Iain Henderson
email: henderso@insa-toulouse.fr

Sobolev regularity of Gaussian random fields

In this article, we fully characterize the measurable Gaussian processes pU pxqqxPD whose sample paths lie in the Sobolev space of integer order W m,p pDq, m P N0, 1 ă p ă `8, where D is an arbitrary open set of R d . The result is phrased in terms of a form of Sobolev regularity of the covariance function on the diagonal. This is then linked to the existence of suitable Mercer or otherwise nuclear decompositions of the integral operators associated to the covariance function and its cross-derivatives. In the Hilbert case p " 2, additional links are made w.r.t. the Mercer decompositions of the said integral operators, their trace and the imbedding of the RKHS in W m,2 pDq. We provide simple examples and partially recover recent results pertaining to the Sobolev regularity of Gaussian processes.

Introduction

Sobolev spaces W m,p pDq are central tools in modern mathematics, most notably in the study of partial differential equations (PDEs). These spaces are built upon the notion of weak derivative: v is the weak derivative of u in the direction x i if for all smooth compactly supported function φ P C 8 c pDq,

ż D upxq Bφ Bx i pxqdx " ´żD vpxqφpxqdx. (1.1)
Weak derivatives generalize classical, pointwise defined derivatives. In particular, there are cases where weak derivatives are well defined and pointwise differentiation otherwise fails (see e.g. [START_REF] Evans | Partial Differential Equations[END_REF], Examples 3 and 4 p. 260). The popularity of Sobolev spaces is justified by a number of reasons: first, they are separable reflexive Banach spaces when 1 ă p ă `8, and separable Hilbert spaces when p " 2 ( [1], Theorem 3.6 p. 61). Through duality, this allows for geometrical interpretations of PDEs which in turn lead to numerous quantitative theoretical results in the study of PDEs [START_REF] Evans | Partial Differential Equations[END_REF]. Second, as the Sobolev norm is defined through integrals of powers of the function and its weak derivatives, it is easily interpreted as an energy functional of the said function, which complies with physical interpretations of PDEs. This is a desirable feature as PDEs are generally used for describing physical phenomena. Finally, Sobolev spaces are useful for practical purposes as they are the natural mathematical framework for the celebrated finite element method when seeking numerical solutions to PDEs ( [START_REF] Brenner | The mathematical theory of finite element methods[END_REF], Chapter 1). When a function of interest u : D Ñ R is unknown, it may be relevant to model it as a sample path of a random field pU pxqq xPD , say a Gaussian process, whose realizations lie in a suitable function space. This is e.g. frequent in Bayesian inference of functions [START_REF] Van Der | Information rates of nonparametric Gaussian process methods[END_REF]. Such suitable spaces can indeed happen to be Sobolev spaces, for example when u describes a physical quantity. The question at hand in this article is thus the following: when do the sample paths of a given Gaussian process lie in some Sobolev space? This question is closely linked to the recent attention that Gaussian processes have drawn for tackling machine learning problems arising from PDE models; see e.g. [START_REF] Lange-Hegermann | Algorithmic linearly constrained Gaussian processes[END_REF][START_REF] Owhadi | Bayesian numerical homogenization[END_REF][START_REF] Raissi | Machine learning of linear differential equations using Gaussian processes[END_REF][START_REF] Wahlstrom | Modeling magnetic fields using Gaussian processes[END_REF]. Notably (see [START_REF] Chen | Solving and learning nonlinear PDEs with Gaussian processes[END_REF]), Gaussian processes seem to provide a numerically competitive and mathematically tractable alternative to the now widespread "physics informed neural networks" (PINNs, [START_REF] Raissi | Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[END_REF]). For the moment though, the machine learning techniques involving Gaussian processes have only been studied within the framework of spaces of functions with classical smoothness : C 0 , C 1 , etc. As argued before, these spaces are often not as well-suited for studying PDEs as Sobolev spaces.

Relevant literature Though weak differentiability is more general, it is less direct to check than classical differentiability. Weak derivatives are defined implicitly and in the most general case, ensuring Sobolev regularity is not usually done by directly verifying that an integral or a series is finite, as would be the case in L p spaces; variational or boundedness criteria are used instead (see Lemma 2.1).

In many important cases however, handy characterizations of such regularity do exist, which have effectively been used to bypass the implicit definition of Sobolev regularity and provide results on the sample path regularity of Gaussian processes. When D " R d , the space W m,p pR d q can be characterized in terms of a sufficient decay of of the Fourier transform ( [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF], Theorem 3 p. 135; [START_REF] Evans | Partial Differential Equations[END_REF], Section 5.8.5; [START_REF] Adams | Pure and Applied Mathematics[END_REF], Section 7.63). Still in the case D " R d , Sobolev regularity is equivalent to the convergence of its de la Vallée Poussin expansion in a suitable space ( [START_REF] Nikol'skii | Approximation of functions of several variables and imbedding theorems[END_REF], Section 8.9). This fact has been the first to be employed for characterizing the Sobolev regularity of stationary Gaussian processes indexed by the unit cube of R d in [START_REF] Cirel'son | Norms of Gaussian sample functions[END_REF][START_REF] Ibragimov | Conditions for Gaussian homogeneous fields to belong to classes H r p[END_REF], in terms of the spectral measure of its covariance. For some Banach spaces, explicit Schauder bases are known and lying in such spaces can be translated as the convergence of some coordinate series. This has been exploited in [START_REF] Ciesielski | Quelques espaces fonctionnels associés à des processus gaussiens[END_REF] for studying the Besov and Besov-Orlicz regularity of one dimensional Gaussian processes (they are natural generalizations of Sobolev regularity, [START_REF] Adams | Pure and Applied Mathematics[END_REF]), and the fractional Brownian motion in particular. Note that in this one dimensional framework, those spaces only contain continuous functions ( [START_REF] Ciesielski | Quelques espaces fonctionnels associés à des processus gaussiens[END_REF], Lemme III.3); a fact which, as we will see, can be quite restrictive. Wavelet analysis is also available for describing Sobolev regularity ( [START_REF] Adams | Pure and Applied Mathematics[END_REF], Section 7.70) and has been used for studying the smoothness of the Brownian motion [START_REF] Ciesielski | Modulus of smoothness of the Brownian paths in the L p norm. Constructive theory of functions[END_REF][START_REF] Roynette | Mouvement Brownien et espaces de Besov[END_REF]. More complex notions such as the existence of an underlying Dirichlet structure have been put to use in [START_REF] Kerkyacharian | Regularity of Gaussian processes on Dirichlet spaces[END_REF]. The latter work deals with Besov B s 8,8 regularity, s ą 0, on compact metric spaces, and relies on a convergence analysis of suitable spectral coefficients, based on the so called Littlewood-Paley decomposition. Note that when 0 ă s ă 1 and D is sufficiently regular, B s 8,8 pDq exactly corresponds to the space of Hölder continuous functions C s pDq (e.g. [START_REF] Edmunds | Function spaces, entropy numbers, differential operators[END_REF], p. 26). In [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF], Karhunen-Loève expansions are used to study whether or not the sample paths of a general second order random process lie in interpolation spaces between the reproducing kernel Hilbert space (RKHS, Section 4.1 below) of the process and L 2 pνq, where ν is a σ-finite measure. This is then applied to study the H s pDq :"W s,2 pDq regularity of the corresponding sample paths when s ą d{2 (Corollaries 4.5 and 5.7 in [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF]), with applications to Gaussian processes in particular (H s is a fractional Sobolev space). Note that RKHS are also popular function spaces in the machine learning community [START_REF] Berlinet | Reproducing Kernel Hilbert Spaces in Probability and Statistics[END_REF]. Using the notion of mean square derivatives, [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF] shows that the sample paths of a general second order random field lie in H m pDq under an integrability condition of the symmetric cross derivatives of the kernel over the diagonal ( [42], Theorem 1). For the suitable definition and use of the mean square derivatives of the process, [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF] additionally requires that the covariance function be continuous over the diagonal as well as its symmetric cross derivatives.

A first study of the H m regularity of Gaussian processes To make things more explicit, let us apply some of the results described above on two examples, namely centered Gaussian processes pU pxqq xPD whose covariance function (or "kernel") is either Matérn ( [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF], pp. 84-85) or finite rank. This will allow us to identify situations in which those previous results may be extended.

Gaussian processes with Matérn kernels are widely used in machine learning for approximating finitely smooth functions, therefore it is quite natural to study this particular case. In the case where the domain D is a bounded open set whose boundary verifies the strong local Lipschitz condition, it is in fact known that the RKHS of Matérn kernels of real order ν ą 0 are exactly H ν`d{2 pDq. In this case, Corollary 4.5 from [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF] is optimal (see [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF], Example 4.8 and Theorem 4.4): the sample paths of the associated Gaussian process lie in the Besov space B s 2,2 pDq for all s ă ν and not in B s 2,2 pDq for all s ě ν. For such domains, it is also true that B s 2,2 pDq " H s pDq. As stated in [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF], p. 370, this result suggests that the sample paths are about d{2 less smooth than the functions in the RKHS. A limitation of the result described above is the regularity assumption on D, which is necessary for [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF] as its results rely on the existence of suitable extension operators to assert that B s 2,2 pDq " H s pDq (see [START_REF] Adams | Pure and Applied Mathematics[END_REF], p. 230). Without such regularity assumptions, this equality does not hold anymore (although generalizations to less regular open sets exist, [START_REF] Devore | Besov spaces on domains in R d[END_REF], Theorem 6.7). Hence the seemingly simple case of Sobolev spaces of integer order defined on arbitrary open sets is left undealt with.

In this regard, it is instructive to investigate the consequences of [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF], Theorem 1, which does not make any regularity assumptions over the open set D. Its statement is as follow, given a centered random field pU pxqq xPD with continuous covariance function k (we refer to Section 2.1.2 for notations). If for all |α| ď m, the weak derivative B α,α k exists, is continuous on the diagonal of D ˆD and ş D B α,α kpx, xqdx ă `8, then the sample paths of U lie in H m pDq almost surely. Matérn kernels are stationary, meaning that they are of the form kpx, x 1 q " k S px´x 1 q for some real valued function k S . For such kernels, the criterion from Theorem 1 in [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF] essentially reduces to the condition that the pointwise derivatives pB 2α k S qp0q exists for all |α| ď m, as well as D being bounded. In fact, when m " 0, we are awkwardly left with the condition that k be continuous and D be bounded. In hindsight, this criterion is not surprising, as the sample paths of a stationary process "look similar at all locations" ( [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF], p. 4), and thus expecting them to be square integrable over some unbounded domain is not reasonable. This example suggests that focusing on stationary Gaussian processes somewhat conceals the real nature of accurate Sobolev regularity criteria for the sample paths of a Gaussian process. We thus turn to non stationary kernels.

Perhaps the simplest non stationary Gaussian processes are those with finite rank covariance functions, i.e. processes of the form U pxqpωq " ř n i"1 ξ i pωqf i pxq, where pξ i q are standard Gaussian random variables (which we assume independent) and pf i q are measurable functions. The covariance function of the latter is kpx, x 1 q " ř n i"1 f i pxqf i px 1 q. In this case, it is rather clear that the property that PpU P H m pDqq " 1 is equivalent to having pf i q Ă H m pDq (see Example 3.3 for a rigorous proof of this statement). It is also true that Sobolev functions may happen to be discontinuous (e.g. they may have local singularities, [START_REF] Evans | Partial Differential Equations[END_REF], p. 22 and Example 4 p. 260). Hence, the continuity assumptions over k as well as its derivatives required in [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF] seem unnecessarily restrictive with reference to the criterion "pf i q Ă H m pDq". This also prevents the criterion from [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF] to be also necessary. Concerning the results described in [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF], the RKHS of k is now equal to Spanpf 1 , ..., f n q, which is only embedded in H m pDq if pf i q Ă H m pDq. Corollary 4.5 from [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF] now only states that the sample paths lie in H m´d{2´ε pDq for all ε ą 0, which is clearly suboptimal (it also requires that m ą d{2 to be non trivial). In fact, for this process, the sample paths have the same regularity as the functions in the RKHS. This shows that the rule of thumb according to which the sample paths are about d{2 less smooth than functions in the RKHS (a fact which is tight for Matérn processes) can be quite misleading. It is better understood as a lower bound on the regularity of the realizations of a Gaussian process.

Concerning our previous study of Matérn kernels however, it is noteworthy that the imbedding of H t pDq in H s pDq is Hilbert-Schmidt precisely when t ´s ą d{2 (recall that t ´s ą d{2 is the correct criterion for Matérn kernels; see Example 4.5 below for more details on such embeddings). Likewise, the integrals from [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF] exactly correspond to traces of very specific integral operators (see Section 2.1.3 for operator theoretic definitions). These observations suggest the existence of a purely spectral criterion for characterizing the H m pDq regularity of Gaussian processes, which could encompass the results from both [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF] and [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF], as well as finite rank Gaussian processes. In fact, we provide such a spectral criterion in Proposition 4.4.

