Spectral functions related to some fractional stochastic differential equations

In this paper we consider fractional higher-order stochastic differential equations of the form \begin{align*} \left( \mu + c_\alpha \frac{d^\alpha}{d(-t)^\alpha} \right)^\beta X(t) = \mathcal{E}(t) , \quad t\geq 0,\; \mu>0,\; \beta>0,\; \alpha \in (0,1) \cup \mathbb{N} \end{align*} where $\mathcal{E}(t)$ is a Gaussian white noise. We derive stochastic processes satisfying the above equations of which we obtain explicitly the covariance functions and the spectral functions.


Introduction
In this paper we consider fractional stochastic ordinary differential equations of different form where the stochastic component is represented by a Gaussian white noise. Most of the fractional equations considered here are related to the higher-order heat equations and thus are connected with pseudoprecesses.
The first part of the paper considers the following stochastic differential equation where d α d(−t) α represents the upper-Weyl fractional derivative. We obtain a representation of the solution to (1.1) in the form where h α (z, s), z, s ≥ 0, is the density function of a positively skewed stable process H α (t), t ≥ 0 of order α ∈ (0, 1), that is with Laplace transform ∞ 0 e −ξz h α (z, s)dz = e −sξ α , ξ ≥ 0.
For (1.2), we obtain the spectral function and the related covariance function. The second type of stochastic differential equations we consider has the form µ + (−1) n ∂ 2n ∂t 2n β X(t) = E(t), β > 0, µ > 0, n ≥ 1, (1.4) where E(t) is a Gaussian white noise. The representation of the solution to (1.4) is where u 2n (x, w), x ∈ R, w ≥ 0 is the fundamental solution to 2n-th order heat equation The autocovariance function of the process (1.5) can be written as where W 2β is a gamma r.v. with parameters µ and 2β. The spectral function f (τ ) associated with (1.7) has the fine form 4w √ 4πw and, from (1.7) we obtain an explicit form of the covariance function in terms of the modified Bessel functions. In connection with the equations of the form (1.6) the so-called pseudoprocesses, first introduced at the beginning of the Sixties ( [6]) have been constructed. The solutions to (1.6) are sign-varying and their structure has been explored by means of the steepest descent method ([9; 1]) and their representation has been recently given by [10].
For the fractional odd-order stochastic differential equation the solution has the structure The solutions u 2n+1 and u 2n are substantially different in their behaviour and structure as shown in [10] and [7]. A special attention has been devoted to the case n = 1 for which (1.10) takes the interesting form where Ai(·) is the first-type Airy function . The process X 3 can also be represented as where the mean value must be meant w.r.t. Y 3 (W β ) and Y 3 is the pseudoprocess related to equation ∂u ∂t = − ∂ 3 u ∂x 3 (1.14) and W β is a Gamma-distributed r.v. with parameters β, µ independent from Y 3 . The autocovariance function of X 3 has the following form where W 2β is the sum of two independent r.v.'s W β . For the solution to the general odd-order stochastic equation we obtain the covariance function Of course, the Fourier transform of (1.16) becomes (1.17) Stochastic fractional differential equations similar to those dealt with here have been analysed in [2], [3] and [5]. In our paper we consider equations where different operators are involved.

