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Spectral asymptotics for a class of integro-differential equations arising in the theory of fractional Gaussian processes

We study the spectral problems for integro-differential equations arising in the theory of Gaussian processes similar to the fractional Brownian motion. We generalize the method of Chigansky–Kleptsyna and obtain the two-term eigenvalues asymptotics for such equations. Application to the small ball probabilities in L2-norm is given.


Introduction
The spectral analysis of Gaussian processes is intensively studied in last two decades, in particular, in the context of the problem of small deviation asymptotics in Hilbert norm for such processes.
It is known, see [14], that to obtain the logarithmic L 2 -small ball asymptotics of a Gaussian process X, we need just one-term asymptotics of the counting function for the eigenvalues of its covariance operator. However, to manage the exact asymptotics (up to a constant), we need at least two-term asymptotics of the eigenvalues with the remainder estimate ( [11], see also [9]).
The last problem is quite delicate and was solved only for several special processes. Most of them are so-called Green Gaussian processes. This means that the covariance function G X is the Green function for the ordinary differential operator (ODO) subject to proper homogenuous boundary conditions. This class contains many classical processes, e.g., the integrated Brownian motion, the Slepian process and the Ornstein-Uhlenbeck process. The special nature of the Green Gaussian processes allows to use the well-developed techniques of spectral theory for ODOs, see, e.g., [13]. This approach was elaborated in [18], [15] and was used in a number of papers, see, e.g., [20] and references therein. We mention also the papers [16], [21], [24] where the twoterm spectral asymptotics was obtained for finite-dimensional perturbations of the Green Gaussian processes.
The case of fractional Gaussian processes is much more complicated. Until recently, only the main term of spectral asymptotics was known, and thus, only logarithmic small ball asymptotics was obtained for such processes. Namely, in the pioneer paper [4] the one-term spectral asymptotics was calculated for the fractional Brownian motion (FBM) W H , that is the zero mean-value Gaussian process with covariance function (here H ∈ (0, 1) is the so-called Hurst index, the case H = 1 2 corresponds to the standard Wiener process).
A general approach was developed in [19]. This approach is based on the powerful theorems on spectral asymptotics of integral operators [1], see also [2,Appendix 7], and covers many fractional processes. However, it also gives only the one-term eigenvalues asymptotics.
By the Laplace transform ψ(z) = 1 0 ψ(y) exp(−zy) dy the problem (1) was reduced to the Riemann-Hilbert problem which, in turn, was solved asymptotically using the ideas of [25], [23] (see some additional references in [6]). In this way the two-term asymptotics of the eigenvalues with the remainder estimate was obtained for the FBM in the full range of the Hurst index. On this base, the exact L 2 -small ball asymptotics for W H was established for the first time, along with some other applications. Notice also that the eigenfunctions asymptotics for FBM was obtained in [6] as well.
In later papers [7], [8] similar results were obtained for some other fractional Gaussian processes.
In this paper we suggest a slightly more general point of view and consider the eigenproblem (1) with general self-adjoint boundary conditions. This gives a unified approach to the previous results and covers several new fractional Gaussian processes.
The paper is organized as follows. In Section 2 we calculate the two-term spectral asymptotics of the problem (1) with arbitrary self-adjoint boundary conditions that do not contain the spectral parameter. Here we mainly follow the line of [6]. It turns out that there are three possible "shifts" of the second term in the asymptotics depending on the sum of orders of boundary conditions. It is well known, see [18,Theorem 7.1] and [15,Theorem 1.1], that this parameter drives the second term of spectral asymptotics for ODOs of arbitrary order. We conjecture that this is the case also for general eigenproblems of the type (1) with ODO of arbitrary order in the right-hand side.
In Section 3 we consider a more general eigenproblem with self-adjoint boundary conditions. We prove that the additional term in (2) can be considered as a weak perturbation of the problem (1) which does not influence upon the two-term eigenvalues asymptotics. In Section 4 we give several examples of fractional Gaussian processes covered by the results of Sections 2 and 3.
Finally, in Section 5 we collect the results on L 2 -small ball probabilities for the fractional processes considered in Section 4.

Analysis of the problem (1) with general boundary conditions
First we consider in detail the case α < 1. In this case the problem (1) reads as follows: with the same boundary conditions.