As a final comment, none of the articles previously mentioned except [START_REF] Ciesielski | Quelques espaces fonctionnels associés à des processus gaussiens[END_REF][START_REF] Kerkyacharian | Regularity of Gaussian processes on Dirichlet spaces[END_REF] deal with spaces of non Hilbert type. General Sobolev spaces W m,p pDq, with 1 ă p ă `8, are particularly useful when studying nonlinear PDEs, e.g. whose nonlinearity is of the form |u| r u (see [START_REF] Tao | Nonlinear dispersive equations: local and global analysis[END_REF], Section 3.9 for examples). As such, we will also focus on this general setting.

General assumptions

The purpose of this article is to uncover necessary and sufficient characterizations of the Sobolev regularity of nonnegative integer order of a given Gaussian process, in terms of its covariance function. In an attempt to make them as general as possible, we set the following targets and assumptions.

piq The covariance function of the Gaussian process will only be assumed measurable, as in [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF]. This contrasts with some of the previously mentioned works [START_REF] Ciesielski | Quelques espaces fonctionnels associés à des processus gaussiens[END_REF][START_REF] Kerkyacharian | Regularity of Gaussian processes on Dirichlet spaces[END_REF][START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF], where the covariance function is assumed continuous. As previously observed though, it seems that assuming the continuity of the covariance (and thus more or less that of the sample paths, [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF] p. 31) to examine some Sobolev regularity of potentially low order is an unnatural assumption. This is especially true as the dimension of D increases, since W m,p pDq is embedded in C 0 B pDq, the Banach space of continuous and bounded functions over D, only when m ą d{p ( [1], Theorems 4.12 and 7.34). piiq We will not make any regularity or shape assumptions on the open set D. Indeed, Sobolev spaces of integer order are easily defined over arbitrary open sets D Ă R d , and thus some results should hold within this general setting. As a result though, we will not deal with fractional Sobolev spaces nor Besov spaces. Indeed, those spaces may have some pathological properties without additional hypotheses on D, namely enjoying a Lipschitz boundary or the cone condition (see e.g. [START_REF] Di Nezza | Hitchhiker's guide to the fractional Sobolev spaces[END_REF], Example 9.1). We will see that elementary characterizations of Sobolev regularity (Lemmas 2.1 and 2.4) will prove to be enough for our purpose.

piiiq Our results should lie outside of the assumption that m ą d{p, where m, p and d correspond to the notation W m,p pDq, D Ă R d . Indeed, several previous results concerning the Sobolev regularity of a given Gaussian process concern the spaces H m pDq, D Ă R d , only in the case m ą d{2. This is convenient because it ensures that H m pDq is continuously embedded in C 0 B pDq when D is smooth enough, which suppresses the ambiguity of choosing a representative of a function in H m pDq. However, m ą d{2 excludes the useful spaces H 1 pR 2 q and H 1 pR 3 q, which are central in the study of many important second order PDEs such as the wave equation, the heat equation, Laplace's equation or Schrödinger's equation.

Our characterizations of measurable Gaussian processes with sample paths in W m,p pDq is phrased in terms of a form of Sobolev regularity of the covariance function on the diagonal. It is then linked to the existence of suitable Mercer or otherwise nuclear decompositions of the integral operators associated to the covariance function and its symmetric weak cross-derivatives. In the Hilbert case p " 2, additional links are made w.r.t. the Mercer decompositions of the said integral operators, their trace and the Hilbert-Schmidt nature of the imbedding of the RKHS in H m pDq. Our results are strongly reminiscient of those found in [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF], where we removed of the continuity assumptions over the covariance in a suitable fashion.

The article is organized as follow. In Section 2, we introduce the necessary notions for properly stating our results as well as some useful lemmas directly related to these notions. In Sections 3 and 4, we state and prove the main results of this article, which treat the general case p P p1, `8q and the special case p " 2 respectively. In Section 5, we conclude and provide some further outlooks. We prove the intermediary lemmas used in the main proofs in Section 6.

Notations Given a Banach space X, X ˚denotes its topological dual. Given x P X and l P X ˚, we denote the duality bracket as follow: lpxq " xl, xy X ˚,X . BpXq denotes the Borel σalgebra of X for its norm topology. Given two linear operators A :

X 1 Ñ Y 1 and B : X 2 Ñ Y 2 , A b B : X 1 b X 2 Ñ Y 1 b Y 2 denotes
their tensor product which verifies pA b Bqpa b bq " pAaq b pBbq. Given two real valued functions f and g, f b g denotes their tensor product defined by pf b gqpx, yq " f pxqgpyq. Given h P R d , |h| denotes its Euclidean norm. Given p P p1, `8q, q will always denote its conjugate: 1{p `1{q " 1 i.e. q " p{pp ´1q. As usual, when D is an open set of R d , we identify the dual of L p pDq with L q pDq. Explicitly, if f P L p pDq and g P L q pDq, we have

xf, gy L p ,L q " ż D f pxqgpxqdx " xg, f y L q ,L p . (1.2)
When there is no risk of confusion, we will write }f } p :" }f } L p pDq . If H is a Hilbert space, x¨, ¨yH denotes its inner product. We denote N :" t1, 2, ...u the set of natural numbers and

N 0 :" N Y t0u. Given an open set D Ă R d , we write D 0 Ť D if D 0 Ă D and D 0 is compact. L 1
loc pDq denotes the space of equivalence classes of locally integrable functions over D, i.e. such that ş K |f pxq|dx ă `8 for all K Ť D. Elements of L 1 loc pDq are identified when they are equal almost everywhere w.r.t. the Lebesgue measure. Given an equivalence class f P L 1 loc pDq, a representative of f is a function p f : D Ñ R such that the equivalence class of p f in L 1 loc pDq is f . We will sometimes denote f and p f with the same symbol, e.g. f . Given a function k defined over D ˆD, E k denotes the associated integral operator (if well defined):

pE k f qpxq " ż D kpx, yqf pyqdy.
(1.

3)

The input and output spaces of E k will be specified on a case-by-case basis.

Background

This section is dedicated to the introduction of the necessary notions required for understanding the main results of the paper, as well as their proofs. It is divided in two parts. Section 2.1 contains the definitions necessary for understanding the statements of Propositions 3.1, 3.6 and 4.4, which constitute the core results of the article. These definitions concern measurable Gaussian processes, Sobolev spaces as well as certain notions from operator theory. Section 2.2 describes most of the tools that we will use in Sections 3 and 4. In particular, it contains a first series of propositions and lemmas (either new or already well-known) which all play a central role in the proofs of Propositions 3.1, 3.6 and 4.4. These results are all directly related to the notions introduced in Section 2.1. They concern several characterizations of Sobolev regularity for locally integrable functions, as well as certain facts about integrals of measurable Gaussian processes, Gaussian sequences and measurable Gaussian processes with L p integrable sample paths. Several characterizations of Gaussian measures over L p spaces are also given, in terms of measurable Gaussian processes in particular.

Preliminary definitions

Measurable Gaussian processes

Throughout this article, pΩ, F, Pq denotes the same probability space. Given p P p1, `8q, L p pPq denotes the space of real valued random variables X such that Er|X| p s ă `8.

If pE, Bq is a measurable space, the law P X of a random variable X : Ω Ñ E is the pushforward measure of P through X, which is defined by P X pBq :" PpX ´1pBqq for all measurable set B P B ( [7], Section 3.7).

A Gaussian process ( [2], Section 1.2) pU pxqq xPD is a family of Gaussian random variables defined over pΩ, F, Pq such that for all n P N, pa 1 , ..., a n q P R n and px 1 , ..., x n q P D n , ř n i"1 a i U px i q is a Gaussian random variable. The law it induces over the function space R D endowed with its product σ-algebra is uniquely determined by its mean and covariance functions, mpxq " ErU pxqs and kpx, x 1 q " CovpU pxq, U px 1 qq ( [START_REF] Jørgensen | Probability With a View Towards Statistics[END_REF], Section 9.8). We then write pU pxqq xPD " GP pm, kq. The covariance function k is positive definite over D, meaning that for all nonnegative integer n and px 1 , ...x n q P D n , the matrix pkpx i , x j qq 1ďi,jďn is nonnegative definite. Conversely, given a positive definite function over an arbitrary set D, there exists a centered Gaussian process indexed by D with the this function as its covariance function ( [START_REF] Adler | Random Fields and Geometry[END_REF], p. 11). We will often denote σpxq :" kpx, xq 1{2 . Given ω P Ω, the corresponding sample path (or realization) of pU pxqq xPD is the deterministic function U ω : D Ñ R defined by U ω pxq :" U pxqpωq. A Gaussian process is said to be measurable if the map pΩ ˆD, F b BpDqq Ñ pR, BpRqq, pω, xq Þ Ñ U pxqpωq is measurable. If pU pxqq xPD is measurable, then from Fubini's theorem the maps of the form x Þ Ñ kpx, x 1 q, x Þ Ñ kpx, xq, etc, are measurable. If a general random field is continuous in probability, then there exists a measurable modification of this random field ( [20], Theorem 2.6 p. 61). This property is e.g. ensured when the covariance function is continuous, from Tchebychev's inequality. Tedious extensions of this result exist ( [19], Theorem 2.3). Finally, the measurability property is ensured for processes defined by series of the form U pxqpωq :" ř j ξ j pωqe j pxq where pξ j q are random variables and pe j q are measurable functions such that the series converges in a suitable space. See [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF] for similar remarks. We further discuss the measurability property of Gaussian processes in Remark 2.10.

Weak derivatives and Sobolev spaces

Let α " pα 1 , ..., α d q P N d 0 . We denote B α " B α1 x1 ...B α d x d the α th derivative, and |α| :"

ř d i"1 |α i |.
In this article, the statement "let |α| ď m" will mean "let α " pα 1 , ..., α d q P N d 0 be such that |α| ď m". Given a function k defined on D ˆD, B α,α k denotes its symmetric cross derivative:

B α,α kpx, yq :" B α1 x1 ...B α d x d B α1 y1 ...B α d y d kpx, yq (formally, B α,α " B α b B α ). A function u P L 1 loc pDq has v P L 1
loc pDq for its α th weak derivative if ( [1], Section 1.62)

@φ P C 8 c pDq, ż D upxqB α φpxqdx " p´1q |α| ż D vpxqφpxqdx. (2.1)
If it exists, v is then unique in L 1 loc pDq and is denoted v " B α u. Let p P r1, `8s. The Sobolev space W m,p pDq is defined as ( [1], Section 3.2) W m,p pDq " tu P L p pDq : @ |α| ď m, B α u exists and B α u P L p pDqu.

(2.2)

Sobolev spaces are Banach spaces for the norm }u} W m,p :" p ř |α|ďm }B α u} p p q 1{p ; they are separable when p ‰ `8 ( [1], Theorem 3.6 p. 61). When p " 2, we denote H m pDq :" W m,2 pDq. H m pDq is a Hilbert space for the following inner product xu, vy H m pDq :"

ÿ |α|ďm xB α u, B α vy L 2 pDq .
(2.3)

Note that we made no assumptions on the regularity of the open set D.

Notions from operator theory

The following reminders can be found in [START_REF] Bogachev | Gaussian measures[END_REF], Section A.2. Let H 1 and H 2 be two Hilbert spaces, and X and Y two Banach spaces.

piq A linear operator T : X Ñ Y is bounded if }T } :" sup }x} X "1 }T x} Y ă `8. A bounded operator T : X Ñ Y is compact if T pBq is a compact set of Y
, where B is the closed unit ball of X. When X " Y , the spectrum of a compact operator is purely discrete, and can be reordered as a sequence pλ n q nPN which converges to 0. ă `8 for one orthonormal basis pf n q of H 1 , in which case the value of this sum does not depend on the orthonormal basis at hand. The Hilbert-Schmidt norm of T , defined as the square root of the sum above, is then also equal to the discrete ℓ 2 norm of its singular values: , where the sequence ps n q still corresponds to the singular values of T . One can then define its trace as the following linear functional, which is independent of the choice of basis pe n q, and equal to the (absolutely convergent) series of the eigenvalues pµ n q of T (Lidskii's theorem): [START_REF] Brislawn | Kernels of trace class operators[END_REF]; see also Proposition 2.9 and Lemma 3.8 below. If T : H 1 Ñ H 1 is bounded, self-adjoint and nonnegative, then we define its trace as the possibly infinite series of nonnegative scalars TrpT q :" ř nPN xT e n , e n y. pivq ( [START_REF] Linde | Characterization of certain classes of Banach spaces by properties of Gaussian measures[END_REF], p. 160) A bounded operator T : X Ñ Y is nuclear if there exists sequences px n q Ă X ˚and py n q Ă Y with ř `8 n"1 }x n } X ˚}y n } Y ă `8 such that T x " ř `8 n"1 xx n , xy X ˚,X y n for all x P X. In this case, we write abusively T " ř `8 n"1 x n b y n . The nuclear norm of T is then defined as νpT q :" inf

piiq If T : H 1 Ñ H 2 is compact, then T ˚T : H 1 Ñ H 1 is compact, self-
}T } 2 HS " ÿ nPN }T f n } 2 H2 " ÿ nPN }T e n }
TrpT q :" ÿ nPN xT e n ,
" `8 ÿ n"1 }x n } X ˚}y n } Y such that T " `8 ÿ n"1 x n b y n * . (2.8)
A bounded operator K : X ˚Ñ X is symmetric if for all x, y P X ˚, xx, Ryy " xy, Rxy, and nonnegative if xx, Rxy ě 0. When X " Y " H where H is a separable Hilbert space, the sets of trace class and nuclear operators coincide; moreover, the same can be said for the trace functional (2.7) and the nuclear norm (2.8) if T has a nonnegative spectrum : νpT q " TrpT q.

