Spectral analysis of stable processes on the positive half-line

We study the spectral expansion of the semigroup of a general stable process killed on the first exit from the positive half-line. Starting with the Wiener-Hopf factorization we obtain the q-resolvent density for the killed process, from which we derive the spectral expansion of the semigroup via the inverse Laplace transform. The eigenfunctions and co-eigenfunctions are given rather explicitly in terms of the double sine function and they give rise to a pair of integral transforms which generalize the classical Fourier sine transform. Our results provide the first explicit example of a spectral expansion of the semigroup of a non-symmetric Levy process killed on the first exit form the positive half-line.


Introduction
Consider the Brownian motion process W = {W t } t≥0 started from W 0 = x > 0 and let T 0 be the first time when W hits zero. It is well-known (see [3,Appendix 1.3]) that the distribution of T 0 can be computed as P x (T 0 > t) = 2 π ∞ 0 e −tλ 2 sin(λx)λ −1 dλ, (1) and that the transition probability density of the process W killed at T 0 has spectral representation p t (x, y) = 2 π ∞ 0 e −tλ 2 sin(λx) sin(λy)dλ.
To give a functional-analytic point of view, let us introduce the semigroup of the killed process as the family of operators {P t } t≥0 defined by and let us define the Fourier sine transform as Πu(λ) = 2/π ∞ 0 u(x) sin(λx)dx, u ∈ L 1 (R + ). (4) It is well-known that Π can be extended to an isometry in L 2 (R + ) and then formula (1) gives us a spectral representation of the semigroup in L 2 (R + ): The function sin(λx) plays an important role in the above expressions. It is the eigenfunction of the one-dimensional Laplace operator ∆ (which is simply the second derivative operator, ∆u = u ), with a Dirichlet boundary condition at zero. Note that ∆ is the infinitesimal generator of the scaled Brownian motion X t = √ 2W t . Brownian motion is a very special process, because it enjoys many useful properties: it is a diffusion process, it is a Lévy process (a process with stationary and independent increments), it is also a self-similar process. The list of other Markov processes with an explicit spectral representation of the semigroup is quite short. We mention here the well-known case of one-dimensional diffusion processes [18], branching processes [19], a family of symmetric Lévy processes obtained as a time change of Brownian motion [15] and the recent work of Patie and Savov on non-selfadjoint Markov semigroups [20].
Our goal in this paper is to generalize results (1), (2) and (5) to strictly stable Lévy processes (which we call simply stable processes) killed on the first exit from (0, ∞). This class consists of Lévy processes which satisfy the scaling (or, self-similarity) property: for any c > 0 the process {cX t } t≥0 (started from X 0 = 0) has the same distribution as {X c α t } t≥0 . Such processes exist when the stability parameter α belongs to the interval (0, 2]. When α = 2 we recover the scaled Brownian motion and for α < 2 we obtain a two-parameter family of processes with jumps that we will discuss in detail in the next section. The scaling property of stable processes is the main reason why they are so popular among researchers and why they appear so frequently in various applications originating in Physics, Chemistry and Biology. Stable processes (or Lévy flights, as they are also known in these fields) occur in modelling such diverse phenomena as fluctuations and transport in plasma, turbulent diffusions, seismic series and earthquakes, signal processing and financial time series (see review article [7] for a comprehensive list of applications).
Our set of admissible parameters excludes processes with one-sided jumps: subordinators or negative subordinators (α ∈ (0, 1) and ρ ∈ {1, 0}) and the spectrally-negative and spectrally-positive processes (α ∈ (1, 2) and αρ = 1 or α(1 − ρ) = 1, respectively). The first two cases are not interesting, as the processes have monotone paths and the semigroup of the process on the positive half-line is rather simple, and the case of spectrally-positive/negative processes is covered separately in Section 3.1.
Let us consider some special cases of stable processes. When α = 2 we necessarily have ρ = 1/2 and the process X in this case is simply the scaled Brownian motion X t = √ 2W t . When α = 1 we can rewrite the characteristic exponent in the form Ψ(z) = sin(πρ)|z| + i cos(πρ)z, so that the process X can be written as X t = sin(πρ)Z t − cos(πρ)t, where Z t is the Cauchy process. A stable process with ρ = 1/2 is symmetric (has the same distribution asX) and can be obtained as a subordinated Brownian motion. More precisely, let S t be an α/2-stable subordinator defined by the Laplace transform E[exp(−zS t )] = exp(−tz α/2 ) and independent of the Brownian motion W , then the process {X t } t≥0 has the same distribution as { √ 2W St } t≥0 . In the general case (except when α ∈ {1, 2}) a stable process is a pure-jump Lévy process characterized by the density of the Lévy measure ν(x), given by where we have denoted c = Γ(1 + α) sin(παρ)/π andĉ = Γ(1 + α) sin(πα(1 − ρ))/π. The Lévy measure describes the distribution and the intensity of the jumps of the process, and it is connected to the characteristic exponent via the Lévy Khintchine formula where h(x) is the cutoff function, which is needed to ensure the convergence of the integral. Using the cutoff function h(x) ≡ 0 when α < 1 and h(x) ≡ x when α > 1 one can check that the above integral representation for Ψ(z) can be evaluated in closed form as given in (6). This construction also works when α = 1 and ρ = 1/2 (in this case we can take the cutoff function h(x) ≡ sin(x)).