Transformation of the problem
Then (3) becomes The Laplace transform gives On the other hand, So, we obtain Now we look to the relaton on the real line. The equation (4) shows that the right-hand side is continuous on R, and we obtain for t > 0 and t < 0 respectively or, equivalently, with regard of (7), we can rewrite (9) as follows: We also know from definition that Φ(z) and Ψ(z) behave as polynomials of order not greater then one at infinity while they are O(z α−1 ) at the origin.
We introduce the function X 0 (z) with the cut at positive semiaxis such that The first relation in (11) is satisfied by the Sokhotski-Plemelj formula It is easy to see that Using (11) we rewrite (10) as follows: where Φ 0 (t) = Φ(νt) and Ψ 0 (t) = Ψ(νt). Therefore, functions (here we used (12) and integration by parts). Since S(z) and D(z) behave as polynomials of order not greater then one at infinity, the Sokhotski-Plemelj formula yields Substituting z = −t, t > 0, we obtain the integral equations where S(t) = S(−t), D(t) = D(−t), and A is the integral operator with the kernel A(t, s) = exp(−νs)h 0 (s) , s, t ∈ R + . By [6,Lemma 5.6] the operator A is contracting in L 2 (R + ) for ν large enough, and maps arbitrary polynomial into L 2 (R + ). Therefore, these equations are uniquely solvable. Moreover, the relation h 0 (0) = sin(θ 0 (0+)) = sin( π(1−α) 2 ) shows that (see [10, 3.241 and therefore, (13) Denote by p 0 ± (t) and p 1 ± (t) the (unique) solutions of the equations on R + and extend them analytically to C \ R − . Then evidently Since ϕ is an entire function, the following relation should be fulfilled: where z 0 is introduced in (6). So, every eigenvalue of the original problem generates a root of (16) by relation (6). It is easy to show that, vice versa, every root of (16) (except ν = 0, if arises) generates an eigenvalue by relation (6). We multiply (16) by exp(iν/2) and obtain By [6, Lemma 5.5] we have and [6, Lemma 5.7] claims Thus, (17) is equivalent to (here and elsewhere ρ = π(1−α)/4 and we use the notation [a] = a+O(ν −1 ), see [13, §4]). By (5) and (13), all coefficients C j are real, therefore, (17) is equivalent to the real system Now we compare the behavior of Φ(νz) and Ψ(νz) at infinity provided by (15), with (5). By [6, Lemma 5.7] we have Using (5) and (13) we obtain We solve these equations and substitute to (18). This gives (here A = cos ν+ρ 2 and B = sin ν+ρ 2 ). The equations (19) complemented by the boundary conditions of the original problem generate a (4 × 4) homogeneous system. Standard argument based on the Rouché theorem shows that the roots of its determinant are approximations of the solutions of (16) for large |ν|.

Separated boundary conditions
Separated boundary conditions (or Sturm type conditions) for the second order operator can be written as follows: 1 (one of two coefficients in every condition may vanish). We denote by κ the sum of orders of the derivatives in boundary conditions (20). It was mentioned in the Introduction that this quantity plays an important role in the spectral asymptotics of ordinary differential operators, see, e.g., [18]. In our case, evidently, κ ∈ {0, 1, 2}.
This system has nontrivial solutions iff Thus, if we enumerate the roots of (16) in increasing order of absolute values then for some k.
By verbatim repetition of [6, Section 5.1.7] we show that k is independent on α. Therefore, it can be calculated by comparing with the case α = 1 where the original problem becomes standard Sturm-Liouville problem. Thus, we obtain k = 0.
3. Let κ = 2. In this case (20) This system has nontrivial solutions iff and we conclude that in this subcase for some k. Comparing with the case α = 1 we obtain k = 0.
Now we can formulate the final result.
Theorem 2.1 The eigenvalues of the problem (3) supplemented by separated boundary conditions have the following asymptotics as n → ∞: where κ stands for the sum of orders of the derivatives in conditions (20).
This statement easily follows from relation (6) and the obtained asymptotics of ν n .

Almost separated boundary conditions
For the second order operator, almost separated (or separated in the principal order) boundary conditions can be written as follows: Analysis of this case repeats mostly the subcase κ = 2, and the eigenvalues asymptotics coincides with (24) for κ = 2.

Non-separated boundary conditions
For the second order operator, non-separated boundary conditions can be written as follows: In this case (19) is reduced to This system has nontrivial solutions iff Therefore, in this case the sequence ν n can be split into two subsequences ν ′ n , ν ′′ n such that, as n → ∞, for some k ′ , k ′′ . Comparing with the case α = 1 we obtain k ′ − k ′′ = 0, so without loss of generality we can put k ′ = k ′′ = 0. Now we can formulate the final result.

Theorem 2.4
The eigenvalues of the problem (3) supplemented by nonseparated boundary conditions have the following asymptotics as n → ∞: This statement easily follows from relations (6) and (26).

Remark 2.5
In the case βγ = 0 the boundary conditions (25) are in fact separated, with κ = 1. So, two subsequences (26) can be merged, and the result coincides with (22). In general case two subsequences of eigenvalues have the opposite shifts with respect to (22), cf. [15, Theorem 1.1].
In the cases β = ±γ one of the subsequences (26) has the second term as in (21) while another one has the second term as in (23). Notice that in contrast to the case α = 1 two subsequences in (26) cannot be asymptotically close or coincide.

