L\'evy processes linked to the lower-incomplete gamma function

We start by defining a subordinator by means of the lower-incomplete gamma function. It can be considered as an approximation of the stable subordinator, easier to be handled thank to its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study L\'{e}vy processes time-changed by these subordinators, with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion gives a model of anomalous diffusion, which exhibits a sub-diffusive behavior.


Introduction
In the spirit of [2], we consider here a subordinator S α (t), t ≥ 0, defined by means of the lowerincomplete gamma function of parameter α ∈ (0, 1], i.e. γ(α, x) = x 0 e −w w α−1 dw, x > 0. (1.1) We will see that, in the special case α = 1, it reduces to a homogeneous Poisson process, while, in general, it can be represented as a compound Poisson process with positive jumps of size greater than one. Such a process retains many properties of the stable subordinator, e.g. the tail behavior of the distribution and the asymptotic form of the fractional moments, even if it loses the property of self-similarity. A standard reference for the theory of stable processes is [23]. By a slight modification we are led to a new subordinator whose jumps are greater than ǫ > 0, which converges to a stable one in the limit for ǫ → 0. We prove that its density q ǫ (x, t) solves an equation where a perturbation of the Riemann fractional derivative appears. When ǫ → 0, such operator reduces to the Riemann derivative and we obtain the well known equation governing the stable density. For an introduction to fractional derivatives and fractional equations consult [19].
The above framework can be extended to the so-called multivariate subordinators, i.e. multidimensional Lévy processes with increasing marginal components (for their properties and applications see e.g. [8] and [3]).
In order to overcome the drawback of infinite moments of S α , we consider a tempered version of our subordinator, say S α,θ (t), t ≥ 0, where θ ≥ 0 is the tempering parameter, whose distribution displays finite moments of any integer order.
The triplet (a, b, ν) is called the Lévy triplet of the Bernstein function ϕ (see, for example, [25], p.21) and ν(·) is a Lévy measure. Finally, a Bernstein function ϕ is complete if and only if its Lévy measure in (2.2) has a completely monotone density m(·) with respect to the Lebesgue measure, i.e.the following representation holds for a completely monotone function m(·): ϕ(η) = a + bη +

Multivariate subordinators
In the multivariate case, we recall that a subordinator in the sense of [8] and [3] is a d-dimensional Lévy process with increasing marginal components. We denote a multivariate subordinator by The multivariate Lévy measure ν(dx 1 , . . . , dx d ) satisfies the following condition is a multivariate Bernstein function. A d-dimensional subordinator is said to be stable if, using the spherical variables ρ ∈ (0, ∞) and θ ∈ B d−1 (B d−1 denoting the d − 1-dimensional unit sphere), its Lévy measure can be expressed as where M (dθ) is a probability measure on B d−1 In other words, a d-dimensional stable subordinator is a multivariate stable process with increasing marginal components.
In this case, the Bernstein function reads By Laplace inversion, the density q(x, t), x ∈ R d + , t ≥ 0 of a multivariate stable subordinator satisfies the following equation where (∇ x · θ) α is the fractional directional derivative along the unit vector θ, defined as Thus the operator on the right-hand side of (2.5), also studied in [6] and [19], is the average, under the measure M (dθ), of (∇ x · θ) α . For d = 2 we have θ = (cos β, sin β), and the operator takes the following form −k π 2 0 cos β ∂ ∂x 1 + sin β ∂ ∂x 2 α q(x 1 , x 2 , t) M (dβ).

Fractional equation satisfied by the incomplete gamma function
The incomplete Gamma function defined in (1.1) is a Bernstein function. Indeed it is non-negative, C ∞ , null at the origin, with derivatives satisfying and so on. Preliminarily, we show that the lower-incomplete Gamma function (1.1) solves the following integro-differential equation x e −w w β−1 dw is the upper incomplete Gamma function (which is defined for any β, x ∈ R and is real-valued for x ≥ 0). Up to a multiplication by α, the operator on the left-side is the Caputo fractional derivative with tempered kernel (see, e.g., [15]). We observe that (2.6) is a relaxation equation because the solution u(x) = γ(α, x) converges to the stationary solutioñ u(x) = Γ(α) as x → ∞.
As an alternative proof, we recall that the Laplace transform of (2.7) is given by (see [2]); moreover, so that the Laplace transforms of the two sides of (2.6) coincide. We can easily check that, for α = 1, the equation (2.6) reduces to d dx which (for u(0) = 0) is satisfied by u(x) = 1 − e −x = γ(1; x), even though the expression of D λ,ρ t given in (2.7) is not well-defined in this special case.