A stable process X is also a Markov process, and its dynamics can be described by the infinitesimal generator L. This operator is defined for a suitable set of functions u as follows As we have mentioned above, when α = 2 the infinitesimal generator is simply the one-dimensional Laplace operator, Lu = u . In the symmetric case (that is, when ρ = 1/2) the infinitesimal generator is the fractional Laplace operator L = −(−∆) α/2 . In the general case, it is a (non-local) integro-differential operator having the following form: If we denote by F the Fourier transform operator then the infinitesimal generator can be represented in a particularly simple form L = −F −1 Ψ(−z)F (see [2][Proposition 9]). In other words, the infinitesimal generator L is a pseudo-differential operator [27] and the characteristic exponent Ψ is the symbol of this operator.

Main Results
Consider a stable process X started from X 0 = x > 0 and denote by T 0 the first exit time from (0, ∞). Let {P t } t≥0 be the transition semigroup of the process X killed at time T 0 : these operators are defined by (3), but with X t instead of W t . Then P t are sub-Markov operators, in particular, they are contraction operators on L 2 (R + ). The transition probability density p t (x, y) is defined as the integral kernel of the operator P t : It is known that the transition density exists when X is a stable process (this follows from Theorem 3 and Example 4 in [4]).
From the analytical point of view, killing the process on the first exit from the positive half-line is equivalent to imposing a Dirichlet boundary condition on the negative half-line. Thus, in order to find the eigenfunctions of the infinitesimal generator we need to solve the equation Lf (x) = λf (x) for x > 0 with the "boundary condition" f (x) = 0 for x ≤ 0. The problem of computing v(t, x) = P t u(x) (for a suitable u) is equivalent to solving the following parabolic partial integro-differential equation x) subject to (i) the "boundary condition" v(t, x) = 0 for x ≤ 0 and (ii) the initial condition v(0, x) = u(x). Finally, the transition probability density p t (x, y) is simply the fundamental solution to this partial integro-differential equation.
In order to present our results, we need to introduce a certain special function S 2 (z) = S 2 (z; α), called the double sine function [11,12]. This function can be defined via two functional equations and the normalizing condition S 2 ((1 + α)/2) = 1. We collect several equivalent definitions and various properties of the double sine function in Appendix A, here we only mention the following two facts which will be used most frequently in this paper: (i) The function S 2 (z) is a real meromorphic function having poles at points z ∈ {mα+n : m, n ∈ N}; (ii) For every b, c ∈ R have the following asymptotic result Moreover, the above asymptotic result holds uniformly in b and c on compact subsets of R.
Now we introduce two functions G and F , which will play an important role in this paper. For (α, ρ) ∈ A and x ≥ 0 we define and F (x) := e x cos(πρ) sin(x sin(πρ) + πρ When it will be needed to stress the dependence on the parameters α and ρ, we will write F (x; α, ρ) for F (x) and G(x; α, ρ) for G(x). We defineF andĜ in a similar way, exchanging the roles of ρ andρ (so thatF (x) = F (x; α,ρ)).
Note that the integral in (11) converges for x = 0 (and thus for all x > 0): this is easy to establish using (10). Another important observation is that the function G is completely monotone, that is (−1) n G (n) (x) > 0 for all n ∈ Z + and x > 0. Finally, we note that the function F is bounded on (0, ∞) if ρ ≥ 1/2, and it grows exponentially if ρ < 1/2 (with a similar result forF andρ).
The following theorem is our first main result in this paper: here we generalize formulas (1) and (2) which hold in the Brownian motion case. Theorem 1. Let X be a stable process defined by parameters (α, ρ) ∈ A.
(ii) If α > 1 or ρ = 1/2 then for all t, x, y > 0 we have In the symmetric case Theorem 1 was established in [15] (see Example 6.1), and in the non-symmetric case the eigenfunctions F (x) were computed in [16] (in both papers the function G(x) was given in an equivalent integral form).