The case α > 1
Repeating the argument of [6, Sec. 5.2], we arrive at the relation (4) with Following the same line as in previous subsections we again obtain formulae (24) and (27).

A more general problem
As it was explained in the Introduction, we wish to consider the problem (2) as a perturbation of the problem (1). For simplicity only, we assume that the operator −ψ ′′ with given boundary conditions is positive definite, otherwise the argument should be changed in a standard way.
We begin with the estimate for the eigenfunctions of the problem (1) with arbitrary self-adjoint boundary conditions that do not contain the the spectral parameter.
By construction, ψ ′ (z) is an entire function, and we can restore it by integrating over the imaginary axis. Using the relation (4) we obtain where The integral in (28) does not depend on the constant terms in F 1 and F 2 . It was calculated in the proof of [6, Lemma 5.8]. Up to a multiplicative constant, we obtain where ν is related to the eigenvalue λ by (6), θ 0 is introduced in (8), Φ 0 and Ψ 0 are defined after (14), and Taking into account the behavior of ingredients in (29) at zero and infinity we derive 2 ψ ′ (x) = A cos(νx + φ(ν, α)) + F (ν, α, x) , where It is easy to see that Now we introduce the energy space H of the problem (1). For smooth functions h 1 and h 2 satisfying boundary conditions, we set and define the Hilbert space H as the completion of the set of such functions with respect to the norm generated by the scalar product (·, ·) H . It is well known that, depending on the boundary conditions, H coincides either with standard Sobolev space W 1 2 (0, 1) or with its subspace of codimension 1 or 2. Corresponding norm is given by where q(h, h) is a quadratic form of the variables h(0) and h(1).
In a standard way, we rewrite the problems (1) and (2) as the equations in H Kψ = λψ; where K and B are compact self-adjoint operators in H defined by relations We are in the position to formulate an important abstract statement. Denote by λ n the eigenvalues of K enumerated in the decreasing order taking into account the multiplicities, and by ψ n corresponding normalized eigenfunctions. Finally, suppose that where a, c, ε, r > 0, b ∈ R. Then the eigenvalues λ n of generalized eigenproblem Kψ n = λ n ψ n + Bψ n have the same two-term asymptotics as n → ∞: First, let the boundary conditions be separated or almost separated. As we have proved in Section 2, see (24), the eigenvalues of the first equation in (33) satisfy the relation (34) with ε = 1 and r = 3 − α.
To obtain the estimate (35) we need to normalize ψ n . Since all eigenfunctions except the first one change the sign, the relation (30) and the estimate (31) imply uniformly in x ∈ [0, 1]. This implies in view of (32) so for the normalized eigenfunctions we have A = ± √ 2 + O(n −1 ). Finally, taking into account (36) we obtain for any p ∈ L 1 (0, 1). 3 Thus, the estimate (35) fulfils with ε = 1, and Proposition 3.1 ensures the two-term eigenvalues estimate (24) for the problem (2).
For the non-separated boundary conditions the eigenvalues of the operator K are organized in two sequences, see (27). However, the estimate (35) also holds, and Remark 2 in [17] ensures that the asymptotics (27) persists for the problem (2). Now we can formulate the final result of this section.
Theorem 3.2 Let p ∈ L 1 (0, 1). Then the two-term eigenvalues asymptotics of the problem (2) with self-adjoint boundary conditions does not depend on p and is given in (24) or (27), depending on boundary conditions.

Gaussian processes related to the problems (1) and (2) with various boundary conditions
We recall that G stands for the covariance function G W H .

Its covariance function reads
and corresponding operator can be considered as a critical one-dimensional perturbation of the covariance operator of W H , see [16]. In [7] this approach was applied to obtain the two-term spectral asymptotics for B H . Moreover, it was mentioned that the direct method developed in [6] for W H does not produce results quite as explicit as those in [6]. However, we show that it is not the case, and the direct method works as well.
Notice that G B H (0, y) = G B H (1, y) ≡ 0, and therefore any eigenfunction of 1 0 G B H (x, y)ϕ(y) dy = λϕ(x) 3 For some boundary conditions the assumption on p can be weakened.
Then evidently ψ ′ (0) = ψ ′ (1) = 0, and The last term vanishes by the choice of c: and we obtain Differentiation gives (1) with α = 2 − 2H. Since boundary conditions are separated, we obtain the spectral asymptotics (24) with κ = 2. By Remark 2.2, we should exclude zero root of (16). This changes n → n + 1 in the right-hand side of (24) and yields that coincides with the result of [7, Theorem 2.2] and even gives a slightly better estimate of the remainder term.