Definition and properties
We start by considering the subordinator defined by means of the lower-incomplete gamma function, i.e. with Laplace exponent αγ(α; η), for α ∈ (0, 1]. is the Laplace exponent of a finite-activity (or step) subordinator where π is an absolutely continuous Lévy measure, with completely monotone density Proof. The incomplete gamma function γ(α, x) is a Bernstein function, as explained in section 2.3. Hence also αγ(α, x) is a Bernstein function. We now prove that representation (2.2) holds, in this case, for a = b = 0 and for the Lévy measure given in (3.2): indeed we have that where the interchange of the integrals' order is allowed by the absolute convergence of the double integral and the application of the Fubini theorem. In order to prove that S α does not have strictly increasing trajectories, we must show that the integral of the Lévy measure on (0, ∞) is finite. Indeed by (2.2) the last condition, together with a = b = 0, is sufficient to prove that a subordinator is a step process (i.e. it has piecewise sample paths), see [24], p.135: in this case we have that by considering formula (3.191.2) of [10], since α > 0. Finally, it is easy to check, by differentiating, that the density of the Lévy measure in (3.2) is completely monotone.

Remark 2
In the limiting case where α → 1 − the process S α reduces to the Poisson process. Indeed we have that lim α→1 − π(dz) = δ 1 (z)dz, which is the Lévy measure of the Poisson process of rate 1: this can be seen by considering that We underline that the Lévy measure given in (3.2) is different from zero only for z ≥ 1: this means that the subordinator performs almost surely jumps of size greater than one. As a consequence and by considering that its diffusion coefficient is zero, the process S α has also finite variation (see Theorem 21.9 in [24]).
Moreover, the result in (3.3) implies that S α is a Lévy process of type A (see Def.11.9 in [24], p.65) and has finite activity, i.e. its number of jumps is finite on every compact interval for almost all the paths (see Thm. 21.3 in [24]). Thus S α can be represented as a compound Poisson process is a homogeneous Poisson process with rate λ = αΓ(α) and the jumps Z α j are i.i.d. random variables, taking values in [1, +∞), with probability density For α = 1, the jumps are unitary and the process coincides with the standard Poisson. The representation (3.4) can be checked directly as follows: the Laplace transform of the addends Z α j is given by for any j = 1, 2, ..., by formula (3.383.9) in [10] for α < 1. Then, by conditioning, we get Finally, we note that S α is not self-similar, as can be checked from its Laplace transform.
The moments of any integer order of S α are not finite, for any t > 0, since does not converge, for any k ≥ 1, (see [1] p.132). Alternatively, this can be seen by applying the Wald formula and by noting that EZ α j = sin(πα) π +∞ 1 (z − 1) −α dz = +∞, j = 1, 2, ..., The reason can be found in the heaviness of its distribution's tail. Indeed it can be proved that it displays the same power law of the stable subordinator, i.e. P (X α (t) > x) ≃ tx −α Γ(1−α) for large x (see [23], p.17).
However we can study the asymptotic expression of the fractional moment of S α , of order p ≤ α and for large t. We recall that the fractional moments have been introduced and studied by many authors, in order to overcome the problem of infinite integer order moments, especially in the stable case (see, among the others, [17] and [28]); in particular, we will follow the techniques given in [13], which are based on fractional differentiation of the Laplace transform.
Theorem 3 1) Let α ∈ (0, 1), then, for any t ≥ 0 and for x → +∞, we have that 2) Let p ∈ (0, 1], then the fractional moment of order p of the process S α exists, finite, for p ≤ α and it asymptotically behaves as follows where we have taken the Taylor series expansion (up to the first order) and we have considered the asymptotic behavior of the lower incomplete gamma function, i.e.
Formula (3.9) can be easily derived by rewriting (1.1) as follows: By applying the Tauberian theorem (see [9], Thm.4, p.446) we get, for any t ≥ 0, result (3.7). In order to derive the asymptotic behavior of the fractional moment of order p, we apply the Laplace-Erdelyi Theorem to the following integral (see [26], for details). Let x ∈ (x 0 ,x 1 ), with x 0 ,x 1 ∈ R, let moreover h(x) and ϕ(x) be independent of t > 0 and h(x) > h(x 0 ) for all x ∈ (x 0 , x 1 ). Moreover, let the following expansions hold, for under the assumption that the integral (with finite or infinite delimiters) converges absolutely for all sufficiently large t. We only need c 0 = b 0 /µa γ/µ 0 , then, for the expressions of the other c j 's we refer to [26] and [27]. In our case, we have that ϕ( , by using the well-known series expression of the incomplete gamma function (see [11]). Thus we get µ = α and a 0 = 1. By considering (3.10) we thus get which coincides with (3.8).