Our next goal is to study the transition semigroup P t and to establish an analogue of identity (5). In the case of non-symmetric stable processes the situation is bound to be more complicated than in the symmetric case, since one of the functions F ,F is exponentially increasing. In order to properly define the operators which diagonalize the transition semigroup, we first need to introduce a suitable space of test functions. The sets H α are non-empty: for example, u(x) = (1 + x) −x ∈ H 1 and v(x) = exp(−x α ) ∈ H β for β ≥ α > 1 (but not for β < α or α ≤ 1). It is clear that H α ≡ H 1 for α ≤ 1 and H α ⊂ H β for 1 ≤ α < β ≤ 2. The following properties follow easily from the definition: if u and v belong to H α then the same is true for functions for all a > 0,b ≥ 0; (iv) x a e −bx α u(x) for all a ≥ 0, b ≥ 0.
We also record here the following important property: (v) The restrictions u| R + of u ∈ H α are dense in L 2 (R + ).
The above property is easy to prove. Indeed, assuming that there exists v ∈ L 2 (R + ) which is orthogonal to all u ∈ H α , we obtain ∞ 0 v(x)(1 + x) −ax dx = 0 for all a > 0. Substituting e y for (1 + x) x and writing w(y) = v(x) dy dx , we see that ∞ 1 w(y)e −ay dy = 0. Therefore, w(y) = 0 for almost all y > 1, and so v(x) = 0 for almost all x > 0.
Next, we define the following integral operators, which generalize Fourier sine transform (4): We denote byP t the transition semigroup of the dual processX, killed on the first exit from (0, ∞). According to Hunt's switching identity (see [2,Theorem 5]), the kernel of the dual semigroup is given byp t (x, y) = p t (y, x), thusP t is simply the adjoint operator of P t . The next theorem is our second main result.
(i) Π can be extended to a bounded self-adjoint operator Π : is a symmetric operator such thatΠH α is dense in L 2 (R + ). For all u ∈ H α we have (ii) For u ∈ H α and t > 0 we have and ΠP tΠ u = e −tλ α u.
(i) The functions u λ (x) := F (λx) are the eigenfunctions of the semigroup P t , that is P t u λ = e −tλ α u λ .
(ii) The functionsF (λx) are the eigenfunctions of the dual semigroupP t , in the sense that for all v ∈ H α we haveP tΠ v =Πe −tλ α v.
Our third main result gives Laplace and Mellin transforms of the eigenfunctions.
The corresponding result forF (x) can be obtained from the above formula by replacing ρ →ρ.
(ii) If ρ ≥ 1/2 then for Re(z) ∈ (−αρ, 0) The two formulas (19) and (20) can lead to many useful explicit results about the eigenfunctions. For example, when α is irrational, one could use the methods of [9,10,14] and obtain complete asymptotic expansions of F (x) as x → 0 + or x → +∞, as well as convergent power series representations. We leave this investigation for future work, and in this paper we only include the discussion of Doney classes in Section 3: as we will see, in this case all expressions involving the double-sine function reduce to simple finite products.
The paper is organized as follows. In Section 2.1 we review some results from the fluctuation theory of Lévy processes that will be required later on, and we also outline the plan for proving our main results. In Section 2.2 we study the Wiener-Hopf factors of stable processes and establish some preliminary results. In Sections 2.3, 2.4 and 2.5 we prove Theorems 3, 1 and 2, respectively. In Section 3 we simplify all of our formulas in the case when the process belongs to one of Doney classes, while the special case of spectrally one-sided processes is treated in Section 3.1. Finally, in Section 4 we present some concluding remarks.

The plan for proving our main results
Our proofs are based on the Wiener-Hopf factorization, which is a key result in the fluctuation theory of Lévy processes [2,17]. We present this result here for the sake of completeness. We start with a Lévy process X (a one-dimensional process with stationary and independent increments) and we denote by the running supremum/infimum processes. We will denote by e(q) an exponential random variable with expected value 1/q, and assume that e(q) is independent of the process X. The following result is contained in Theorem 6.15(i) and identity (6.28) in [17]: Theorem 4 (The Wiener-Hopf factorization). Let X be a Lévy process started from zero. Then for q > 0 (i) the random variable X e(q) is independent of X e(q) − X e(q) ; (ii) the random variables X e(q) − X e(q) and X e(q) have the same distribution.