2.2 Main tools of the article 2.2.1 Characterization of W m,p -regularity for locally integrable functions

As for pointwise derivatives, finite difference operators can be used for characterizing Sobolev regularity. Given y P R d , introduce the translation operator pτ y uqpxq " upx `yq, which is bounded over L p pR d q. Introduce the associated finite difference operator:

∆ y " τ y ´Id.

(2.9)

The linear subspace of bounded operators over L p pR d q induced by the translation operators is commutative, as τ y1 ˝τy2 " τ y1`y2 " τ y2 ˝τy1 . Let py 1 , ..., y m q P pR d q m , we define the m th order finite difference operator associated to py 1 , ..., y m q to be ∆ py1,...,ymq :" ś m i"1 ∆ yi where the product symbol denotes the composition of operators. When y P R d , the adjoint of ∆ y is also a finite difference operator, which is computable using the change of variable formula. If y P R d , then ∆ ẙ " τ ´y ´Id.

(2.10)

Finally, when α " pα 1 , ..., α d q P N d 0 and h " ph 1 , ..., h d q P pR ˚qd , we denote by δ α h the finite difference approximation of B α defined by

δ α h " d ź i"1 ˆ∆hiei h i ˙αi " ˆ∆h1e1 h 1 ˙α1 ¨¨¨ˆ∆ h d e d h d ˙αd .
(2.11)

Above, pe 1 , ..., e d q is the canonical basis of R d . We use the convention that p∆ hiei q αi " Id if α i " 0. Depending on which one is the most convenient, we will either use ∆ py1,...,ymq or δ α h . We will use the following characterizations of W m,p -regularity; they are "straightforward" generalizations of Proposition 9.3 from [START_REF] Brézis | Functional analysis, Sobolev spaces and partial differential equations[END_REF] to m ě 2. These characterizations have the benefit of being valid without any regularity assumptions on the open set D. We prove Lemma 2.1 in Section 6, as we could not find it stated as such in the literature.

Lemma 2.1. Suppose that u P L 1 loc pDq. Let m P N 0 , p P p1, `8s and introduce q ě 1 the conjugate of p : 1{p `1{q " 1. Then the following statements are equivalent.

(i) u P W m,p pDq (ii) (Variational control) for all α such that |α| ď m, there exists a constant C α such that

@φ P C 8 c pDq, ˇˇż D upxqB α φpxqdx ˇˇď C α }φ} L q pDq .
(2.12)

In this case, the L p norm of B α u is given by

}B α u} L p pDq " sup φPC 8 c pDqzt0u ˇˇˇż D upxq B α φpxq }φ} L q dx ˇˇˇ. (2.13)
(iii) (Finite difference control) there exists a constant C such that for all open set D 0 Ť D, for all ℓ ď m and all py 1 , ..., y ℓ q P pR d q ℓ such that

ř ℓ i"1 |y i | ă distpD 0 , BDq, }∆ py1,...,y ℓ q u} L p pD0q ď C|y 1 | ˆ... ˆ|y ℓ |. (2.14)
Moreover, for all |α| ď m, }B α u} L p pDq ď C for any C verifying equation (2.14). Finally, one can actually take C " }u} W m,p pDq in equation (2.14).

In Point piiiq above, the assumption that ř ℓ i"1 |y i | ă distpD 0 , BDq ensures that the quantity ∆ py1,...,y ℓ q upxq makes sense when x P D 0 . A similar criterion to Point piiiq above is given in [START_REF] Leoni | A first course in Sobolev spaces[END_REF], Theorem 10.55, still in the case m " 1 (as well as in [START_REF] Evans | Partial Differential Equations[END_REF], Section 5.8.2.a, Theorem 3). This theorem only requires the L p control of the ratios δ α h u with |α| ď 1. As such, we could have also stated a version of Lemma 2.1piiiq solely in terms of the ratios δ α h u with |α| ď m.

Sobol, Sectioev reguln 10.5arity and generalized functions

The theory of generalized functions (or distributions) provides a flexible way of characterizing Sobolev regularity, by building a larger space in which partial derivatives are always defined. Given an open set D, denote C 8 c pDq the space of smooth functions with compact support in D. Endow it with its usual LF topology, defined e.g. in [START_REF] Treves | Topological Vector Spaces, Distributions and Kernels[END_REF], Chapter 13. This topology is such that the sequence pφ n q converges to φ in C 8 c pDq if and only if there exists a compact set K Ă D such that Supppφ n q Ă K for all n and @α " pα 1 , ..., α d q P N d 0 , sup

@φ P C 8 c pDq, T pφq " ż D upxqφpxqdx, (2.16)
for some u P L 1 loc pDq, in which case one writes T " T u . Given any function u P L 1 loc pDq and α P N d 0 , its distributional derivative D α u is defined by the following formula ( [START_REF] Treves | Topological Vector Spaces, Distributions and Kernels[END_REF], pp. 248-250):

D α u : φ Þ ÝÑ p´1q |α| ż D B α φpxqupxqdx.
(2.17)

D α u then also lies in D 1 pDq. Sobolev regularity can now be rephrased as follow: u lies in W m,p pDq iff for all |α| ď m, the distributional derivative D α u is in fact a regular generalized function represented by some v α P L p pDq i.e. D α u " T vα . Then v α is unique in L p pDq and B α u " v α in L p pDq, where B α u is the α th weak derivative of u.

Moreover, the control equation (2.12) shows that B α u exists and lies in L p pDq if and only if D α u : C 8 c pDq Ñ R can be extended as a continuous linear form over L q pDq. Ensuring the existence of such extensions will thus be of prime interest for us, and is the topic of the next lemma. Specifically, the next result states that given continuous linear or bilinear forms over C 8 c pDq, the existence of extensions of these maps to L q pDq can be ensured by obtaining suitable estimates on a well chosen countable set E q Ă C 8 c pDq. Restricting ourselves to E q will allow us to eliminate any measurability issues when introducing the supremum of certain random variables indexed by E q , as a countable supremum of random variables remains a random variable (i.e. a measurable map). Below, we write } ¨}q :" } ¨}L q pDq for short.

Lemma 2.2 (Extending continuous linear and bilinear forms over C 8

c pDq to L p pDq). Let p P p1, `8q. There exists a countable Q´vector space E q " tΦ q n , n P N 0 u Ă C 8 c pDq with the following property.

(i) A distribution T P D 1 pDq is a regular distribution, T " T v , for some v P L p pDq iff it verifies the countable estimate for some constant C ą 0 @φ P E q , |T pφq| ď C}φ} q , (2.18)

or equivalently, sup nPN |T pΦ q n q|{}Φ q n } q ă `8 (here, setting Φ q 0 " 0 without loss of generality). This is equivalent to T admitting an extension over L q pDq which is then uniquely given by T pf q " ş D f pxqvpxqdx. Moreover,

sup nPN |T pΦ q n q| }Φ q n } q " sup φPC 8 c pDq |T pφq| }φ} q , (2.19)
whether these quantities are finite or not.

(ii) Let b be a continuous bilinear form over C 8 c pDq. Then b can be extended to a continuous bilinear form over L q pDq iff it verifies the countable estimate @φ, ψ P E q , |bpφ, ψq| ď C}φ} q }ψ} q .

(2.20)

In this case, such an extension is unique and there will exist a unique bounded operator B : L q pDq Ñ L p pDq verifying the following identity @φ, ψ P C 8 c pDq, bpφ, ψq " xBφ, ψy L p ,L q .

(2.21)

The proof of this result can be found in Section 6. It relies on Lemma 2.3 below, which is interesting in itself.

Lemma 2.3. C 8

c pDq is sequentially separable, i.e. there exists a countable subset F Ă C 8 c pDq such that for all φ P C 8 c pDq, there exists a sequence pφ n q Ă F such that φ n Ñ φ in C 8 c pDq for its LF topology. 1A short proof of this result can be found in [START_REF] Henderson | Characterization of the second order random fields subject to linear distributional pde constraints[END_REF], Lemma 3.6. Given the set E q provided by Lemma 2.2, we next define the countable set F q to be F q :" tφ{}φ} q , φ P E q , φ ‰ 0u " tf q n , n P Nu Ă S q p0, 1q.

(2.22)

Above, pf q n q nPN is an enumeration of F q and S q p0, 1q is the unit sphere of L q pDq. The next lemma is then a direct consequence of Lemmas 2.1 and 2.2.

Lemma 2.4 (Countable characterization of Sobolev regularity). Let p P p1, `8q. For any u P L 1 loc pDq, u lies in W m,p pDq iff for all multi index α such that |α| ď m, there exists a constant C α such that

@φ P E q , ˇˇˇż D upxqB α φpxqdx ˇˇˇď C α }φ} q , (2.23)
or equivalently, in terms of the set F q defined in equation

(2.22), sup φPFq ˇˇˇˇż D upxqB α φpxqdx ˇˇˇˇ" sup nPN ˇˇˇˇż D upxqB α f q n pxqdx ˇˇˇˇă `8. (2.24)
Moreover,

sup φPFq ˇˇˇˇż D upxqB α φpxqdx ˇˇˇˇ" sup φPC 8 c pDqzt0u ˇˇˇˇż D upxq B α φpxq }φ} q dx ˇˇˇˇ, (2.25)
whether these quantities are finite or not. If one of them is finite, then it is equal to }B α u} L p pDq .

This lemma provides us with a somewhat explicit countable criteria for Sobolev regularity, which is valid whatever the open set D.

Integrals of measurable Gaussian processes

We will need the following lemma pertaining to the sample path-wise integration of Gaussian processes.

Lemma 2.5. Let D Ă R d be an open set. Let pU pxqq xPD " GP p0, kq be a measurable centered Gaussian process such that its standard deviation function σ lies in L 1 loc pDq. Then the sample paths of U lie in L 1 loc pDq almost surely and given φ P C 8 c pDq and α P N d 0 , the map defined by

U α φ : Ω Q ω Þ ÝÑ p´1q |α| ż D U ω pxqB α φpxqdx (2.26)
is a Gaussian random variable. Moreover, for all p P p1, `8q, pU α φ q φPFq is a centered Gaussian sequence (i.e. a Gaussian process indexed by a countable set), where F q is defined in equation (2.22).

Following Lemma 2.4, in order to study the W m,p regularity of Gaussian processes, we will be interested in the boundedness of the Gaussian sequence pU α φ q φPFq given |α| ď m. As such, the next result pertaining to bounded Gaussian sequences will be quite helpful. It can be seen as a weak form of Fernique's theorem ( [6], Theorem 2.8.5, p. 75). (2.27)

In particular, Er|U | p s ă `8 for all p P N.

Gaussian measures over Banach spaces and L p spaces

This section will be helpful for providing a necessary and sufficient condition according to which PpU P W m,p pDq " 1q, in terms of the "spectral" properties of the covariance kernel of U . A Gaussian measure µ ( [6], Definition 2.2.1) over a Banach space X is a measure over its Borel σ-algebra such that given any x ˚P X ˚, the pushforward measure of µ through the functional x ˚is a Gaussian measure over R (see Section 2.1.1 for a definition of the pushforward). Gaussian measures are equipped with a mean vector a µ P X ˚˚and a covariance operator K µ : X ˚Ñ X ˚˚, defined in [START_REF] Bogachev | Gaussian measures[END_REF], Definition 2.2.7. When X is separable, µ is Radon ( [START_REF] Bogachev | Gaussian measures[END_REF], p. 125). This implies that a µ lies in X and that the covariance operator K µ maps X to X ( [6], Theorem 3.2.3). The vector a µ and the covariance operator K µ are defined by the following formulas

@x P X ˚, xa µ , xy " ż X xx, zyµpdzq, (2.28)
@x, y P X ˚, xy, K µ xy " ż X xx ´aµ , zy xy ´aµ , zyµpdzq.

