A Gaussian random field on a bounded domain is characterized by its mean and its covariance. Depending on the research community, the covariance is described by a covariance kernel or a covariance operator . More specifically, given a symmetric continuous covariance kernel , the corresponding covariance operator is positive semidefinite and self-adjoint on the Hilbert space and given by
for , . Our main goal in this paper is to, given the regularity of the kernel , derive regularity estimates for in terms of certain smoothness spaces related to elliptic operators with boundary conditions on .
Our motivation to analyze the coupling of these two formulations in detail comes from the theory and approximation of solutions to stochastic partial differential equations (SPDEs). While the analysis of these equations and their numerical approximations is mainly done in Hilbert spaces, e.g., certain fractional order spaces related to the differential operator in the equation, with a -Wiener process as driving noise, algorithms that generate this driving noise in practice are often based on the covariance kernel . The class of Matérn kernels is a popular example in spatial statistics. Surprisingly, to the best of our knowledge, such results are not available in the literature.
To be able to put our abstract results and their consequences in a more specific context, let us consider a linear stochastic reaction-diffusion equation with additive noise
on a convex bounded domain , with boundary . Here the functions fulfill an ellipticity condition, is some smooth initial function and homogeneous boundary conditions of either Dirichlet or Neumann type are considered. The stochastic noise term is Gaussian, white in time and correlated by a symmetric continuous covariance kernel in space. This can be seen as a simplified version of equations considered for the modeling of sea surface temperature and other geophysical spatio-temporal processes on some spatial domain [29, Chapter 6]. This equation is considered in the context of  as a stochastic differential equation of Itô type on the Hilbert space of square integrable functions on . The stochastic partial differential equation (1) is then written in the form
for . The unbounded linear operator on is densely defined, self-adjoint and positive definite with a compact inverse, see Section 2.2 for precise assumptions. The stochastic term is an -valued
-Wiener process on a complete filtered probability space. Here is a positive semidefinite self-adjoint operator on with kernel . If is pointwise defined and jointly measurable with respect to the product -algebra (with denoting the Borel -algebra on ) then is the covariance function of the random field . In general, there is no analytic solution to (2) so numerical approximations have to be computed. It is then vital to understand how various regularity properties of influence the behavior of and its approximation, since this can determine the convergence rate of the numerical approximations. We discuss this in concrete terms in Section 5.
The research field on SPDEs of the form (2) has been very active in the 21st century. There is a substantial body of literature, both from theoretical [8, 29] as well as numerical [15, 23, 28] perspectives. For SPDEs on domains without boundary (i.e., when is replaced by Euclidean space, a torus or a sphere) the question of how regularity properties of influence is well understood, especially in the homogeneous case [10, 16, 26, 32]. This refers to the case that only depends on the difference between two points and in , examples including the class of Matérn kernels, see Remark 4.3are explicitly known. In particular it is common to consider the case that and commute, see, e.g., [8, Section 5.5.1]. One of the few instances in which an author instead considers the properties of as a function on when deriving connections between properties of and properties of can be found in . The main result of this paper is [2, Theorem 4.2], which states that has to satisfy the boundary conditions of in a certain sense in order for and to commute. In practice, this excludes the physically relevant case of homogeneous noise from approaches such as that of [8, Section 5.5.1], see [2, Corollary 4.9]. Our approach to the problem consists instead of deriving sufficient conditions on for which the associated symmetric operator fulfills estimates of the form
for fractional powers of and suitable constants . By and we denote the spaces of trace-class and Hilbert–Schmidt operators, respectively. We consider both homogeneous and non-homogeneous kernels .
-valued random variabletakes values in the subspace of the fractional Sobolev space at all times . This is a commonly encountered assumption in the literature, particularly when analyzing numerical approximation schemes for SPDEs, see Section 5. The condition also has implications for the qualitative behavior of . It guarantees that takes values in [8, Proposition 6.18] with sample paths continuous in for arbitrary [8, Theorem 5.15]. In particular, if (3) holds with , is a strong solution (in the PDE sense) to (2) as opposed to just a weak solution, cf. [8, Theorem 5.40]. This means that the process takes values in and has the intuitive representation
for . Moreover, if (3) holds with , a classical Sobolev embedding theorem (see, e.g., [30, Section 8]) ensures that takes values in the space of Hölder continuous functions on with exponent -a.s.. The evaluation functional is continuous on this Banach space. It follows that for each , is a smooth random field on as opposed just an abstract random variable in a Hilbert space, c.f. [14, Sections 7.4-7.5]. In summary, there could be many reasons why one would like to know for which the estimate (3) is satisfied for a given kernel . In the case of non-homogeneous kernels, we consider Hölder conditions on the kernel when deriving the estimate (3). This is a natural choice given our proof technique, which is based on the fact that , the image of the square root of , coincides with the reproducing kernel Hilbert space of functions on associated with the kernel . For the same reason, in the case of homogeneous kernels
, we consider a decay condition on the Fourier transform of, which is used in practice when generating the driving noise with fast Fourier transforms (cf. ). While this interpretation of as a reproducing kernel Hilbert space is well-known, we are not aware that it has been used to find estimates of the form (3) before.