Let us explain how we will use the above Wiener-Hopf factorization result in order to obtain information about the transition probability density of the process killed on the first exit from (0, ∞). Assume that X 0 = 0 and denote by f X (x) and f X (x) the probability density functions of X e(1) and −X e(1) , respectively (we will prove later that these densities exist and express them in terms of the double sine function S 2 ). Due to the scaling property of stable processes, the random variable X e(q) has density q 1/α f X (xq 1/α ), with a similar result for X e(q) . Let us denote by H q (x, y, z) the joint density of (X e(q) , X e(q) ) for the process started at X 0 = x: According to the Theorem 4, the pair (X e(q) , X e(q) ) has the same distribution as (x + S q − I q , x − I q ), where S q and I q are independent random variables having the same distribution as X e(q) −x and x−X e(q) , respectively. Therefore the function H q (x, y, z) can be written as follows Let us denote by h t (x, y, z) the joint density of (X t , X t ) for the process started from X 0 = x, that is This function is related to the transition probability density and the semigroup of the killed process via the following identities where we have assumed that u is bounded on (0, ∞). At the same time, by conditioning on the random variable e(q) (which has exponential distribution with the density qe −qt 1 {t>0} ) we see that and therefore the function q → q −1 H q (x, y, z) is the Laplace transform of t → h t (x, y, z). Our plan for proving Theorem 1(ii) and 2(ii) is to invert the Laplace transform in (23) and to obtain with f X (x) and f X (x) expressed in terms of the double sine function, and then deform the contour of integration, so that the vertical line iR is transformed into Hankel's contour (beginning at −∞, going around 0 in counter-clockwise direction and ending at −∞, see Lemma 2 below). As we will see, a lot of effort is required for justifying this transformation of the contour of integration.

Remark 1.
The function H q (x, y, z) is closely related to resolvent operators, defined as It is clear from (22) that R q is an integral operator with the kernel

Some preliminary results
In the next theorem we identify the Wiener-Hopf factors for a stable process X, which are defined as the Laplace transform of positive random variables X e(1) and −X e(1) : andφ(z) can be obtained from the above formula by replacing ρ →ρ.
The above result was established in [14] in the case when α = 1 (see Theorem 4 in [14] and formula (A.2) below). Here we present a much simpler proof which also covers the case α = 1.

Proof of Theorem 5:
The proof is based on the Wiener-Hopf factorization result, Theorem 4. Writing X e(1) = X e(1) + (X e(1) − X e(1) ) and using properties (i) and (ii) we obtain the following factorization This is a classical Riemann-Hilbert problem: we need to find a function φ(−iz) analytic in the upper half-plane Im(z) > 0 and continuous in the closed upper half-plane and another functionφ(iz) having the same properties but in the lower half-plane Im(z) < 0 which meet at the boundary Im(z) = 0 as prescribed by (27).
Let us define by f (z) the function in the right-hand side of (26), and byf (z) the same function, but with ρ replaced byρ. First we will verify that the functions f andf satisfy (27). Assume that z > 0 and let us denote w = α ln(z)/(2πi). Then where we have used identity (A.7) in the form Next, according to the second functional equation in (9) we have .
Using the above result and identity (A.7) in the form .
Thus we have verified that that the functions f andf satisfy (27) for z > 0, and the proof for z < 0 follows by taking the complex conjugate in the above equation.
To prove that our candidate solutions f andf are in fact the Wiener-Hopf factors, we need to apply a certain uniqueness argument. Using the fact that S 2 (z) is analytic and zero-free in the strip 0 < Re(α) < 1 + α we check that the functions f andf are analytic and zero-free in the half-plane Re(z) ≥ 0. The asymptotic result (10) easily gives us z −1 ln(f (z)) → 0 as z → ∞ uniformly in the halfplane Re(z) ≥ 0 (with a similar result forf ). To finish the proof we only need to apply the uniqueness result given in [13, Theorem 1(f)] and conclude that φ ≡ f . Proposition 1. Assume that (α, ρ) ∈ A. The random variable X e(1) is a mixture of exponentials and its density is given by The function f X (x) (the density of −X e(1) ) can be obtained from above equations by replacing ρ →ρ.
Proof. According to [23,Theorem 2], the random variable X e(1) is a mixture of exponentials, therefore we can write its density in the form (28) with some positive measure µ(du). Applying Fubini's Theorem we see that thus the Wiener-Hopf factor φ(z) given by (26) is the Stieltjes transform of the measure µ(du). Using the well-known result on the inversion of Stieltjes transform we conclude that the measure µ(du) has a density µ(u) which can be found via From formula (26) we find (as before, denoting w = iα ln(u)/(2π)) Applying the second functional equation in (9) we check that Combining the above two formulas we finally obtain φ(e πi u) = u −αρ/2 |S 2 (1/2 + α + αρ/2 + iα ln(u)/(2π))| 2 u −α/2 + e −πiαρ u α/2 , and now the desired result (29) follows easily from (30) and the above equation.