(2.29)

Any operator K : X ˚Ñ X ˚˚which is the covariance operator of a Gaussian measure is called a Gaussian covariance operator. In Propositions 2.7 and 2.8, we present useful characterizations of Gaussian measures µ over two important classes of Banach spaces: spaces of type 2 and cotype 2 respectively. For a definition of spaces of type 2 and cotype 2, see e.g. [START_REF] Chobanjan | Gaussian characterizations of certain Banach spaces[END_REF]. In this article, we will only use the fact that L p pDq is of type 2 when p ě 2, and cotype 2 when 1 ď p ď 2 (see [START_REF] Bogachev | Gaussian measures[END_REF], p. 152). Moreover we will restrict ourselves to the case where X is separable. As this implies that µ is Radon, this removes problems pertaining to extensions of measures otherwise considered in [START_REF] Linde | Characterization of certain classes of Banach spaces by properties of Gaussian measures[END_REF] and [START_REF] Chobanjan | Gaussian characterizations of certain Banach spaces[END_REF].

Proposition 2.7 ( [32], Theorem 4 or [START_REF] Bogachev | Gaussian measures[END_REF], Remark 3.11.24). Let X be a separable Banach space of type 2, and let µ be a Gaussian measure over X. Then its covariance operator is symmetric, nonnegative and nuclear. Conversely, given any a P X and any symmetric, nonnegative and nuclear operator K : X ˚Ñ X, there exists a Gaussian measure over X with mean vector a and covariance operator K.

Denote ℓ 2 the Hilbert space of square summable sequences.

Proposition 2.8 ( [12], Theorem 4.1 and Corollary 4.1). Let X be a separable Banach space of cotype 2, and let µ be a Gaussian measure over X. Then there exists a continuous linear map A : l 2 Ñ X and a symmetric, nonnegative and trace-class operator S : l 2 Ñ l 2 such that covariance operator of µ is given by ASA ˚(in particular, the covariance operator of µ is nuclear). In other words, µ is the pushforward measure of a Gaussian measure µ 0 over ℓ 2 through some bounded linear map A. Conversely, given any a P X and any operator of the form ASA ˚where A : ℓ 2 Ñ X is a bounded linear map and S a symmetric, nonnegative and trace class operator over ℓ 2 , there exists a Gaussian measure over X with mean vector a and covariance operator K.

In practice, we will replace ℓ 2 with L 2 pDq, which are isomorphic Hilbert spaces. The propositions 2.7 and 2.8 generalize the case where X is a separable Hilbert space, which can be found in [START_REF] Bogachev | Gaussian measures[END_REF], Theorem 2.3.1. We finish with the following handy result describing centered Gaussian measures over L p -spaces. piq Let µ be a centered Gaussian measure over L p pDq where 1 ď p ă `8 and D Ă R d is an open set. Then there exists a function k P L p pD ˆDq such that the covariance operator of µ is E k : L q pDq Ñ L p pDq, the integral operator associated to k. Moreover, there exists a centered measurable Gaussian process pU pxqq xPD whose covariance function k verifies k " k in L p pD ˆDq, and whose sample paths lie in L p pDq a.s.. Setting σpxq " kpx, xq 1{2 , k verifies

ż D kpx, xq p{2 dx " ż D σpxq p dx ă `8.
(2.30)

Additionally, P U " µ, where P U is the pushforward of P through the Borel-measurable map ω Þ Ñ U ω P L p pDq. Conversely, given any measurable nonnegative definite function k verifying (2.30), the corresponding integral operator E k : L q pDq Ñ L p pDq is the covariance operator of a centered Gaussian measure µ over L p pDq.

piiq Given a centered measurable Gaussian process pU pxqq xPD whose covariance function we denote k, the condition (2.30) is equivalent to pU pxqq xPD having its sample paths lie in L p pDq a.s.. This result is quite strong, as it ensures the existence of a representative in L p pD ˆDq of the kernel of any Gaussian covariance operator, which is the covariance function of a measurable Gaussian process. This will enable us to remove awkward measurability issues w.r.t. k over its diagonal and equation (2.30). Without the use of an underlying measurable Gaussian process, these issues are not trivial to deal with, see e.g. [START_REF] Brislawn | Kernels of trace class operators[END_REF] for an analysis of the Hilbert case p " 2.

Remark 2.10. Proposition 2.9 shows that the assumption that a given Gaussian process is measurable is slightly less demanding that it might seem. As observed in Section 2.1.1, the existence of a measurable modification of a general random field is difficult outside of it being continuous in probability. For a Gaussian process pU pxqq xPD " GP p0, k u q however, Propositions 2.7, 2.8 and 2.9 shows that the measurability of its covariance function over D ˆD and the integrability of its standard deviation in L p pDq (or equivalently, suitable nuclear decompositions of its associated integral operator E k ) ensure the existence of a measurable Gaussian process pV pxqq xPD " GP p0, k v q with the same covariance function in L 1 loc pD ˆDq. Consequently, k u " k v a.e. on D ˆD. Note though that the process V need not be a modification of U . Since k u " k v a.e., we only have that U and V have the same finite dimensional marginals "almost everywhere" in the sense of the Lebesgue measure: for all n P N and almost every px 1 , ..., x n q P D n , pU px 1 q, ..., U px n qq and pV px 1 q, ..., V px n qq have the same law.

Throughout this article, we will only consider centered Gaussian processes pErU pxqs " 0q and Gaussian measures pa µ " 0q. Generalizations of the results of this article to non centered Gaussian processes are straightforward.

3 Sobolev regularity for Gaussian processes : the general case, 1 ă p ă `8

We can now state our first result, which deals with W m,p pDq-regularity of Gaussian processes, given any p P p1, `8q and any open set D Ă R d . (ii) (Integral criteria) For all |α| ď m, the distributional derivative B α,α k lies in L p pD ˆDq and admits a representative k α in L p pD ˆDq which is the covariance function of a measurable Gaussian process. For all such k α , denoting σ α pxq :" k α px, xq 1{2 , we have ż D σ α pxq p dx ă `8.

(3.1)

(iii) (Covariance structure) For all |α| ď m, the distributional derivative B α,α k lies in L p pD Dq and the associated integral operator E α k : L q pDq Ñ L p pDq defined by

E α k f pxq " ż D B α,α kpx, yqf pyqdy (3.2)
is symmetric, nonnegative and nuclear: there exists pλ α n q Ă R `and pψ α n q Ă L p pDq such that

`8 ÿ n"0 λ α n }ψ α n } 2 p ă `8, B α,α kpx, yq " `8 ÿ n"0 λ α n ψ α n pxqψ α n pyq in L p pD ˆDq. (3.3)
If 1 ď p ď 2, then one can choose pλ α n q such that ř n λ α n ă `8, and there exists a bounded operator A α : L 2 pDq Ñ L p pDq and an orthonormal basis pϕ α n q of L 2 pDq such that ψ α n " A α ϕ α n for all n ě 0 (in particular, we have the uniform bound }ψ α n } p ď }A α }).

The proposition above shows that a suitable L p control of the function B α,α k over the diagonal is necessary and sufficient for ensuring the Sobolev regularity of the sample paths of the Gaussian process with covariance function k. Formally speaking, the function px, yq Þ Ñ B α,α kpx, yq is the covariance function of the differentiated process, pω, xq Þ Ñ B α U ω pxq. This is formal only, as the weak derivative of the sample paths are only defined up to a set of Lebesgue measure zero, and thus there is no obvious way of defining the joint map pω, xq Þ Ñ B α U ω pxq. Note also that the idea of ensuring a suitable control of this covariance function near its diagonal is not with reminding more standard results pertaining to the differentiability in the mean square sense of a random process (see e.g. [START_REF] Adler | Random Fields and Geometry[END_REF], Section 1.4.2). See [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF] for similar remarks on the Sobolev regularity of random fields.

Observe also that there is an asymmetry between Point piiq and Point piiiq of Proposition 3.1, as one depends on whether p is lower or greater than 2 while the other does not. Moreover, both points rely on the finiteness of some quantity, so explicit bounds should be sought so that Point piiq controls Point piiiq and conversely. This is the content of Proposition 3.6.

Finally, observe that the integrability criteria piiq cannot be expected to hold for any positive definite representative kα of B α,α k, even if kα is measurable on its diagonal. For example, set kα px, yq :" k α px, yq `δx,y where δ x,y is the Kronecker delta, which verifies kα " B α,α k in L p pD ˆDq. But if D has infinite Lebesgue measure, it is also clear that ş D kα px, xq p{2 dx ě ş D δ x,x dx " `8. Lemma 3.8 describes a natural set of "admissible" representatives for which Point piiq holds, in the case p ě 2. Remark 3.2. Under the assumption that pU pxqq xPD is measurable, the statement that its sample paths lie in some Sobolev space is not up to a modification of the process. This is a consequence of Lemmas 2.4, 2.5 and 2.6, which show that the Sobolev regularity of its paths is fully determined by the finite dimensional marginals of the process (see equation (3.6)). This contrasts with more classical results, e.g. pertaining to the continuity of the process ( [3], Section 1.4.1). Still, ensuring the measurability of the process is not really straightforward (see Remark 2.10; in fact, this property may happen to be only ensured up to a modification of the initial process).

Example 3.3 (Finite rank covariance functions). Let p P p1, `8q, m P N 0 and n P N. Consider f 1 , ..., f n P L p pDq and choose representatives of those functions in L p pDq, also denoted by f 1 , ..., f n , so that they may be understood as functions in the classical sense. Consider the covariance function kpx, x 1 q :" ř n i"1 f i pxqf i px 1 q. Assume that f 1 , ..., f n P W m,p pDq, then for all |α| ď m, the weak derivative B α,α k is obviously given by

B α,α kpx, x 1 q " n ÿ i"1 B α f i pxqB α f i px 1 q in L p pD ˆDq, (3.4)
and the associated integral operators fulfill the criterion piiiq of Proposition 3.1. Thus the corresponding measurable Gaussian process has its sample paths in W m,p pDq almost surely. Note that this was obvious in the first place, since this Gaussian process can be written as U pxq " ř n i"1 ξ i f i pxq where ξ 1 , ..., ξ n are independent standard Gaussian random variables (checking that the covariance function is the right one is trivial). Conversely, assume that the sample paths of the associated measurable Gaussian process lie in W m,p pDq almost surely. Then the function k verifies the criterion piiiq of Proposition 3.1, and in particular B α,α k P L p pD ˆDq. One can then show that f 1 , ..., f n P W m,p pDq (copy the proof of Lemma 4.9, Point piq).

While this example can easily be studied and solved on its own, we observed in the introduction that (surprisingly) this example fell out of the scope of the previous results pertaining to the Sobolev regularity of Gaussian processes. Indeed, the stationarity assumptions of [START_REF] Cirel'son | Norms of Gaussian sample functions[END_REF][START_REF] Ibragimov | Conditions for Gaussian homogeneous fields to belong to classes H r p[END_REF] are not met. Since the domain D is not assumed to be endowed with an underlying Dirichlet structure, the results from [START_REF] Kerkyacharian | Regularity of Gaussian processes on Dirichlet spaces[END_REF] pertaining to the B s 8,8 regularity of Gaussian processes do not apply. In fact, in our setting, D is not even assumed to be compact and k is not assumed continuous, contrarily to [START_REF] Kerkyacharian | Regularity of Gaussian processes on Dirichlet spaces[END_REF]. Likewise, the one dimensional framework of [START_REF] Ciesielski | Quelques espaces fonctionnels associés à des processus gaussiens[END_REF] is too restrictive for our example. In the case where p " 2, the continuity assumptions of the covariance function as well as its cross derivatives over the diagonal required in [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF] are also not fulfilled. Still in the case where p " 2, the RKHS associated to k (see the upcoming Section 4.1) is equal to Spanpf 1 , ..., f n q, which is a subspace of H m pDq; without further assumptions on f 1 , ..., f n , it is not a subspace of H m`ε pDq for any ε ą 0. Thus the results from [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF], Corollary 4.5, only ensure the suboptimal fact that the sample paths lie in H m´d{2´η pDq for all η ą 0. Moreover, this result only holds under additional regularity assumptions over D.

Proof. (Proposition 3.1) We show piq ùñ piiq & piiiq, piiq ùñ piq and piiiq ùñ piiq. piq ùñ piiq & piiiq : Assume piq and let |α| ď m. We first prove that the map N α : pΩ, F, Pq Ñ pR, BpRqq, ω Þ Ñ }B α U ω } L p pDq is measurable. Indeed, given φ P F q (see equation (2.22) for the definition of F q ), the map

U α φ : ω Þ ÝÑ ż D B α U ω pxqφpxqdx " p´1q |α| ż D U ω pxqB α φpxqdx (3.5)
is a real valued random variable (this follows from Lemma 2.5). From Lemma 2.4, one also has

ˆω Þ Ñ }B α U ω } L p pDq ˙" sup φPFq |U α φ |. (3.6)
The supremum being taken over a countable set, N α is indeed a measurable map. Given any u P L p pDq, a slight modification of this proof shows that ω Þ Ñ }B α U ω ´u} L p pDq is also measurable. We can now show the map T α : pΩ, F, Pq Ñ pL p pDq, BpL p pDqqq, ω Þ Ñ B α U ω is measurable. Let u P L p pDq, r ą 0 and B " Bpu, rq be an open ball in L p pDq. Then from the measurability of ω Þ Ñ }B α U ω ´u} L p pDq ,

T ´1 α pBq " tω P Ω : }B α U ω ´u} L p pDq ă ru P F. (3.7)
Since L p pDq is a separable metric space, its Borel σ-algebra is generated by the open balls of L p pDq (see e.g. [START_REF] Bogachev | Measure theory[END_REF], Exercise 6.10.28). Thus T α is Borel-measurable and the pushforward of P through T α induces a (centered) probability measure µ α over the Banach space L p pDq. We show that it is Gaussian. Let v P L q pDq and denote T v the associated linear form over L p pDq.