Remark 4
The fractional moment of order p converges, for t → +∞, to the value obtained in the stable case, for any t (see [18]).

Link to stable subordinators
The one-dimensional case We now purpose a slight generalization of the previous results, in order to provide an approximation of a stable subordinator: while the previously defined subordinator S α performs jumps greater than 1, we now consider a lower bound for the jump size equal to ǫ > 0. We thus define the following Lévy measure with support on (ǫ, ∞) and with density The corresponding Laplace exponent has the form ϕ ε (η) = α ǫ α γ(α; ηǫ). Indeed, By a simple change of variable, the Laplace exponent can also be expressed as η α multiplied by a correction factor depending on ǫ: is such that O ǫ (η) → 1, as ǫ → 0. Thus, in the limit as ǫ → 0, the related subordinator S (ε) α := S (ε) α (t), t ≥ 0 converges to a α-stable subordinator, since By considering that α is a compound Poisson process, i.e.
where N ǫ (t) is a Poisson process with intensity αΓ(α)ǫ −α and Z ǫ j has density α is a compound Poisson approximation of a stable subordinator. Therefore, it can be useful in many applications, since it is easier to be handled with respect to the stable subordinator, thanks to its finite activity.
As far as the governing equation is concerned, we can show that the transition density of S (ε) α satisfies a fractional equation, which generalizes the governing equation (2.4) of the stable subordinator. In particular, the fractional derivative on the right side is corrected by means of the following operator where h : R + → R is a function such that the above integral converges, while e −y∂x denotes (with a little abuse of notation), the translation operator. Note that (3.13) tends to the identity operator as ǫ → 0, since Thus we can check that the density q ε := q ε (x, t), x, t ≥ 0, of S (ε) α solves the following equation by applying the Laplace transform to both members, which gives where O ǫ (η) has been defined in (3.12).

Remark 5
The approximation presented above could be applied to the fractional derivative with timedependent order, i.e.
∂ ∂x α(t) , where α(t) takes values in (0, 1). Such operator governs a timeinhomogeneous version of the stable subordinator (see, for example, [4] and [22]), which could be approximated by considering the time-dependent Lévy measure π ǫ (x, t) = α(t) The multivariate case Following the lines of the one-dimensional case, we look for a compound Poisson approximation for a multivariate stable subordinator, which we introduced in sect. 2.2. We define the family of Lévy measures and, by the same calculations as in (3.11), we obtain the following family of Bernstein functions (the symbol η denotes the vector (η 1 , . . . , η d ) and · denotes the scalar product) where the corrective term O ǫ (η · θ) = α ǫ α ǫ 0 e −η·θy y α−1 dy tends to 1 as ǫ → 0. By Laplace inversion, the density q ǫ (x, t) of our process satisfies tends to the identity operator in the limit ǫ → 0.

The tempered subordinator S α,θ
In order to avoid the inconvenience of infinite moments of S α , we define a tempered counterpart of the latter.
Theorem 6 Let η, θ > 0 and α ∈ (0, 1], then the function is the Laplace exponent of a tempered subordinator S α,θ := {S α,θ (t), t ≥ 0}, with Lévy triplet (0, 0, π θ ) and (absolutely continuous) Lévy measure π θ , with density The sample paths of S α,θ are not strictly increasing; the mean and variance of S α,θ read, respectively, Proof. It is immediate to check that (4.1) is a Bernstein function (as a consequence of Theorem 1). Moreover we can prove that the representation (2.2) holds, in this case, for a = b = 0 and for the Lévy density given in (4.2): indeed we have that which coincides with (4.1). Also in this case the Lévy measure is finite, since with respect to η and considering the relationship Remark 7 It is easy to check that the mean and variance of S α,θ , given in (4.3), tend to infinity, as θ → 0, as expected from (3.6).