Let us denote by R the Riemann surface of the logarithm function. In what follows we will often consider functions defined on R or on sectors in R. In particular, we have ln(e ic z) = ic + ln(z) for all c ∈ R and z ∈ R. Next, we state two simple lemmas, which will be used often in this paper. Both of these results are well-known and can be easily established by Cauchy Residue Theorem.
Lemma 1 (Rotating the contour of integration). Assume that the function f (z), z ∈ R, is analytic in the sector − < arg(z) < b + for some b > 0 and > 0, except for a finite number of poles at points z = z i lying in the sector 0 < arg(z) < b. Assume also that for some δ > 0 we have
The proof of the above result is very simple -one only needs to evaluate the integral distinguishing the two cases λ > 1 and λ < 1. We leave all the details to the reader.

Proof of Theorem 3
Proof of Theorem 3(i): We denote .
The above two equations give us Applying the the second functional equation in (9) we check that Combining the above two results with (34) we obtain formula (32).

Proof of Theorem 3(ii):
Let us again denote by f (z) the Laplace transform of F (x), given by (19). From formulas (10) and (19) we check that the upper bound (33) is true for all z ∈ (0, ∞). At the same time, when ρ ≥ 1/2 the function F (x) is bounded and according to Lemma 4 it satisfies F (x) = O(x αρ ) as x → 0 + . Therefore we can apply Fubini's Theorem and conclude that where both integrals converge absolutely for Re(s) ∈ (−αρ, 0). The integral identity (A.11) implies Formula (20) follows from the above two equations by using (A.7), the second functional equation in (9) and applying the reflection formula for the gamma function.

Proof of Theorem 1
In the next lemma we collect some properties of the function µ(u) defined by (29).
Note that d du [2 sin(π(1/2 + αρ/2 + iα ln(u)/(2π))] where in the last step we have used the functional equation (9) and formula (A.6). Thus we have proved (37) for u = u − , and the result for u = u + follows by taking the complex conjugate. The result in item (ii) follows at once from (10) and (39).
(iii) For x > 0 The corresponding results for f X (x) can be obtained by changing ρ →ρ andF → F .
This ends the proof of item (ii).
Combining the above result with (47) and (11) we get Using formulas (9)  The above four formulas combined with the definition ofF (x) (see formula (12)) imply the desired identity (41).
Remark 2. The proof of part (i) of Lemma 6 shows that f X (x) can in fact be extended to an analytic function on the whole of R, though this result will not be used anywhere in this paper. Also, note that (41) implies the following results, which establish a direct link between the spectral theory of stable processes on the half-line and the Wiener-Hopf factorization theory.
Let us consider what happens to the asymptotic behavior of the function f X (x) (as |x| → +∞) as we increase | arg(x)|. Lemma 6(ii) tells us that there is a transition from power-type decay |q| −α−1 to exponential growth exp(−ξ|q|) which occurs at the critical level | arg(x)| = σ := π(1/α − ρ + 1/2). For values of | arg(x)| < σ we have ξ > 0, thus we have a power-type decay; precisely at the critical level | arg(x)| = σ we have ξ = 0 which results in a bounded oscillatory behavior; finally, when | arg(x)| exceeds the critical level σ we have ξ ∈ (−1, 0) and f X (x) exponentially increases and oscillates. It is easy to check that for all (α, ρ) ∈ A we have σ > π/(2α). Using this result we rewrite the upper bound in Lemma 6(ii) in the following less informative (but more useful) form: There exist constants C = C(α, ρ) > 0 and θ = θ(α, ρ) > 0 such that for all q in the sector | arg(q)| < π/2 + θ we have and for all q in the sector π/2 + θ < | arg(q)| ≤ π we have The corresponding results for f X (x) can be obtained by changing ρ →ρ.
The next corollary will play a crucial role in the proof of Theorem 1(ii) and Theorem 2(ii).
Corollary 3. Assume that (α, ρ) ∈ A. Then for all x, y > 0 we have Proof. First of all, we use formulas (9) and (A.7) to check that Next, we use identity (41) and compute To finish the proof one needs to integrate the above expression in z ∈ (0, min(x, y)) and use the fact that F (0+) =F (0+) = 0 (see Lemma 4).