Centered FBM. This process is defined as
and its covariance function reads Notice that Differentiation gives (1) with α = 2 − 2H. Since boundary conditions are separated, we obtain the spectral asymptotics (24) with κ = 0. This yields 3. Centered Brownian bridge. This process is defined similarly: and its covariance function reads As in the previous example, We define ψ by formula (37). Then evidently ψ(0) = ψ(1), ψ ′ (0) = ψ ′ (1), and The second term vanishes by (38), and differentiation gives (1) with α = 2 − 2H. Since boundary conditions are periodic, we obtain the spectral asymptotics (27) with β = −γ. By Remark 2.5, we should exclude zero root of (16). This changes n → n + 1 in the right-hand side of (27) and yields 4. Fractional Slepian process. The conventional Slepian process S on [0, 1] can be defined in several ways: 1) S is a zero mean-value stationary Gaussian process with correlation where W 1 and W 2 are independent Wiener processes.
Fractional Slepian processes defined by analogy in these three ways are different, see, e.g. [12] for the first one. We define the process S H as the mixture of two independent FBMs: Its covariance function reads We emphasize that, as in the case H = 1/2, the following relation holds: Notice also that G S H (0, y) + G S H (1, y) ≡ 1. By symmetry of the kernel, any eigenfunction of In the first case we define Evidently ψ(0) + ψ(1) = 0, and that is equivalent to In the second case we define Then evidently ψ(0) = ψ(1) = 0 and ψ ′ (0) + ψ ′ (1) = 0, so the boundary conditions (41) are satisfied as well. Further, The double substitution is equal to ψ(1) and vanishes after differentiation. So, we obtain with α = 2 − 2H and boundary conditions (41). The boundary conditions in this problem are non-separated but contain the spectral parameter λ. So, formula (27) is not applicable. However, the basic scheme runs without essential changes. We change λ → λ/2 in (6) and arrive at (19). Using (41) we rewrite (19) as follows: This system has nontrivial solutions iff and we conclude for some k. Comparing this result with the eigenvalues of the conventional Slepian process, see [22], we obtain k = 1, and

5.
The spectral analysis of some other mixtures of fractional processes can be reduced to the problems of the same type, cf. [20,Section 2] for the mixtures of the Green Gaussian processes. We consider here only one modification of the previous example.
Let (1)). This process is a one-dimensional perturbation of the fractional Slepian process, and its covariance function reads 2 this perturbation is non-critical, see [16], and the two-term spectral asymptotics does not change. The case γ = 1 2 is critical and should be studied separately, cf. [ To manage the equation for eigenfunctions in the case γ = 1 we notice that ϕ(0) + ϕ(1) = 0 and make the change of function (40), (42). This gives (43) with α = 2 − 2H and anti-periodic boundary conditions So we obtain the spectral asymptotics (27) with β = γ. This yields 6. The fractional Ornstein-Uhlenbeck process beginning at zero. In the fractional setting, the Ornstein-Uhlenbeck process can be defined in a number of nonequivalent ways, see, e.g., [5]. Following [7], we consider the solution of the Langevin equation driven by the FBM: where β ∈ R is the drift parameter and ξ ∼ N (0, σ 2 ) is the initial condition independent of W H . The covariance function is given by the formula We begin with σ = 0, i.e. ξ = 0. This case was considered in [8,Sec. 6]. By separate fine analysis the following expression for eigenvalues was derived: where the sequence ν n satisfies (22) with k = 0 and α = 2 − 2H. We claim that this result is covered by our Theorem 3.2. Indeed, the change of function (cf. the proof of Lemma 6.1 in [8]) reduces the equation to the problem with boundary conditions (ψ ′ − βψ)(0) = ψ(1) = 0. Theorem 3.2 shows that which coincides with (45) taking into account that ν 2 n + β 2 = ν 2 n (1 + O(n −2 )). Thus, the claim follows.
The estimate (35) with ε = 1 persists, and Proposition 3.1 shows that the term β 2 u in (47) does not influence upon the two-term eigenvalues asymptotics.

Application to the small ball probabilities
As it was mentioned in the Introduction, the results of the previous section give rise to the exact (up to a constant) L 2 -small ball asymptotics of all considered Gaussian process. We define two important quantities: D(H) := H (2H + 1) sin π 2H+1 sin(πH)Γ(2H + 1) (2H + 1) sin π .
We substitute the two-term spectral asymptotics into the general result of [18,Theorem 6.2]. For the examples 3 and 5 we use in addition the Lifshits lemma, see, e.g., [15,Lemma 0.1]. This gives us the following statement.
Theorem 5.1 The exact small ball asymptotics for the fractional processes considered in Section 4 read as follows: where the values of B X and D X are collected in the 2. In contrast, L 2 -small ball asymptotics for W H and U H β in the case σ = 0 coincide up to a constant for all H ∈ (0, 1).