The generator equation.
Let us consider the case β 0 = θ = 0. For h ∈ B b (R), where B b (R) denotes the set of real-valued bounded Borel measurable functions, equipped with the sup-norm, the operator T t defined by defines a strongly continuous contraction semigroup on B b (R). If A is the generator of T t , then (5.4) satisfies for h in the domain of A. If σ α (t) is a stable subordinator, then the process X(σ α (t)) induces the subordinate semigroupT In light of the Phillips theorem (see [24] page 212), the semigroup (5.5) satisfies where the fractional power of the operator is defined by at least on the same domain of A. Now, if we employ the subordinator S (ǫ) α , which is an approximation of σ α (see the discussion in section 3.2), we obtain an approximation of equation (5.6). Indeed, using again the Phillips theorem, satisfies the following equation The operator on the right-side is an approximation of the fractional power in (5.7), to which it converges as ǫ → 0. We observe that, in the special case X(t) = t, i.e. when T t is the shift operator, the operator on the right-side is an approximation of the Marchaud fractional derivative, namely
It is easy to check that the jump component of the subordinated process has finite activity for any α ∈ (0, 1), since Moreover we have that The characteristic function of Z(t) is given by By conditioning and considering (4.3), we have that EB(β 0 t + S α,θ (t)) = 0, for any t, θ ≥ 0, and the autocovariance of the subordinated Brownian motion, for any t, τ ≥ 0, reads (5.10) Thus, even if the autocovariance is linear w.r.t. the time argument, the parameters α and θ can be interpreted as a measure of deviation from the standard Brownian dependence structure: in particular, for θ → 0 and for α strictly less than 1, the autocovariance tends to infinity, for any t.

Subordinated fractional Brownian motion
We now consider the process {B H (S α (t)), t ≥ 0}, where B H := {B H (t), t ≥ 0} is the fractional Brownian motion (hereafter FBM) with Hurst parameter H and the subordinator S α is supposed to be independent of it. The FBM B H is defined, for any H ∈ (0, 1) as a self-similar process with index H and with zero-mean Gaussian distribution. Its one dimensional distribution has density x ∈ R, t ≥ 0.
It can be expressed, in terms of the standard Brownian motion B := {B(t), t ≥ 0} , by the following representation For details on the fractional Brownian motion we refer to [16]. It is worth recalling that the FBM exhibits subdiffusive dynamics for H < 1/2 and a superdiffusive one for H > 1/2; indeed the moment of order q of the FBM is given by (6.1) (see, for example, [13]). Different forms of time-changed FBM have been introduced and studied (see [13], [14], [21]). We prove here that the FBM, subordinated by an independent S α , displays the long-range dependence (LRD) property, for H ∈ (0, 1/2); moreover, this behavior depends on α, instead of what happens in the cases of the FBM subordinated by the tempered stable subordinator (studied in [13]) and by the gamma process (analyzed in [14]). Indeed, in the last cases, the LRD rate depends only on the Hurst parameter H.
Since the process is not stationary, we use the following definition of long-range dependence: a process Z(t) is said to have LRD property if, for s > 0 and t > s,.
Theorem 9 Let H ∈ (0, 1/2) and α ≥ 2H. Let Z H (t) := B H (S α (t)), t ≥ 0, (6.3) where B H is the FBM, with Hurst parameter H and S α is supposed to be independent of it. Then Z H has the LRD behavior given in (6.2), with d = 1 − H α . Proof. We notice that the subordinator, being a compound Poisson process has stationary and independent increments. By conditioning and considering (6.1), we get, for q < α/H, We thus evaluate the covariance of the process Z H , as follows , for s < t, By putting K 2H,α := Γ 1 − 2H α /Γ (1 − 2H), we can write E (Z H (t), Z H (s)) ∼ H α K 2H,α st 2H α −1 . Therefore, the correlation function asymptotically behaves as follows, for t → +∞, Note that we have applied (6.4) for q = 2 and thus (6.5) holds for α ≥ 2H, by Theorem 3; as a consequence, the result is limited to the case of a FBM with H < 1/2.

Remark 10
We underline that the values of H ≥ 1/2 are excluded, since, in this range, the E S α (t) 2H is infinite. To overcome this limitation we could have used the tempered subordinator S α,θ (t) (as done in [13], in the stable case); unfortunately, in the tempered case, the function h(x) in (3.10) would be given by h(x) = αγ(α; x + θ) − αγ(α; θ), which cannot be expanded, as requested by the Laplace-Erdelyi Theorem.

Remark 11
We stress that the LRD parameter d is dependent on α, on the contrary of what happens in the case of a FBM subordinated by a tempered stable subordinator or by the gamma process, where the rate d of the LRD depends only on the Hurst parameter H and coincides with that of the fractional Brownian motion itself (see [13] and [14], respectively).
It is evident by (6.4) that var(Z H (t)) ≃ Kt 2H/α , for t → +∞ (where K is a constant depending on α, H) and therefore the process Z H behaves asymptotically as a subdiffusion, according to the parameter α. Indeed, 2H/α is always less than one (since, by assumption, 2H ≤ α) and the subdiffusive behavior is more marked the greater the value of α (for any fixed H).