Proof of Theorem 1, part (i): Let us denote by m t (y) the probability density of the random variable −X t when X 0 = 0 (the existence of this density is known -see [4,5,14], though it also follows easily from the absolute convergence of the integral in formula (51) below). As we have discussed in Section 2.1, the scaling property of stable processes implies that the probability density function of −X e(q) is given by q 1/α−1 f X (yq 1/α ). Conditioning on the exponential random variable e(q) we arrive at the Laplace transform identity ∞ 0 e −qt m t (y)dt = q 1/α−1 f X (yq 1/α ).
We write down m t (y) as the inverse Laplace transform and then pass to Hankel's contour (via Lemma 2 and Corollary 2; here we need α > 1 or ρ ≥ 1/2): Next we compute where the application of Fubini's Theorem is justified due to Corollary 2. The integral in curly brackets can be evaluated using formula (41): where we have used the fact that F (0+) = 0 (Lemma 4) and identity (9). Formula (13) follows by combining the above two identities and changing the variable of integration q = λ α .
Proof of Theorem 1, part (ii): We recall our notation from Section 2.1: the function h t (x, y, z) denotes the joint probability density of random variables (X t , X t ) for a stable process X started from X 0 = x. We start with equation (24), which we reproduce here for convenience: According to Corollary 2, for fixed 0 < z < min(x, y) the integrand is bounded from above by C min(1, |q| −3 ) when q ∈ iR, thus the integral in (52) converges absolutely. Now we plan to pass to Hankel's contour of integration in (52). Corollary 2 tells us that in some sector π/2 ≤ | arg(z)| < π/2 + θ the integrand is bounded from above by C min(1, |q| −3 ). In the sector π/2 + θ ≤ | arg(z)| ≤ π the term e qt is exponentially decaying. When α > 1, the integrand is bounded by Ce 2|q| 1/α +qt , which decays to zero exponentially. When α ≤ 1 and ρ = 1/2 the integrand is bounded by Ce qt , which again decays to zero exponentially. Thus all conditions of Lemma 2 are satisfied and we obtain Assume first that x = y. We use formula (22) and apply Fubini's Theorem to obtain Im e 2πi/α f X ((y − z)e πi/α q 1/α )f X ((x − z)e πi/α q 1/α ) dz q 2/α−1 e −qt dq.
To finish the proof, we compute the integral in curly brackets in (53) using Corollary 3 and change the variable of integration q = λ α . To remove the restriction x = y we note that the right-hand side in (14) defines a continuous function of t, x, y > 0, and we define p t (x, x) by continuity.

Proof of Theorem 2
Recall that we have defined ζ := π 2 min(1, 1/α). Let us define S r to be the "shift operator": S r u(x) = u(r + x), and let Lu(z) := ∞ 0 u(x)e −xz dx denote the Laplace transform of u. Note that u ∈ H α implies S r u ∈ H α for all r ≥ 0. Lemma 7. Let u ∈ H α . Then LS r u(z) is an entire function and for every small > 0 there exists C = C(u, α, ) such that for all r ≥ 0 we have |LS r u(z)| ≤ C min(1, |z| −1 ) in the sector | arg(z)| ≤ ζ + π/2 − .
Proof. The fact that LS r u(z) is an entire function follows from Definition 1, which implies that u(x) is bounded for small x = 0 and decays faster than exp(−δx ln(x)) for large x.
Let us fix β ∈ (−ζ, ζ). By the same estimate as above, we can rotate the contour of integration and change the variable of integration x = e iβ y to obtain According to Definition 1, there exists δ(u, α, ) > 0 and C(u, α, ) > 0 such that |u(r + e iβ y)| ≤ C min(1, y −δy ) for all y > 0. Using the estimate Re(e iβ z) ≥ sin( )|z| in the sector | arg(e iβ z)| ≤ π/2 − we obtain (for all z in this sector) Furthermore, Since β can be chosen arbitrary in the interval (−ζ, ζ), we obtain the desired result.
The proof of the case r > 0 follows exactly the same steps, since the function v r (x) = S r u(x) also belongs to H α . Lemma 7 then tells us that the upper bound in (57) is uniform in r.
The proof of part (ii) follows very similar steps. We choose and β in the same way as above. Let f (z) := 2/π ∞ 0 F (x)e −zx dx. As we have established in formula (19), the function f (z) can be analytically continued to a meromorphic function on R. Moreover, f (z) is analytic in the sector | arg(z)| < πρ and in any smaller sector | arg(z)| < πρ − 2 it satisfies an upper bound |f (z)| ≤ C min(1, |z| −1−αρ ), which follows from (10) and (19).