A stochastic equation involving fractional powers of fractional operators
In this section we consider the following generalization of the Gay and Heyde equation (see [3]) where E(t), t > 0, is a Gaussian white noise with The fractional derivative appearing in (2.1) must be meant, for 0 < α ≤ 1, as For information on fractional derivatives of this form, called also Marchaud derivatives, consult [11, pag. 111]. For λ ≥ 0, we introduce the Laplace transform which can be immediately obtained by considering that for a function f such that e λt f (t) ∈ L 1 ([0, ∞)).
Theorem 2.1. The representation of a solution to the equation (2.1) can be written as Proof. The solution to the equation (2.1) can be obtained as follows where h α (z, s) is the probability law of H α (s), s > 0. In the last step of (2.9) we used the translation property This is because In view of the Taylor expansion which holds for a bounded and continuous function φ : [0, ∞) → [0, ∞). Since we can find a sequence of r.v.'s {a j } j∈N and an orthonormal set, say {φ j } j∈N , for which (2.13) holds true ∀ j and such that we can write (2.10). Therefore, is the formal solution to the fractional equation (2.1) with representation, in mean square sense, given by where δ is the Dirac delta function and from (2.10) we infer that Consult on this point [5]. A direct proof is also possible because from (2.8) we have that (2.18) In the last step we applied (2.10).
Our next step is the evaluation of the Fourier transform of the covariance function of the solution to the differential equation (2.1). Let Proof. The Fourier transform of the covariance function of lag h = t 2 − t 1 of (2.7) is given by By considering the characteristic function of a positively-skewed stable process with law h α , we have that Thus, we obtain that Remark 2.3. In the special case α = 1 the result above simplifies and yields (2.23) We note that for β = 1, (2.23) becomes the spectral function of the Ornstein-Uhlenbeck process. Processes with the spectral function f are dealt with, for example, in [2] where also space-time random fields governed by stochastic equations are considered. The covariance function is given by where K ν is the modified Bessel function with intergal representation given by (see for example [4], formula 3.478). We observe that K ν = K −ν and K 1 We study the covariance of (1.2). Recall that, a stable process S of order α with density g is characterized by g(ξ, t) = Ee iξS(t) = e −σ 2 |ξ| α t , α ∈ (0, 2]. Consider two independent stable processes S 1 (w), S 2 (w), w ≥ 0, with σ 2 1 = 1 and σ 2 2 = 2µ cos πα 2 . Let g 1 (x, w), x ∈ R, w ≥ 0 and g 2 (x, w), x ∈ R, w ≥ 0 be the corresponding density laws. Then, the following result holds true.
and W β is a gamma r.v. with parameters µ 2 , β.

Fractional powers of higher-order operators
We focus our attention on the following equation that is, to the equation (1.4) for n = 1.
Moreover, the spectral function of (3.1) reads and the corresponding covariance function has the form where W 2β is a gamma r.v. with parameters µ, 2β.
Proof. We can formally write so that from (3.1) we have that By observing that We notice that where B(W 2β ) is a Brownian motion with random time W 2β . Thus, we obtain that An alternative representation of the covariance function above reads We now pass to the general even-order fractional equation (1.4).
Theorem 3.2. The representation of a solution to the equation (1.4) can be written as Moreover, the spectral function of (3.7) is and the covariance function reads where W 2β is a gamma r.v. with parameters µ, 2β.
(3.11) We write (3.14) In conclusion, we have that and this confirms (3.7). From (3.7), in view of (2.20), we obtain By following the same arguments as in the previous proof, we get that The spectral density of X(t) is therefore Theorem 3.2 extends the results of Theorem 3.1 when even-order heat-type equations are involved.
We now pass to the study of the equation (1.9) for n = 1 and κ = ∓1, and where Ai(x) is the Airy function and W is a gamma-distributed r.v. with parameters 2β and µ.
Proof. By following the approach adopted above, after some calculation we can write that is the solution to is the solution to The third-order heat type equation has solution, for κ = −1, because of the asymptotic behaviour of the Airy function (see [1] and [9]). The solution to (1.9) with n = 1 (that is κ = −1) is therefore (3.21). The equation (3.25) has solution, for κ = +1, given by Thus, by following the same reasoning as before, we arrive at and we obtain that (3.23) solves (3.17) with κ = +1 is (3.23).
In light of (2.20) we get From the Fourier transform (3.27), we get that Also, we obtain that

Moreover, the covariance function
Cov X (h) = σ 2 µ 2β Eu 2n+1 (κh, W 2β ). has Fourier transform Proof. The proof follows the same lines as in the previous theorem.