Let pϕ n q Ă C 8 c pDq be such that ϕ n Ñ v in L q pDq ( [1], Corollary 2.30) and ω P Ω be such that U ω lies in L 1 loc pDq:

T f pB α U ω q " ż D B α U ω pxqvpxqdx " lim nÑ8 ż D B α U ω pxqϕ n pxqdx (3.8) " lim nÑ8 p´1q |α| ż D U ω pxqB α ϕ n pxqdx. (3.9)
For each value of n, Lemma 2.5 shows that the map ω Þ Ñ p´1q |α| ş D U ω pxqB α ϕ n pxqdx is a Gaussian random variable. Thus ω Þ Ñ T v pB α U ω q is a Gaussian random variable as an a.s. limit of Gaussian random variables. This shows that the pushforward of µ α through T v is Gaussian (see Section 2.1.1 for the pushforward), since for all Borel set B P BpRq, µ α pT ´1 v pBqq " µ α ptg P L p pDq : T v pgq P Buq " Pptω P Ω : T v pB α U ω q P Buq.

(3.10)

Hence, µ α is Gaussian. We next show that B α,α k P L p pD ˆDq and that the covariance operator of µ α is the integral operator E α k : L q pDq Ñ L p pDq with kernel B α,α k. Let D 0 Ť D ˆD and K 0 Ť D be such that D 0 Ă K 0 ˆK0 (for example, set K 1 :" tx P D : Dy P D, px, yq P Ku, K 2 :" ty P D : Dx P D, px, yq P Ku which are both compact subsets of D and K 0 :" K 1 Y K 2 ). Let h " ph 1 , ..., h d q P pR ˚qd be such that ř d i"1 α i h i ă distpK 0 , D 0 q. Use then the bilinearity of the covariance operator:

ż D0 |pδ α h b δ α h qkpx, yq| p dxdy " ż D0 |Erδ α h U pxqδ α h U pyqs| p dxdy (3.11) ď ż K0ˆK0 |Erδ α h U pxqδ α h U pyqs| p dxdy (3.12) ď ż K0ˆK0 Er|δ α h U pxqδ α h U pyq| p sdxdy (3.13) ď E "ˆż K0 |δ α h U pxq| p dx ˙2ȷ " E " }δ α h U } 2p p ı (3.14) ď E " }U } 2p W m,p pDq ı ": C p ă `8. (3.15)
The expectation in equation (3.15) is indeed finite because of the following. Given |α| ď m, equation (3.6) shows that the map ω Þ Ñ }B α U ω } p is the supremum of a Gaussian sequence which is finite a.s. by assumption; Lemma 2.6 then implies that all the moments of this supremum are finite. Writing then }U } W m,p in terms of these L p norms yields equation (3.15). To see that the control (3.15) implies that B α,α k P L p pD ˆDq, we copy below the steps of equations (6.

ˇˇˇż DˆD pδ α h b δ α h qkpx, yqφpx, yqdxdy ˇˇˇď }pδ α h b δ α h qk} p }φ} q ď C}φ} q . (3.16)
Next, use the discrete integration by parts formula,

ż DˆD pδ α h b δ α h qkpx, yqφpx, yqdxdy " ż D kpx, yqpδ α h b δ α h q ˚φpx, yqdxdy. (3.17)
When h Ñ 0, observe that pδ α h b δ α h q ˚φpx, yq Ñ B α,α φpx, yq pointwise. Use Lebesgue's dominated convergence theorem and equation (3.16) to obtain ˇˇˇż DˆD kpx, yqB α,α φpx, yqdxdy ˇˇˇď C}φ} q ,

which indeed shows that B α,α k P L p pD ˆDq, from Riesz' lemma. We now identify K α , the covariance operator of µ α , in terms of B α,α k. Let f, g P L q pDq and using the density of C 8 c pDq in L q pDq ( [1], Corollary 2.30), let pf n q, pg n q Ă C 8 c pDq be two sequences such that f n Ñ f and g n Ñ g, both in L q pDq. Then (explanation below),

xf, K α gy L q ,L p " lim nÑ8 xf n , K α g n y L q ,L p (3.19) " lim nÑ8 ż L p pDq xf n , hy L q ,L p xg n , hy L q ,L p dµ α phq " lim nÑ8 ż Ω xf n , B α U ω y L q ,L p xg n , B α U ω y L q ,L p dPpωq (3.20) " lim nÑ8 ż Ω xB α f n , U ω y L q ,L p xB α g n , U ω y L q ,L p dPpωq " lim nÑ8 ż DˆD B α f n pxqB α g n pyqkpx, yqdxdy (3.21)
"

lim nÑ8 ż DˆD f n pxqg n pyqB α,α kpx, yqdxdy " ż DˆD f pxqgpyqB α,α kpx, yqdxdy " xf, E α k gy L q ,L p (3.22)
We used the sequential continuity of K α in equation (3.19), the transfer theorem for pushforward measure integration ( [7], Theorem 3.6.1) in equation (3.20) and Fubini's theorem in equation (3.21). Thus K α " E α k . According to Proposition 2.9, since µ α is a Gaussian measure over L p pDq, there exists a representative k α of B α,α k in L p pD ˆDq which is the covariance function of a measurable Gaussian process. Note σ α pxq " k α px, xq 1{2 , then the same proposition shows that

ż D σ α pxq p dx ă `8, (3.23)
which shows piiq. By Corollary 3.5.11 from [START_REF] Bogachev | Gaussian measures[END_REF], E α k is nuclear and admits a symmetric nonnegative representation as the one in equation (3.3). If 1 ď p ď 2, then L p pDq is of cotype 2 and since we have shown that E α k is a Gaussian covariance operator, from Proposition 2.8 there exists a bounded operator A α : L 2 pDq Ñ L p pDq and a trace class operator

S α : L 2 pDq Ñ L 2 pDq such that E α k " A α S α A α. Introduce a Mercer decomposition of S α (equation (2.6)): S α " ř n λ α n ϕ α n b ϕ α n .
Use the continuity of A α and A α to obtain that B α,α kpx, yq " ř n λ α n pA α ϕ α n qpxqpA α ϕ α n qpyq in L p pD ˆDq, which finishes to prove piiiq. piiq ùñ piq : from Proposition 2.9, let pV α q be a centered measurable Gaussian process with covariance function k α . Then its sample paths lie in L p pDq a.s. and the Gaussian measure it induces over L p pDq through the map ω Þ Ñ V α ω P L p pDq is the centered Gaussian measure with covariance operator E α k . Given φ P C 8 c pDq, denote V α φ the following random variable

V α φ : ω Þ Ñ ż D V α ω pxqφpxqdx. (3.24)
From Lemma 2.5, pV α φ q φPFq is a Gaussian sequence. It is also centered and using Fubini's theorem to permute E and ş , we have that

ErV α φ V α ψ s " ż DˆD φpyqψpxqk α px, yqdxdy " ż DˆD φpyqψpxqB α,α kpx, yqdxdy " ż DˆD B α φpyqB α ψpxqkpx, yqdxdy. (3.25)
ErU α φ U α ψ s " ż DˆD B α φpyqB α ψpxqkpx, yqdxdy. (3.26)
Having the same mean and covariance, the two Gaussian sequences pV α φ q φPFq and pU α φ q φPFq have the same finite dimensional marginals. One checks in an elementary fashion that their countable suprema over F q then have the same probability law (e.g. by showing that they have the same cumulative distribution function). Recalling from Lemma 2.4 that }V α ω } p " sup φPFq |V α φ pωq|, we obtain that

1 " Pp}V α ω } p ă `8q " Pp sup φPFq |V α φ | ă `8q " Pp sup φPFq |U α φ | ă `8q, (3.27)
which shows that B α U P L p pDq almost surely. This is true for all |α| ď m, which shows piq. piiiq ùñ piiq : if piiiq, then from either Proposition 2.7 or 2.8 depending on whether p ď 2 or p ě 2, there exists a Gaussian measure over L p pDq whose covariance operator is E α k as defined in equation (3.2). Proposition 2.9 yields piiq.

Remark 3.4 (Distributing derivatives over nuclear decompositions). In Point piiiq of Proposition 3.1, it is very tempting to distribute the cross derivative B α,α over the nuclear decomposition of k, thus setting λ α n " λ 0 n and ψ α n " B α ψ 0 n . While we can show that B α ψ n P L p pDq (copy the proof of Lemma 4.9piq), it is not clear whether the obtained decomposition converges in L p pD ˆDq, or that it corresponds to a nuclear one, i.e.

ř n λ n }B α ψ n } 2 p ă `8 (it is not even clear in what sense this derivative can be distributed, apart from the distributional sense). When p " 2, it turns out that this is true (see the upcoming Proposition 4.4): distributing derivatives on nuclear decompositions yield nuclear decompositions, as soon as nuclear decompositions of the differentiated kernel exist. The following formal computation shows that we should expect this property to hold also when 1 ă p ă 2. Assume formally that the derivative can be distributed pointwise, and introduce the functions v α pxq :" ř n λ n B α ψ n pxq 2 and σ α pxq :" v α pxq 1{2 . From Proposition 3.1, we expect that }σ α } p ă `8. When 1 ď p ă 2, the reverse Minkowski inequality in L p{2 pDq (see [START_REF] Adams | Pure and Applied Mathematics[END_REF], Theorem 2.13 p. 28) then yields

`8 ÿ n"0 λ n }B α ψ n } 2 p " `8 ÿ n"0 λ n }B α ψ 2 n } p{2 ď › › › › `8 ÿ n"0 λ n B α ψ 2 n › › › › p{2 " }v α } p{2 " }σ α } 2 p ă `8, (3.28)
so that the series

ř n λ n }B α ψ n } 2 p converges.
From this, it is then readily checked that the equality B α,α k " ř n λ n B α ψ n b B α ψ n holds in L p pD ˆDq, which is then a nuclear decomposition of B α,α k. Recall however that in spaces of cotype 2, it is not sufficient to require that a self-adjoint, nonnegative operator be nuclear for it to be a Gaussian covariance operator (Proposition 2.8). When p ą 2 though, the following proposition shows that this property does not hold anymore: distributing derivatives on nuclear decompositions does not, in general, yield nuclear decompositions, even if nuclear decompositions of the differentiated kernel exist. Proposition 3.5. Let p ą 2 and D " p0, 1q. There exists an explicit covariance function k : D ˆD Ñ R of a measurable Gaussian process with the following properties.

piq There exists a countable set N Ă D ˆD such that for all px, yq P pD ˆDqzN , the classical derivative B x B y kpx, yq exists. Moreover, the map px, yq Þ Ñ pB x B y kqpx, yq can be extended over D ˆD as a function (still denoted B x B y k) which is the covariance function of a measurable Gaussian process. piiq ş 1 0 kpx, xq p{2 dx ă `8 and ş 1 0 pB x B y kqpx, xq p{2 dx ă `8, hence nuclear decompositions of both k and B x B y k exist in L p pD ˆDq, from Propositions 2.7 and 2.9 (note that those two integral equations imply that both k and B x B y k lie in L p pD ˆDq, from the inequality |hpx, yq| p ď hpx, xq p{2 hpy, yq p{2 if h is positive definite, so that Proposition 2.9 indeed applies).

piiiq There exists pψ n q Ă W 1,p pDq such that k " ř 8 n"0 ψ n bψ n in L p pDˆDq, ř `8 n"0 }ψ n } 2 p ă `8 but ř `8 n"0 }ψ 1 n } 2 p " `8. Hence pψ n q provides a nuclear decomposition of k, while pψ 1 n q does not for B x B y k (even though such decompositions exist!).

The proof of Proposition 3.5 is deferred to Section 6. Note that a general procedure for obtaining nuclear decompositions of Gaussian covariance operators (hence of suitably integrable covariance functions in the case of L p spaces), based on a gradual "orthonormalization" procedure, can be found in the proof of Theorem 3.5.10 of [START_REF] Bogachev | Gaussian measures[END_REF].

The following proposition deals with the apparent asymmetry in p between Points piiq and piiiq of Proposition 3.1. We recall that the nuclear norm νpT q is defined in equation (2.8). Contrarily to Proposition 3.1, we do not exclude p " 1.