Note that e −x F (x) and e λx u(x) are in L 2 (R + ). We express Πu using Plancherel's theorem for Laplace transform as follows The contour of integration is deformed in the same way as in (59) and we obtain Applying Lemma 3 and the above mentioned estimate |f (we −iβ )| ≤ C min(1, |w| −1−αρ ) we get the desired result (58).
Proof. The fact that both operators Π andΠ are symmetric follows easily from their definition (15). Let us denote by M the Mellin transform operator: It is well-known that the Mellin transform operator is an isometry between L 2 (R + ) and L 2 (R). Since the operator Π is a Mellin convolution, Theorem 3(ii) and a standard computation leads to identity Formula (20) and location of poles/zeros of the double sine function imply that MF (z) has no poles on the real line, and the asymptotic relation (10) tells us that |MF (z)| exp((ρ − 1/2)π|z|) → 1 as z → ∞. Thus for ρ ≥ 1/2 the function MF (z) is bounded and it induces a bounded multiplication operator on L 2 (R), thus the operator Πu = M −1 [MF × M −1 u] is a bounded (self-adjoint) operator on L 2 (R + ). The fact thatΠ maps H α into L 2 (R + ) follows from Lemma 8(ii). Let us prove that the imagê ΠH α is dense in L 2 (R + ). Assume to the contrary thatΠH α is not dense in L 2 (R + ). Then there must exist a nonzero function w ∈ L 2 (R + ) such that ∞ 0 w(x)Πu(x)dx = 0 for all u ∈ H α . Consider any nonzero function u ∈ H α and define u a (x) = u(ax) for a > 0. As we have mentioned in the paragraph following Definition 1, we have u a ∈ H α for all a > 0. Let v(λ) =Πu(λ). Then it is clear from the definition ofΠ thatΠu a (λ) = a −1 v(a −1 λ), so that all functions v(a −1 x) belong toΠH α and we have (61) Next, due to the fact that v =Πu and the estimate (58) we see that Mv(z) is analytic in a strip |Im(z)| < 1/2, thus Mv(z) is nonzero almost everywhere in this strip. Equation (61) then implies that Mw(−z) = 0 for almost all z ∈ R, therefore w ≡ 0 and we arrive at a contradiction.
Proof of Theorem 2: Given the result in Lemma 9, it remains to establish formulas (16), (17) and (18). Our plan is to prove (17) using similar method as in the proof of Theorem 1(ii) and then to derive the remaining statements (16) and (18) as corollaries of (17). When ρ = 1/2 formula (17) follows at once from (14) and Fubini's Theorem (whose application is justified since the functions F andF are bounded in this case). Thus we will only consider the case when ρ > 1/2.
We recall that the operator Ξ is defined by (56) and S r is the "shift" operator from Lemma 7. Let us fix u ∈ H α . We begin by rewriting formula (22): where we have defined The Laplace transform of t → k t (x, z) can be computed by applying Fubini's Theorem to the above formula and using (21), (23) and (56): Let us denote w = (x − z)|q| 1/α . According to Corollary 2 and Lemma 8, the function q → K q (x, z) is analytic in the sector | arg(q)| < π + and it is bounded in the sector | arg(z)| ≤ π by |K q (x, z)| ≤ C(u, α, ρ) × |q| 2/α−1 min(w αρ−1 , w −α−1 ) × min(|q| ρ−1/α , |q| −1/α ). (64) For fixed z and x the upper bound (64) implies the estimate: At the same time, using the trivial bounds min(w αρ−1 , w −α−1 ) ≤ w αρ−1 and min(|q| ρ−1/α , |q| −1/α ) ≤ q ρ−1/α we obtain from (64) the following result: Next, we fix x, z satisfying 0 < z < x, use the upper bound (65) and express k t (x, z) as the inverse Laplace transform Next, we use the same upper bound (65) and Lemma 2 and pass to Hankel's contour of integration in the above integral: Combining this result with formula (62) and applying Fubini's Theorem we obtain The use of Fubini's Theorem in the previous step is justified since the function (z, q) → K qe πi (x, z)e −qt is absolutely integrable on (0, x) × (0, ∞) -this follows easily from the upper bound (66). Next, we rewrite the integral in square brackets in (67) as follows In the second step we have again applied Fubini's Theorem: to justify its use note that according to Corollary 2 we have |f X ((x − z)q 1/α e πi/α )f X ((y − z)q 1/α e πi/α )u(y)| ≤ C × |x − z| αρ−1 × |y − z| αρ−1 e y × |u(y)| for some constant C = C(α, ρ, q), and the function in the right-hand side of the above equation is integrable over the region {(z, y) ∈ R 2 : 0 < z < min(x, y)}. In the third step of (68) we have applied Corollary 3. Combining formulas (67) and (68) and changing the variable of integration q = λ α we obtain the desired result (17). Formula (16) follows from (17) by taking the limit t → 0 + : Then P t u(x) → u(x) for all x > 0 (since u is continuous and a Lévy process X killed on the first exit from (0, ∞) is continuous in probability), while the right-hand side of (17) converges to ΠΠu.