The estimate (4) does not, as far as we know, have an immediate interpretation in terms of regularity properties of or . It is, however, important for analyzing weak errors of approximations to certain SPDEs, such as the stochastic wave equation. It has also been used in the recent work  to derive higher convergence rates for approximations of the covariance operator of SPDE solutions. There is an immediate connection between the condition on in (4) and regularity of : the condition is true with if and only if
is an element of the Hilbert tensor product space, see [6, 36]. Instead of exploiting this connection, we consider Hölder or Fourier transform conditions on also in this case. The reason for this is partly that these conditions are easier to check in applications compared to the rather abstract tensor product condition. We also want to ensure easy comparisons between the estimates (3) and (4) under the same conditions on .
The outline of the paper is as follows. The next section contains an introduction to the necessary mathematical background along with our assumptions on and . This includes short introductions to fractional powers of elliptic operators on bounded domains and reproducing kernel Hilbert spaces along with the proofs of some preliminary lemmas. In Section 3 we derive the estimates (3) and (4) under Hölder conditions on a non-homogeneous kernel , while in Section 4 we do the same under a decay condition on the Fourier transform of a homogeneous kernel . Section 5 concludes the paper with a discussion of the implication of our results for the numerical analysis of SPDEs on domains with boundary. The applications we discuss are not limited to stochastic reaction-diffusion equations but include several other SPDEs involving elliptic operators on bounded domains, such as stochastic Volterra equations or stochastic wave equations.
Throughout the paper, we adopt the notion of generic constants, i.e., the symbol is used to denote a positive and finite number which may vary from occurrence to occurrence and is independent of any parameter of interest. We use the expression to denote the existence of a generic constant such that .
In this section, we introduce our notation and reiterate some important results that we use in Sections 3-4. The material mainly comes from [1, Chapter 1], [12, Section 1.3-1.4], [23, Appendix B] and [41, Chapters 1-2]. We give explicit references for vital or nonstandard results.
2.1. Trace class and Hilbert–Schmidt operators
Let and be real separable Hilbert spaces. By we denote the space of linear and bounded operators from to and by and the subspaces of trace class and Hilbert–Schmidt operators, respectively. There are several equivalent definitions of these spaces in the literature. Here, we simply note that they are separable Banach spaces of compact operators with norm characterized by
is the sequence of singular values of, i.e., the eigenvalues of . Note that if , , , then and
Similarly, if , , then and
If and , then and
The space is a separable Hilbert space with inner product
for and an arbitrary orthonormal basis of . For a positive semidefinite symmetric operator , .
2.2. Fractional powers of elliptic operators on bounded domains
Consider a Gelfand triple of real separable Hilbert spaces with dense and continuous embeddings. We assume that forms an adjoint pair with duality product such that for all , . Given and , can be constructed by identifying with its continuous dual space and letting be the completion of under the norm of . Next, let be a continuous, symmetric and coercive bilinear form, i.e., it is linear in both arguments, and there are constants such that and for all . Then, there exists a unique isomorphism such that for all . Viewing as an operator on , it is densely defined and closed. We write for its restriction to , which is then a densely defined, closed, self-adjoint and positive definite operator with domain . It has a self-adjoint inverse , which we assume to be compact.
Applying the spectral theorem to , we obtain the existence of a sequence of positive nondecreasing eigenvalues of , along with an accompanying orthonormal basis of eigenfunctions in . For , we define fractional powers of by
We write . Note that . For , we let be defined by (9), and we write for the completion of under the norm . Equivalently, we can write
Then for all . Regardless of the sign of , is a Hilbert space with inner product . For , is isometrically isomorphic to [23, Theorem B.8]. In particular, for ,
For , we have where the embedding is dense and continuous. Since and are orthonormal bases of and , respectively, we may represent the embedding by
It follows that is compact. One can show that for all and ,
in the sense of complex interpolation, see, e.g.,. Lemma 2.1 in  allows us to, for all , extend to an operator in , and we will do so without changing notation.