Proposition 3.6. Let µ be a centered Gaussian measure over L p pDq, where 1 ď p ă `8. Let k P L p pD ˆDq be the kernel of its covariance operator pK µ " E k q, chosen such that k is also the covariance function of a measurable Gaussian process pU pxqq xPD , from Proposition 2.9. Define σpxq " kpx, xq 1{2 and set C p " 2 p{2 Γppp `1q{2q{ ? π pC p " Er|X| p s where X " N p0, 1qq. Then the following bounds hold.

• if 1 ď p ă 2, there exists a symmetric, nonnegative and trace class operator S over L 2 pDq and a bounded operator A : L 2 pDq Ñ L p pDq such that E k " ASA ˚. Moreover,

νpE k q ď inf A,S s.t. E k "ASA ˚}A} 2 νpSq ď }σ} 2 p ď C ´2{p p inf A,S s.t. E k "ASA ˚}A} 2 νpSq.
(3.29)

• if 2 ď p ă `8, then E k is symmetric, nonnegative and nuclear, and

C ´2{p p νpE k q ď }σ} 2 p ď νpE k q. (3.30)
Observe that if p " 2 then C 2 " 1 and equation (3.30) yields }σ} 2 2 " νpE k q " TrpE k q. It is expected that the nuclear norm of E k cannot directly appear on the right hand side of equation (3.29), as not all nuclear operators are Gaussian covariance operators when 1 ď p ă 2 (Proposition 2.8). Proposition 3.6 in fact suggests that for general Banach spaces X of cotype 2, the following map defined over the set of Gaussian covariance operators B :

X ˚Ñ X, B Þ Ñ inf A,S s.t. B"ASA ˚}A} 2 νpSq, (3.31)
is the natural measurement of the "size" of such operators. When X is of type 2, this would be the case for the nuclear norm B Þ Ñ νpBq.

Remark 3.7. Proposition 3.6 is interesting from an application point of view because it states that the operator norms appearing in this proposition, as well as the L p norm of the standard deviation function σ, are suitable quantities for quantitatively controlling the L p norm of the sample paths of the underlying Gaussian process. Explicitly, we have the following L p control in expectation: Er}U } p p s " C p }σ} p p (see equation (3.33)). Applying this fact recursively, we obtain that the W m,p -Sobolev norm of the sample paths of the Gaussian process in question is controlled as follow, denoting σ α pxq " B α,α kpx, xq 1{2 (choosing a representative of B α,α k which is the covariance of a measurable Gaussian process)

E " }U } p W m,p ‰ " C p ÿ |α|ďm }σ α } p p . (3.32)
If such a control cannot be obtained, then it means that the sample paths of U do not lie in W m,p pDq in the first place. Finally, we have the following asymptotic behaviour of the constant when p Ñ `8 : C ´2{p p " expp1q{pp ´1q.

Proof. (Proposition 3.6) We begin with the following general fact concerning the measurable Gaussian process pU pxqq xPD , observing from Fubini's theorem that

Er}U } p p s " E " ż D |U pxq| p dx ȷ " ż D Er|U pxq| p sdx " ż D C p σpxq p dx " C p }σ} p p , (3.33)
where C p " 2 p{2 Γppp `1q{2q{ ? π. Indeed, given X " N p0, σ 2 q, then Er|X| p s " C p σ p . Suppose now that 1 ď p ă 2. Let µ 0 be a Gaussian measure on L 2 pDq and A : L 2 pDq Ñ L p pDq a bounded operator such that µ " µ 0A (pushforward of µ 0 through A, see Section 2.1.1) and S the trace class covariance operator associated to µ 0 (see Proposition 2.8). Recall also that from Proposition 2.9, µ " P U . Then (explanation below), . One also verifies that A ˚:

C p }σ} p p " Er}U } p p s " ż Ω }U ω } p p Ppdωq " ż L p pDq }f } p p µpdf q (3.
L q pDq Ñ L 2 pDq is given by A ˚f pxq " f pxqσpxq 1´p{2 , with }A} " }A ˚}. Introduce the functions k 0 px, yq :" kpx, yqσpxq p{2´1 σpyq p{2´1 , and σ 0 pxq " k 0 px, xq 1{2 ; k 0 is the covariance function of the measurable Gaussian process V pxq :" σpxq p{2´1 U pxq, and verifies

}σ 0 } 2 2 " ż D σ 0 pxq 2 dx " ż D k 0 px, xqdx " ż D σpxq p dx " }σ} p p ă `8. (3.44)
Therefore E k0 , the integral operator over L 2 pDq associated to k 0 , is trace class (Proposition 3.1piiq). Observe also that k " pA b Aqk 0 which also yields that

E k " AE k0 A ˚. Thus, inf A,S s.t. E k "ASA ˚}A} 2 νpSq ď }A} 2 νpE k0 q ď }σ} 2´p p }σ} p p " }σ} 2 p . (3.45)
Combining equations (3.39) and (3.45) yields the desired result of equation (3.29). Suppose now that p ě 2. Recall that µ " P U . We successively use the transfer theorem for pushforward measure integration, Jensen's inequality for probability measures pp{2 ě 1q and the nuclear norm estimate from [START_REF] Linde | Characterization of certain classes of Banach spaces by properties of Gaussian measures[END_REF], Theorem 3:

Er}U } p p s " ż Ω }U ω } p p Ppdωq " ż L p pDq }f } p p µpdf q " ż L p pDq }f } 2ˆp{2 p µpdf q (3.46) ě ˆżL p pDq }f } 2 p µpdf q ˙p{2 ě νpE k q p{2 , (3.47)
which together with equation (3.33) yields }σ} 2 p ě C ´2{p p νpE k q. We now prove the last remaining inequality, i.e. }σ} 2 p ď νpE k q. For this, consider kpx, yq " ř n µ n ψ n pxqϕ n pyq, a nuclear representation of k in L p pD ˆDq, with }ψ n } p " }ϕ n } p " 1 and ř n |µ n | ă `8. Denote by v the function v : x Þ Ñ ř `8 n"0 µ n ψ n pxqϕ n pxq. Minkowski's inequality in L p{2 pDq shows that x Þ Ñ ř `8 n"0 |µ n ψ n pxqϕ n pxq| is finite a.e. and in fact that v P L p{2 pDq:

}v} p{2 " › › › › `8 ÿ n"0 µ n ψ n ϕ n › › › › p{2 ď `8 ÿ n"0 |µ n | ˆ› › ψ n ϕ n › › p{2 ď `8 ÿ n"0 |µ n | ˆ}ψ n } p }ϕ n } p " `8 ÿ n"0 |µ n | ă `8. (3.48)
In equation (3.48) above, we used used the Cauchy-Schwarz inequality on }ϕ n ψ n } p{2 . From the nuclear decomposition of k, it is very tempting to write }σ} 2 p " }v} p{2 , but unfortunately the diagonal of D ˆD has a null Lebesgue measure. This equality turns out to be true but this fact is non trivial and deferred to Lemma 3.8 below. From this lemma and equation (3.48) which holds whatever the nuclear decomposition of E k , taking the infimum over all nuclear representations of E k in equation (3.48) yields }σ} 2 p ď νpE k q. This finishes the proof.

The next lemma, which was key in the proof of equation (3.30), states that evaluating the L p{2 -norm of the diagonal of a nuclear representation of a Gaussian covariance operator K in L p pDq, p ě 2, yields the same result as evaluating L p{2 -norm of the diagonal of the covariance function k of any measurable Gaussian process pU pxqq xPD such that E k " K. This fact is not obvious at all, as the diagonal of D ˆD has null Lebesgue measure and different representatives of k in L p pD ˆDq have no reason a priori to agree on sets of null measure. However, the assumptions that the representation is nuclear and that U is measurable turn out to be strong enough to yield the desired conclusion. The proof ideas for this result should largely be credited to [START_REF] Brislawn | Kernels of trace class operators[END_REF]; we generalized them in a straightforward fashion from L 2 pDq to L p pDq and applied them to the Gaussian process pU pxqq xPD of Proposition 3.6. They are based on the Hardy-Littlewood maximal inequality. Lemma 3.8. Let 2 ď p ă `8, D Ă R d be an open set and pU pxqq xPD " GP p0, kq be a measurable Gaussian process whose sample paths lie in L p pDq a.s.. From Propositions 2.9 and 2.7, E k : L q pDq Ñ L p pDq is nuclear and there exists sequences pµ n q Ă R, pψ n q, pϕ n q Ă L p pDq such that k " ř n µ n ψ n b ϕ n in L p pD ˆDq, with }ψ n } p " }ϕ n } p " 1 and

ř n |µ n | ă `8. Then x Þ Ñ ř 8 n"0 |µ n ψ n pxqϕ n pxq| is finite a.e. and v : x Þ Ñ ř 8 n"0 µ n ψ n pxqϕ n pxq is nonnegative a.e.. Moreover, }σ} p p " ż D kpx, xq p{2 dx " ż D ˆ`8 ÿ n"0 µ n ψ n pxqϕ n pxq ˙p{2 dx " }v} p{2 p{2 . (3.49)
A remarkable consequence of this result is that the L p{2 -norm of the diagonal of a nuclear representation of E k " ř n µ n ψ n b ϕ n is invariant w.r.t. said nuclear decomposition, while its finiteness fully characterizes the nuclearity of E k (Proposition 3.6piiq); the same invariance property does not hold for ř n |µ n |, hence the need to define the nuclear norm of E k as the infimum over such quantities.

Proof of Lemma 3.8. We first prove the statement when D " R d . We begin with some definitions and observations. For r ą 0, denote C r :" r´r, rs d and C r pxq :" x `Cr . For f P L p pR d q (resp. g P L p pR d ˆRd q), denote its average over C r pxq (resp. C r pxq ˆCr pxq) as

A pdq r f pxq :" 1 |C r | ż Crpxq f ptqdt, A p2dq r gpxq :" 1 |C r | 2 ż Crpxq ż Crpxq gps, tqdsdt (3.50)
The functions A pdq r f and A p2dq r g are defined pointwise and continuous. The point of averaging over cubes rather than balls is that we have A M pdq f (resp. M p2dq g) is measurable, nonnegative and defined pointwise over R d (resp. R d ˆRd ). For all x P R d , we obviously have the pointwise majoration

|A pdq r f pxq| ď M pdq f pxq, (3.51)
and likewise for M p2dq g. A key point for us will be the Hardy-Littlewood maximal theorem ( [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF], Theorem 1 p. 5), which states that there exists a constant S p ą 0 such that for all f P L p pR d q,

}M pdq f } p ď S p }f } p . (3.52)
This theorem allows a first general observation, given f P L p pR d q. Indeed, the Lebesgue differentiation theorem ( [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF], Corollary 1 p. 5) states that A Lebesgue's dominated convergence theorem in L p pDq yields that we also have the convergence

}A pdq r f ´f } L p pR d q Ý ÝÝ Ñ rÑ0 0. (3.54)
We will also use that the nonlinear operator M is submutliplicative and subadditive:

M p2dq pψ
› › › › `8 ÿ n"0 |µ n |M pdq ψ n M pdq ϕ n › › › › p{2 ď `8 ÿ n"0 |µ n | ˆ› › M pdq ψ n M pdq ϕ n › › p{2 ď `8 ÿ n"0 |µ n | ˆ› › M pdq ψ n › › p › › M pdq ϕ n › › p (3.60) ď `8 ÿ n"0 |µ n | ˆS2 p }ψ n } p }ϕ n } p " S 2 p `8 ÿ n"0 |µ n | ă `8. (3.61)
We used the Cauchy-Schwarz inequality in equation (3.60). Choose now a conull set T Ă R d , on which the Lebesgue differentiation theorem applies for all ψ n and ϕ n , and on which x Þ Ñ ř n |µ n |M pdq ψ n pxqM pdq ϕ n pxq is finite (such a set exists from the finiteness of its L p{2 -norm). For all x P T , the Lebesgue dominated convergence theorem for the discrete measure ř nPN0 δ n (using the domination (3.59)) yields the equality (3.58).

We now focus on the Gaussian process pU pxqq xPR d . Since its sample paths U ω lie in L p pR d q almost surely, equation (3.54) yields that for almost every ω P Ω,

}A pdq r U ω ´Uω } p p Ý ÝÝ Ñ rÑ0 0. (3.62)
We also have that for every such ω P Ω and r ą 0,

}A pdq r U ω ´Uω } p ď }A pdq r U ω } p `}U ω } p ď }M pdq U ω } p `}U ω } p ď pS p `1q}U ω } p , (3.63)
and from Fubini's theorem, the right-hand side of equation (3.63) lies in L p pPq:

Erω Þ Ñ }U ω } p p s " Er}U } p p s " ż R d
Er|U pxq| p sdx " C p }σ} p p ă `8. In particular, using the reverse triangle inequality on the norm V Þ Ñ Er}V } p p s 1{p , we have

E " }A pdq r U } p p ‰ Ý ÝÝ Ñ rÑ0 E " }U } p p ‰ " C p }σ} p p . (3.66)
We then wish to use equations (3.66) and (3.58) to prove the desired result. For this, observe that from the linearity of the operator A as for Lemma 2.5, and the expression of its covariance function follows from the measurability of U and Fubini's theorem. Fubini's theorem and the fact that Er|X| p s " C p s p if X " N p0, s 2 q then lead to

Er}A pdq r U } p p s " ż R d Er|A pdq r U pxq| p sdx " C p ż R d ´Ap2dq r kpx, xq ¯p{2
dx.