Let us now prove formula (18). We denote by (u, v) the inner product in L 2 (R + ). We start with v, w ∈ H α and use formulas (16) and (17) and the fact that operators Π andΠ are symmetric to compute In the last step we have used the fact that e −tλ α w ∈ H α . Since (69) is valid for all v ∈ H α and H α is dense in L 2 (R + ) we conclude thatP tΠ w =Πe −tλ α w. Therefore, ΠP tΠ w = ΠΠe −tλ α w = e −tλ α w, as desired.
Proof of Corollary 1: Part (ii) of Corollary 1 was established above (see the discussion following (69)).
To prove part (i), we start with u ∈ H α , v ∈ L 2 (R + ) and denoteΠu = w. According to (16) we have Πw = u. We use formula (18) and compute Since w ranges overΠH α , and the latter is a dense subset of L 2 (R + ) due to Theorem 2(i), we see that (70) implies that for all v ∈ L 2 (R + ) we have P t Πv = Πe −tλ α v. Application of Fubini's Theorem (note that u λ (x) = F (λx) is bounded when ρ ≥ 1/2) gives Since v ∈ L 2 (R + ) is arbitrary, we have P t u λ (x) = e −tλ α u λ (x) for almost all x, λ > 0. By continuity, this relation holds for all x, λ > 0, as desired.

Doney classes
Doney [6] has introduced classes of stable processes for which the Wiener-Hopf factors can be computed explicitly in terms of q-Pochhammer symbols. Let k, l ∈ Z and let us define by C k,l the class of stable processes with parameters (α, ρ) satisfying These classes include the spectrally-positive (C 0,1 ) and the spectrally-negative processes (C −1,−1 ). Note that X ∈ C k,l if and only ifX ∈ C −k−1,−l . Let us define the q-Pochhammer symbol (1 − aq −j ) −1 , n < 0, and (a; q) 0 = 1.
The main result of Doney [6] is the following formula for the Wiener-Hopf factor: for a process X ∈ C k,l we have where q := e 2πiα andq := e − 2πi α . We note that this formula follows from our general result (Theorem 5) and formula (A.8). It is not surprising that all expressions involving the eigenfunctions F (x) also simplify considerably for Doney classes C k,l .
Proposition 2. Assume that X ∈ C k,l .

Spectrally one-sided processes
The spectrally-negative (spectrally-positive) processes belong to classes C 0,1 (respectively, C −1,−1 ). All the formulas given in Proposition 2 for general Doney classes C k,l remain valid, except that in the spectrally-negative case αρ = 1 we should take G(x) ≡ 0. Let us consider this case in more detail.
In the spectrally-negative case it is well-known that the random variable X e(1) has exponential distribution with parameter one, so that f X (x) = e −x , x > 0.

Conclusion
Our methods for establishing Theorems 1, 2 and 3 are fundamentally based on the Wiener-Hopf factorization (Theorem 4), which comes from the fluctuation theory of Lévy processes. We hope that the methods developed in this paper will help to study spectral properties of semigroups of more general Lévy processes on a half-line (see [16] for some results in this direction). Of course, the case of stable processes is a very special one, as stable processes lie in the intersection of the class of Lévy processes and the class of self-similar Markov processes. We would like to mention that the spectral properties of the semigroup of general positive self-similar Markov processes (which include stable processes killed on the first exit from (0, ∞)) are studied in [21] using a different approach (based on Lamperti transformation and Mellin transform techniques).
Finally, we would like to emphasize the importance of the double sine function S 2 (z; α) in the study of stable processes. This function appears in the definition (12) of the eigenfunctions, in formulas (19) and (20) that give the Laplace and Mellin transform of the eigenfunctions and in formula (26), which gives the Wiener-Hopf factors of stable processes. So far, apart from some number-theoretic applications [12,26], this curious special function has appeared mostly in the Physics literature [8,22,24,29], where it is used in studying quantum topology and cluster algebras. It is an interesting question whether this appearance of the double sine function in our study of stable processes is simply a coincidence or there is indeed a deeper connection between stable processes on the half-line and quantum topology and/or cluster algebras. (iv) The double gamma function G(z; α) of Barnes [1]: .