simplifies when α = 2, and the right-hand side equals which agrees with [3, p.440].
1.4. Outline of the paper. The remainder of the paper is divided as follows. Section 2 provides some notation as well as a key proposition which is of independent interest. Section 3, 4 and 5 give the proofs of the main Theorems, respectively on the normalized excursion, the meander, and L conditioned to stay positive. Finally, Section 6 is an appendix on the Wright's function Φ α , where we compute its asymptotic expansion at infinity.

Notations.
We start by recalling the definition of the considered processes, for which we mainly refer to Chaumont [4]. We assume that L is defined on the Skorokhod space of càdlàg processes. We denote by P x its law when L 0 = x, with the convention that P = P 0 , and by (F t , t ≥ 0) its natural filtration. Define, for z ∈ R, (1) We denote by P t x,y the law of the bridge of L of length t, going from x to y. (2) We denote by P ↑ x the law of L started at x > 0 and conditioned to stay positive. It is classically given by the h-transform : This law admits a weak limit in the Skorokhod sense as x ↓ 0 which we shall denote P ↑ 0 . (3) We denote by P (me) the law of the meander of L, which is given by the limit (4) Finally, we recall that, since L is spectrally positive, the law of the stable excursion of length t is equivalent to the bridge of L conditioned to stay positive, and starting and ending at 0. From Lemma 4 in [4], this law may be for instance defined by The key proposition. The proofs of Theorems 1, 3 and 5 will rely heavily on the following proposition, which gives the joint Laplace transform of the pair T 0 , Proposition 6. For z, λ ≥ 0 and µ > 0 we have : Proof. We first assume that λ = 0. The law of T 0 0 L s ds has been studied by Letemplier & Simon in [15]. In particular, they obtain the Mellin transform : Replacing ν by −ν and using the definition of the Gamma function, we obtain : .
We now invert this Mellin transform following Janson [10] : hence, by scaling, we thus obtain which is Proposition 6 when λ = 0. We now deal with the general case. Let x > y > 0.
Applying the Markov property, we deduce from the absence of negative jumps that Furthermore, by translation, we also have : Finally, setting x − y = z > 0 and λ = µy > 0, and plugging (2 which ends the proof of Proposition 6.
Remark 7. Proposition 6 was proven in the Brownian case by Lefebvre [14], by applying the Feynman-Kac formula. A generalization where T 0 is replaced by the exit time from an interval was obtained by Lachal [13] by a similar method.
3. The area under a spectrally positive stable excursion 3.1. Proof of Theorem 1 : the double Laplace transform. Notice first that by monotone convergence, the absolute continuity formula (2.2) remains true at s = t : Next, starting from Proposition 6, we may write Recall now from Sato [22,Theorem 46.3] that since L has no negative jumps, T 0 is a positive stable random variable of index 1/α, i.e. for z > 0 : Dividing (3.2) by z and letting z ↓ 0, we deduce from (3.1) and (3.3) that Theorem 1 now follows by the complement formula for the Gamma function and the scaling property.

3.2.
Proof of Corollary 1 : study of the positive moments. To get information on the moments, we shall work with a slight modification of the formula of Theorem 1. Indeed, using the decomposition and it remains to compute the asymptotic expansion of both sides as x → 0. Using the asymptotic expansions of Φ α and Φ ′ α given in Proposition 10 in the Appendix, we get for p ≥ 1.
By identification, we thus obtain the announced recurrence relation : 3.3. Proof of Corollary 1 : asymptotics. To study the asymptotics of (Ω n ), observe first that by definition, this sequence is positive, and so is the sequence (c (α) n ). Therefore, we deduce from the recurrence relation (1.4) and from Corollary 11 in the Appendix that there exists a finite constant κ 1 such that for n large enough which gives the announced upper bound.
It does not seem easy to obtain a lower bound for Ω n from the recurrence relation (1.4). Instead, we shall rather study directly the tail of the survival function of A ex . To do so, we recall the following absolute continuity formula for the normalized excursion, see [4, Formula (11)]: where c is a normalization constant and, from Monrad & Silverstein [20, Formula (3.25)], j * 1−s is a measurable function which admits the asymptotics : where γ and η are two positive constants. Using this absolute continuity formula, we first deduce that : Applying the Markov property, we further obtain where we used in the last equality the absolute continuity formula (2.1) for P ↑ L 1/4 . Now, since L 1/2 ≥ inf 0≤u≤ 1 2 L u > 2x, we deduce from the asymptotics of j * 1/4 that for x large enough there exist two positive constants γ and η such that where, in the third inequality, we have used the independent increments property of L. The lower bound now follows by taking the logarithm on each side, and using the limits and, see Chaumont [4, p.12], 1 n follows then as before from Kasahara's Tauberian theorem of exponential type [11,Theorem 4].
3.4. Study of some (fractional) negative moments. As was observed by Flajolet & Louchard [6] for the classic Airy distribution, one may also compute by recurrence some specific negative moments. These moments are related to the asymptotic expansion of Φ α at λ = 0, which is given by its very definition as a series. Corollary 8. Let us set for n ≥ 1 : Then, the sequence (∆ n ) follows the recurrence relation In particular, the first value is given by : Proof. Starting from Theorem 1 and multiplying both sides by Φ α , we obtain We now divide this equality by z and let z ↓ 0. Using (3.3), the scaling property and the definition of the meander, the left-hand side converges towards Let us set, to simplify the notation : From (1.1), we deduce that for λ ∈ C : which shows that Φ α and Ψ α are closely related to the integral of L. Notice also that