(3.68)

We will finally apply Lebesgue's dominated convergence theorem on equation (3.68) when r goes to zero, using the limit given in equation (3.58)

¯p{2 dx " C p ż R d ˆ`8 ÿ n"0 µ n ψ n pxqϕ n pxq ˙p{2 dx,
which, together with equation (3.66), finishes the proof.

To deal with the general case where D is only an open subset of R d , extend any function f P L p pDq to a function f P L p pR d q by setting f pxq " f pxq if x P D, f pxq " 0 elsewhere. f remains measurable, and all the arguments and results stated above are preserved.

4 Sobolev regularity for Gaussian processes : the Hilbert space case, p " 2

In the case p " 2, we provide an alternative proof of the integral and spectral criteria of Proposition 3.1, based on the study of the "ellipsoids" of Hilbert spaces (see Section 4.2). These geometrical objects are well understood in relation with Gaussian processes (see [START_REF] Dudley | The sizes of compact subsets of Hilbert space and continuity of Gaussian processes[END_REF] or [START_REF] Talagrand | Upper and lower bounds for stochastic processes[END_REF], Section 2.5). Compared with the general case p P p1, `8q, we draw additional links between the different Mercer decompositions of the kernels B α,α k, the evaluation of the trace of E α k and the Hilbert-Schmidt nature of the imbedding of the reproducing kernel Hilbert space (see Section 4.1 below) associated to k in H m pDq.

Reproducing Kernel Hilbert Spaces (RKHS, [5])

Consider a general set D and a positive definite function k : D ˆD Ñ R, i.e. such that given any n P N and px 1 , ..., x n q P D n , the matrix pkpx i , x j qq 1ďi,jďn is nonnegative definite. One can then build a Hilbert space H k of functions defined over D which contains the functions kpx, ¨q, x P D and verifies the reproducing identities xkpx, ¨q, kpx 1 , ¨qy H k " kpx, x 1 q @x,

x 1 P D, (4.1) xkpx, ¨q, f y H k " f pxq @x P D, @f P H k . (4.2)
H k is the RKHS of k. This space is exactly the set of functions of the form f pxq " ř `8 i"1 a i kpx i , xq such that }f } 2 H k " ř `8 i,j"1 a i a j kpx i , x j q ă `8. If for all x P D, kpx, ¨q is measurable, then H k only contains measurable functions. One may then consider imbedding H k in some Sobolev space H m pDq. Recall that in H m pDq, functions are equal up to a set of Lebesgue measure zero. If such an imbedding i : H k Ñ H m pDq is well-defined (i.e. if f P H k then its weak derivatives B α f exist and lie in L 2 pDq for all |α| ď m), we will sometimes use the same notation for f P H k and its equivalence class f P H m pDq; strictly speaking, the latter should be denoted ipf q. It may then happen that i is not injective, as with the RKHS associated to the Kronecker delta kpx, x 1 q " δ x,x 1 (in this case, we even have ipH k q " t0uq.

Remark 4.1. In Proposition 4.4, we will be interested in the Hilbert-Schmidt nature of the imbedding i. However, it may happen that H k is not separable, such as with the RKHS associated to the Kronecker delta δ x,x 1 . This results in additional care required for defining the notion of Hilbert Schmidt operators, as the definition from Section 2.1.3piiq cannot hold. Still, this case is dealt with in Proposition 4.4pivq. See [START_REF] Owhadi | Separability of reproducing kernel spaces[END_REF] and [START_REF] Bogachev | Gaussian measures[END_REF], Remark 3.2.9 p. 103 for discussions on non separable RKHS.

Ellipsoids of Hilbert spaces and canonical Gaussian processes [21]

Let pH; x, y H q be a separable Hilbert space. We introduce pV x q xPH the canonical Gaussian process of H, defined as the centered Gaussian process whose covariance function is the inner product of H :

ErV x V y s " xx, yy H . (4.3) A subset K of H is said to be Gaussian bounded (GB) if Ppsup xPK |V x | ă `8q " 1. (4.4)
The GB property was first introduced for studying the compact sets of Hilbert spaces, see [START_REF] Dudley | The sizes of compact subsets of Hilbert space and continuity of Gaussian processes[END_REF] on that topic. In equation (i) If K is a GB-set, then its closed, convex, symmetric hull is a GB-set.

(ii) The closure of a GB-set is compact.

Given a self-adjoint compact operator T : H Ñ H, introduce a basis of eigenvectors x n and its real eigenvalues λ n , λ n Ñ 0. The image of the closed unit ball of H, B " B H p0, 1q is the following "ellipsoid" ( [START_REF] Dudley | The sizes of compact subsets of Hilbert space and continuity of Gaussian processes[END_REF], p. 312)

T pBq " " ÿ λną0 a n x n s.t. ÿ λną0 a 2 n {λ 2 n ď 1 * . (4.5)
The main result we will use is the following. (ii) (Spectral structure) For all |α| ď m, the distributional derivative B α,α k lies in L 2 pD ˆDq and the associated integral operator

E α k f pxq " ż D B α,
λ n ϕ n pxqϕ n pyq in L 2 pD ˆDq, (4.8)
where pλ n q is a nonnegative sequence and pϕ n q is an orthonormal basis of L 2 pDq. Moreover, for all |α| ď m and for all n P N 0 such that λ n ‰ 0, B α ϕ n P L 2 pDq, B α,α k P L 2 pD ˆDq and the following facts hold:

`8 ÿ n"0 λ n }B α ϕ n } 2 2 ă `8, B α,α kpx, yq " `8 ÿ n"0 λ n B α ϕ n pxqB α ϕ n pyq in L 2 pD ˆDq. (4.9)
In this case, E α k in equation (4.6) is well-defined and trace class, with TrpE α k q " ř `8 n"0 λ n }B α ϕ n } 2 2 . (iv) (imbedding of the RKHS) H k Ă H m pDq, the corresponding natural imbedding i : H k Ñ H m pDq is continuous and ii ˚: H m pDq Ñ H m pDq is trace class. Equivalently, kerpiq K endowed with the topology of H k is a separable Hilbert space and j :" i | kerpiq K : kerpiq K Ñ H m pDq is Hilbert-Schmidt. Moreover, the Hilbert-Schmidt norm of j (see Section 2.1.3piiq and piiiq) is given by

}j} 2 HS " Trpii ˚q " ÿ |α|ďm TrpE α k q. ( 4.10)
Note that Point pivq above agrees with the definition of Hilbert-Schmidt operators on general, non necessarily separable Hilbert spaces ( [6], p. 367). Before proving this result, we discuss Proposition 4.4 in relation with previous results from the literature. First, point pivq is not without reminding Driscoll's theorem ( [START_REF] Kanagawa | Gaussian processes and kernel methods: A review on connections and equivalences[END_REF], Theorem 4.9) which is well-known in the machine learning/RKHS community; this theorem states the following. Let k and r be two positive definite functions defined over D, and let U " GP p0, kq. Suppose that H k Ă H r with a Hilbert-Schmidt imbedding, then the sample paths of U lie in H r almost surely.

Second, Proposition 4.4 and equation (4.7) in particular, is a generalization of Theorem 1 from [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF] in the case of Gaussian processes; By removing the assumption in [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF] that the covariance function be continuous on its diagonal as well as its symmetric cross derivatives, the sufficient condition derived in [START_REF] Scheuerer | Regularity of the sample paths of a general second order random field[END_REF] becomes also necessary. Finally, Proposition 4.4 shows that if p " 2, then in the nuclear decomposition of E α k (see Proposition 3.1piiiqq one can choose λ α n " λ n and ψ α n " B α ψ n . It is not obvious that this should hold when p ă 2 and it in fact fails when p ą 2 (see Remark 3.4 and Proposition 3.5).

Example 4.5 (Hilbert-Schmidt imbeddings of Sobolev spaces). Proposition 4.4 can be compared with the results found in [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF] and its Corollary 4.5 in particular. This corollary states that if D Ă R d is sufficiently smooth, if H k Ă H t pDq with a continuous imbedding and if t ą d{2, then the sample paths of the centered Gaussian process with covariance function k lie in H m pDq for all real number m P r0, t ´d{2q. For example, this holds when k is a Matérn covariance function of order t ´d{2; its RKHS is then exactly H t pDq ( [START_REF] Steinwart | Convergence types and rates in generic Karhunen-Loeve expansions with applications to sample path properties[END_REF], Example 4.8).

In the particular case where in addition m is an integer, we recover this result from Proposition 4.4. Indeed, it is known that when m P p0, t ´d{2q, the imbedding of H t pDq in H m pDq is Hilbert-Schmidt. When the involved indexes are nonnegative integers, this is known as Maurin's theorem ( [START_REF] Adams | Pure and Applied Mathematics[END_REF], Theorem 6.61, p. 202). Maurin's theorem is generalized to fractional exponents in [START_REF] Triebel | Über die Approximationszahlen der Einbettungsoperatoren JpB r p,q pΩq Ñ B r 1 p 1 ,q 1 pSqq[END_REF], Folgerung 1 p. 310 (in German) or [START_REF] Lai | Noyaux d'Agmon[END_REF], Proposition 7.1 (in French). If H k Ă H t pDq with a continuous imbedding, then the inclusion map of H k in H m pDq is Hilbert-Schmidt for all m P r0, t ´d{2q X N 0 . From Proposition 4.4pivq, we obtain that the sample paths of the corresponding Gaussian process indeed lie in H m pDq.

However, not all RKHS that are subspaces of H m pDq with a Hilbert-Schmidt imbedding are contained in some H t pDq with t ą m `d{2, as the following trivial example shows. Fix any ε ą 0 and consider the rank one kernel kpx, x 1 q " f pxqf px 1 q where f is chosen such that f P H m pDq and f R H m`ε pDq (choose a representative of f in L 2 pDq so that f is a function in the classical sense). Then H k " Spanpf q and the imbedding of H k in H m pDq is Hilbert-Schmidt since it is rank one; but H k Ć H m`ε pDq. Proposition 4.4 yields that the associated trivial Gaussian process U pxqpωq " ξpωqf pxq where ξ " N p0, 1q has its sample paths in H m pDq (it was obvious in the first place).

Example 4.6 (One dimensional case). We build a covariance function which is not pointwise differentiable at any pq, q 1 q P Q ˆQ, and such that the corresponding Gaussian process has its sample paths in H 1 pRq. Let h a pxq :" maxp0, 1 ´|x ´a|q be the hat function centered around a P R. It lies in H 1 pRq but it is not differentiable at x " a, a ´1 and a `1. Let pq n q be an enumeration of Q. Then the following positive definite function over R kpx, x 1 q :"

`8 ÿ n"0 1 2 n h qn pxqh qn px 1 q, ( 4.11)
is not differentiable in the classical sense at each point px, x 1 q of the form pq n , q m q, but the map ii ˚, with i : H k Ñ H 1 pRq the canonical imbedding, is trace-class (use equations (4.9) and (4.10)):

Trpii ˚q " TrpE k q `TrpE 1 k q (4.12) ď `8 ÿ n"0 1 2 n }h qn } 2 2 ``8 ÿ n"0 1 2 n }h 1 qn } 2 2 (4.13) ď `8 ÿ n"0 1 2 n ``8 ÿ n"0 1 2 n ˆ22 " 10. (4.14)
Before proving Proposition 4.4, we will require a number of lemmas concerning the Mercer decomposition of Hilbert-Schmidt operators over L 2 pDq. They are proved in Section 6. Suppose that it verifies the estimate

@φ, ψ P E 2 , |b α pφ, ψq| ď C α }φ} 2 }ψ} 2 , (4.16)
where E 2 is the set given in Lemma 2.2. Then b α can be extended to a continuous bilinear form over L 2 pDq and there exists a unique bounded, self-adjoint and nonnegative operator F α k :

L 2 pDq ÝÑ L 2 pDq such that @φ, ψ P C 8 c pDq, b α pφ, ψq " xF α k φ, ψy L 2 pDq . (4.17)
Lemma 4.8. Let k P L 2 pD ˆDq be a positive definite function and α a multi-index. Suppose that the weak derivative B α,α k exists and lies in L 2 pD ˆDq. Then the bilinear form b α from equation (4.15) verifies the estimate (4.16) with C α " }B α,α k} 2 . Introduce F α k , the bounded operator from Lemma 4.7. Introduce also E α k , the integral operator defined on L 2 pDq associated to B α,α k,

pE α k f qpxq " ż D B α,α kpx, yqf pyqdy. (4.18)
Then E α k " F α k and E α k is self-adjoint and nonnegative. Lemma 4.9. Let k P L 2 pD ˆDq be a positive definite function and E k be its associated nonnegative definite Hilbert-Schmidt operator. Let kpx, yq "

`8 ÿ i"0 λ i ϕ i pxqϕ i pyq (4.19)
be a symmetric, nonnegative expansion of k in L 2 pD ˆDq where pλ i q is a nonnegative sequence decreasing to 0; it may or may not be its Mercer expansion (i.e. pϕ i q may or may not be an orthonormal basis of L 2 pDq; they are still assumed to be elements of L 2 pDq though).