The fact that is a set of interpolation spaces allows us to prove the following lemma for a symmetric operator .
Let be symmetric. Then the following three claims are equivalent for and :
for all .
Assume first that . If , then by symmetry of ,
so that can be continuously extended to . Conversely, if extends to , the operator norm is finite and therefore
so that . Let us now write for the adjoint of with respect to , i.e., the operator in defined by
for . Let us similarly write for the adjoint of with respect to . We have for , that
so that by density of in , . This implies
for , hence . Since the eigenvalues of are the same as those of [14, Section 4.3], the singular values of and are the same, so that (5) implies the equivalence of 1 and 2 for . For the proof is the same, except that we take in (10).
In the spirit of the previous result, let us also note that the equality in (3) is true when is also assumed to be positive semidefinite (so that is well-defined). Indeed, the right hand side guarantees that extends to so for
Therefore also extends to . The operator is symmetric and positive semidefinite on , so (since on an eigenfunction of ) we have
Before we put the abstract framework above into a concrete setting, we note some properties of fractional Sobolev spaces. We denote by and the classical Sobolev spaces of order , on a bounded domain with Lipschitz boundary and , , respectively. The norm of is given by
where is the weak derivative with respect to a multiindex and the norm of is defined in the same way. When there is no risk of confusion we write for and we set . For , , , we use the same notation for the fractional Sobolev space as for the classical Sobolev space. The space is equipped with the Sobolev–Slobodeckij norm
for . Since is Lipschitz, this can equivalently [12, page 25] be characterized by
Here the norm of the fractional Sobolev space is given by
where is the Fourier transform of .
The embedding of into is dense and, since is Lipschitz, compact for all [12, Theorem 18.104.22.168]. This last fact gives us yet another way to characterize and extend the definition to negative . Since, for , is densely and continuously embedded in , there is a positive definite self-adjoint operator such that and [27, Section 1.2]. The compactness of the embedding implies [37, Sections 4.5.2-4.5.3] that is compact and the spaces , , can now be constructed by the spectral decomposition of the operator. The space is, like , defined as the completion of with respect to the norm . By [41, Theorem 1.35] we obtain, for , . We define for by , and note that this definition is independent of due to the fact that . By this characterization of the norm of , , we can repeat the proof of Lemma 2.1 to obtain the following analogous result.
Let be symmetric. Then the following three claims are equivalent for and :
for all .
The same characterization also allows us to deduce the following lemma, which tells us when the embedding , , is Hilbert–Schmidt and of trace class.
For , the embedding fulfills:
We pick some such that and write for the orthonormal basis of eigenfunctions of the operator that is associated with the space . Then
which gives , being considered as an operator on . Therefore, [36, Lemma 3] implies that the singular values of coincide with the eigenvalues of . By the same argument, the singular values of coincide with the eigenvalues of the same operator. It therefore suffices to show the claim for .
The result is proven for in , specifically as a consequence of [36, Lemma 2, Satz 2]. For the general case, we first note that when is Lipschitz, there is an operator , such that [12, Theorem 22.214.171.124]. For an arbitrary bounded domain such that , we have
By definition of the norm (12), we note that is bounded, and clearly the restriction is, too. Therefore, sufficiency of and , respectively, follows as a consequence of (6) and (7). Necessity follows by an analogous argument: if then for and any domain with . ∎
We now let the spaces , , obtain a concrete meaning by taking to be a bilinear form on given by
where denotes weak differentiation with respect to , . The coefficients , , are functions fulfilling . Moreover, we assume that there is a constant such that for all and almost all , . The function is non-negative almost everywhere on .
We shall consider two cases of boundary conditions for . In the first case (Dirichlet boundary conditions) we take . Here denotes the trace operator, an extension of the mapping . It is well-defined on for [41, Theorem 1.3.9]. In the second case (Neumann boundary conditions) we take , and assume in addition that there is a constant such that almost everywhere on . Then is a symmetric, bilinear form on . The operator is regarded as a realization of the strongly elliptic operator
on , with boundary conditions on in the first case and in the second case. Here
with being the outward unit normal to , is well-defined as an element in for , . Since the embedding of into is compact and maps into , the assumption that this operator is compact is true.
Next, we relate the spaces to , and for this we need to assume that is convex. In the case of Dirichlet boundary conditions, we have