4.2.
Proof of Theorem 3 : first moment and asymptotics. To simplify the following computation, we set .
we deduce that and the value E [A me ] will follow by letting λ → +∞ and applying the monotone convergence theorem on the left-hand side. Since the asymptotics of Φ α and Φ ′ α are given in Proposition 10 in the Appendix, it only remains to study those of Ψ α . To this end, we observe from (4.3) that Applying Watson's lemma, we deduce from the definition (4.2) of F α that which implies, since Φ α decreases exponentially fast, the asymptotic expansion : Therefore, going back to the definition of H α , we deduce that As a consequence, we obtain the limit from which we deduce the value of the first moment Next, to compute the asymptotics of P(A me > x), we shall work with Mellin transforms, which is a convenient tool when dealing with stable processes. Using Formula (1.5) and applying the Fubini-Tonelli theorem, we have for ν ∈ 0, 1 − 1 α , Integrating twice by parts to remove the singularities at ν = 1 − 1 α and ν = 2, we further obtain where, as λ → +∞, the integrand on the left-hand side is equivalent to The Mellin transform on the left-hand side thus admits a simple pole at ν = 2 + α − 1 α , hence so does the Mellin transform on the right-hand side. Applying the converse mapping theorem, we deduce that, as λ → +∞, which yields the announced asymptotics.

5.
The area under L conditioned to stay positive.

5.1.
Proof of Theorem 5 : the double Laplace transform. We proceed as for the meander. Applying the Markov property, we first have Dividing both sides by z and letting z ↓ 0 then yields by definition of P ↑ Observe next that, integrating by parts the definition (4.2) of F α , we also have As a consequence, we deduce from (4.3) that and the result follows as before by analytic continuation.

5.2.
Proof of Theorem 5 : asymptotics. The asymptotics of the tail of A ↑ is easy to obtain as we can work with Laplace transforms and use repeatedly Karamata's Tauberian theorem. Indeed, since we are dealing with monotone integrands, there is the asymptotics for α < 2 : x 1−α .

Appendix on M-Wright's functions
We gather and prove in this section several useful formulae for the M-Wright's function Φ α and its derivative.
Applying Cauchy's integral theorem, we may deform the path of integration to pass by the imaginary axis, and thus obtain the integral representation : 1+α cos cos πα 2 z 1+α 1 + α − zx dz. (6.1) 6.2. Asymptotic expansion of Φ α and Φ ′ α . We now study the asymptotics of Φ α (x) as x → +∞. A general (theoretical) asymptotic expansion for φ was computed by Wright [26] (see also [7, Theorem 2.1.3]), but it seems difficult to extract from his formula an explicit expression for the coefficients. This will be our objective here.
We start with a simple lemma. We may now compute the asymptotic expansion of Φ α and Φ ′ α . Proposition 10. We have the asymptotic expansions as x → +∞ : where d (α) 0 = 1 and for p ≥ 1 : Proof. Since these asymptotics are already known for the Airy function, we shall assume in the following that α < 2. We start with the asymptotic expansion of Φ α . Coming back to the formula (6.1), we may write, using the change of variable z = x 1 α y :