(i) If the partial mixed weak derivative B α,α k exists and lies in L 2 pD ˆDq, then for all i P N 0 such that λ i ‰ 0, B α ϕ i P L 2 pDq.

(ii) Assume that for all i P N 0 such that λ i ‰ 0, B α ϕ i P L 2 pDq, and that the bilinear form b α from equation (4.15) verifies the estimate (4.16). Let F α k be the bounded operator from Lemma 4.7. Then

TrpF α k q " `8 ÿ i"0 λ i }B α ϕ i } 2 L 2 pDq , (4.20)
whether these quantities are finite or not. If in equation (4.20), either one of them is finite, then the series of functions

ř iPN0 λ i B α ϕ i pxqB α ϕ i pyq is norm convergent in L 2 pD ˆDq (i.e. ř iPN0 λ i }B α ϕ i b B α ϕ i } L 2 ă `8), B α,α k lies in L 2
pD ˆDq and we have the following equality:

B α,α kpx, yq " `8 ÿ i"0 λ i B α ϕ i pxqB α ϕ i pyq in L 2 pD ˆDq. (4.21)
Moreover, F α k is the (Hilbert-Schmidt) integral operator with kernel B α,α k, i.e. F α k " E α k " E B α,α k . Finally, equation (4.21) then holds for asymmetric derivatives, as for all |α|, |β| ď m, we also have

ř iPN λ i }B β ϕ i b B α ϕ i } L 2 ă `8.
We can now prove Proposition 4.4.

Proof. (Proposition 4.4) We successively prove piiq ùñ piq, piq ùñ piiq, piiq ùñ piiiq, piiiq ùñ piiq, piiiq ùñ pivq and pivq ùñ piiiq. Before all things, the assumptions and Lemma 2.5 show that the sample paths of U lie in L 1 loc pDq, that the random variable given by the formula

U α φ : Ω Q ω Þ ÝÑ p´1q |α| ż D U pxqpωqB α φpxqdx (4.22)
is well defined and that pU α φ q φPF2 is a Gaussian sequence (see equation (2.22) for the definition of F 2 ). piiq ùñ piq : From Lemma 4.8, E α k is a self-adjoint, nonnegative Hilbert-Schmidt operator; it is actually trace-class by assumption. We can thus define A α :" a E α k , which is a Hilbert-Schmidt, self-adjoint, nonnegative operator. From Proposition 4.3, A α pBq is a GB-set (B is the closed unit ball of L 2 pDq). Therefore, using the canonical Gaussian process of L 2 pDq,

Pp sup ψPAαpBq |V ψ | ă `8q " 1, (4.23)
which, since F 2 Ă B, yields in particular that

Pp sup φPF2 |V Aαpφq | ă `8q " 1. (4.24)
We now observe that the two Gaussian sequences pV Aαpφq q φPF2 and pU α φ q φPF2 have the same finite dimensional marginals. Indeed, they are both centered Gaussian sequences with the same covariance: Since E α k is self-adjoint and nonnegative, we can introduce its square root A α :" a E α k , which is also a bounded, self-adjoint and nonnegative operator. As in equation (4.27), we can introduce pV Aαpφq q φPF2 and observe that pV Aαpφq q φPF2 and pU α φ q φPF2 have the same law. Thus,

ErV Aαpφq V Aαpψq s " xA α pφq, A α pψqy L 2 " xA 2 α pφq, ψy L 2 " xE α k φ, ψy L 2 . (4.25) ErU α φ U α ψ s " E " ż D U pxqB α φpxqdx ż D U pyqB α ψpyqdy ȷ " ż DˆD kpx, yqB α φpxqB α ψpyqdxdy " ż DˆD B α,α kpx, yqφpxqψpyqdxdy " xE α k φ, ψy L 2 . ( 4
Pp sup φPF2 |V Aαpφq | ă `8q " Pp sup φPF2 |U α φ | ă `8q " 1. (4.33)
Therefore, A α pF 2 q is a GB-set. From Proposition 4.2(ii), ConvpA α pF 2 qq is compact. One then checks by elementary considerations that ConvpA α pF 2 qq " A α pBq, where B is the unit ball of L 2 pDq. This shows that A α is a compact operator. But from Proposition 4.2(i), A α pBq " ConvpA α pF 2 qq is also a GB-set. From Proposition 4.3, A α is Hilbert-Schmidt and E α k is traceclass. In particular, E α k is a Hilbert-Schmidt operator with a kernel k α that lies in L 2 pD ˆDq. Moreover, @φ, ψ P For the existence of a representative k α with the desired properties, we refer to the previous Proposition 3.1piiq. Finally, the equality TrpE k q " ş kpx, xqdx when k is also the covariance function of a measurable Gaussian process is e.g. given in the proof of Proposition 3.11.15 of [START_REF] Bogachev | Gaussian measures[END_REF], p. 150. This finishes to prove piiq. piiq ùñ piiiq: If piiq, then from Lemma 4.8 (using the notations from Lemmas 4.7 to 4.9), F α k " E α k . From Lemma 4.9piq, the functions pϕ n q lie in H m pDq and from Lemma 4.9piiq, since E α k " F α k is assumed trace class, equation (4.9) holds, as well as the trace formula.

piiiq ùñ piiq: if ř `8 n"0 λ n }B α ϕ n } 2 2 ă `8, then from Lemma 4.9piiq, F α k " E α k , and still from Lemma 4.9piiq, TrpE α k q ă `8, i.e. E α k is trace class. As previously, for the existence of a representative k α with the desired properties, we refer to the equivalence between Points piiq and piiiq from Proposition 3.1. As previously, the trace formula is given in [START_REF] Bogachev | Gaussian measures[END_REF] Furthermore, using the bilinearity of x¨, ¨yH k , we have that }∆ py1,...,y ℓ q kpx, ¨q} 2 H k " rp∆ py1,...,y ℓ q b ∆ py1,...,y ℓ q qkspx, xq.

We then deduce that (explanation below)

@f P H k , }∆ py1,...,y ℓ q f } 2 L 2 pD0q " ż D0 `∆py1,...,y ℓ q f ˘pxq 2 dx ď }f } 2 H k ż D0
rp∆ py1,...,y ℓ q b ∆ py1,...,y ℓ q qkspx, xqdx (4.40)

ď }f } 2 H k `8 ÿ i"1 λ i ż D0
p∆ py1,...,y ℓ q ϕ i qpxq 2 dx (4.41)

ď }f } 2 H k `8 ÿ i"1 λ i ´}ϕ i } 2 H m |y 1 | 2 ¨¨¨|y ℓ | 2 ¯(4.42) ď }f } 2 H k ˆÿ |α|ďm TrpE α k q ˙`|y 1 | 2 ¨¨¨|y ℓ | 2 ˘. ( 4
@f P H k , }δ α h f } 2 L 2 pD0q ď }f } 2 H k ˆÿ |α|ďm TrpE α k q ˙. (4.44)
pivq ùñ piiiq: by assumption, ii ˚is a compact self-adjoint nonnegative operator acting on the Hilbert space H m pDq. There exists a decreasing nonnegative sequence pµ j q jPN and an orthonormal basis of eigenvectors of ii ˚, pψ j q jPN such that for all f P H m pDq,

ii ˚pf q " `8 ÿ j"1 µ j xψ j , f y H m ψ j in H m pDq. (4.51)
Since ii ˚is assumed trace class,

ÿ |α|ďm `8 ÿ j"1 µ j }B α ψ j } 2 L 2 " `8 ÿ j"1 µ j }ψ j } 2 H m " `8 ÿ j"1 µ j ă `8. (4.52)
We now show that the following equality holds in L 2 pD ˆDq:

kpx, yq " `8 ÿ j"1 µ j ψ j pxqψ j pyq. (4.53)

In conjunction with equation (4.52), this equation will allow us to use Lemma 4.9piiq, which will imply the point piiiq. First, one easily shows that ř `8 j"1 µ j ψ j b ψ j , the right-hand side of equation ( 4.53), is indeed in L 2 pD ˆDq (e.g. use that ř j µ j ă `8). The upcoming equation (4.63) will then show that k is indeed in L 2 pD ˆDq. Now, decompose ipk x q P H m pDq on the basis pψ j q jPN , given any x P D: ipk x q " `8 ÿ j"1 xψ j , ipk x qy H m ψ j in H m pDq.

(4.54)

In equation (4.54), the scalar xψ j , ipk x qy H m is obtained through the reproducing formula (4.2):

xψ j , ipk x qy H m " xi ˚pψ j q, k x y H k " i ˚pψ j qpxq. (4.55)

Moreover, ψ j is an eigenvector of ii ˚: µ j ψ j " ii ˚pψ j q in H m pDq. In particular, }µ j ψ j ´ii ˚pψ j q} L 2 pDq " 0. (4.56) But the pointwise defined function i ˚pψ j q is a representative of ii ˚pψ j q in H m pDq, since i is the imbedding of H k in H m pDq. Setting S " ř j µ j " Trpii ˚q, one then has (explanation below)

› › › › k ´`8 ÿ j"1 µ j ψ j b ψ j › › › › 2 L 2 pDˆDq " ż DˆD ´kpx, yq ´`8 ÿ j"1
µ j ψ j pxqψ j pyq ¯2dxdy µ j ψ j pyq `µj ´1i ˚pψ j q ´ψj pxq ˘˙2 dydx (4.60)

The following directions are interesting for generalizing the results presented here. First, similar spectral/integral criteria should be obtained for fractional Sobolev and Besov spaces. Second, similar results should be sought to tackle the limit cases p " 1 and p " `8. Linked to the case p " 1, results should be sought for the space of functions of bounded variations ( [START_REF] Brézis | Functional analysis, Sobolev spaces and partial differential equations[END_REF], p. 269), which are important in many problems related to physics. In particular, those spaces are adapted to the study of nonlinear hyperbolic PDEs, where shocks (discontinuities in the solution) may appear and solutions may only be understood in the weak or distributional sense ( [START_REF] Serre | Systems of conservation laws[END_REF], Lemma 2.2.1 and Proposition 2.3.6).

Ackowledgements This research was supported by SHOM (Service Hydrographique et Océanographique de la Marine) project "Machine Learning Methods in Oceanography" no-20CP07. The author warmly thanks Pascal Noble, Olivier Roustant and Rémy Baraille for fruitful discussions, as well as the referees for their constructive feedbacks, which led to an improved version of the manuscript.

Proofs of intermediary results and lemmas

Proof. (Lemma 2.1) This proof follows exactly the lines of the proof of Proposition 9.3 from [START_REF] Brézis | Functional analysis, Sobolev spaces and partial differential equations[END_REF]. piq ðñ piiq: suppose that u P W m,p pDq, use the fact that the distributional derivative D α u is a regular distribution represented by a function that lies in L p pDq, denoted by B α u :

@φ P C 8 c pDq, ż D upxqB α φpxqdx " p´1q |α| ż D B α upxqφpxqdx. (6.1)
Hölder's inequality yields (2.12) with C α " }B α u} L p . Conversely, suppose that (2.12) holds and consider any |α| ď m. Since C 8 c pDq is dense in L q pDq (whatever the open set D, [START_REF] Adams | Pure and Applied Mathematics[END_REF], Section 2.30), equation (2.12) shows that the linear form L α : φ Þ ÝÑ p´1q |α| ş D upxqB α φpxqdx, φ P C 8 c pDq, can be extended to a continuous linear form over L q pDq. From Riesz' representation lemma, there exists v α P L p pDq such that L α pφq " xv α , φy L p ,L q for all φ P L q pDq. In particular, this is valid for all φ P C 8 c pDq, which shows that for all |α| ď m, B α u exists and is equal to v α . Thus u P W m,p pDq. Finally, Hölder's inequality and the density of C 8 c pDq in L q pDq yield }B α u} L p pDq " sup and h " ph 1 , ..., h d q P pR ˚qd be such that ř d i"1 α i h i ă distpD 0 , BDq. Recall that δ α h from equation (2.11) is a finite difference approximation of B α and from piiiq,

ˇˇˇż D δ α h upxqφpxqdx ˇˇˇď }φ} L q pD0q }δ α h u} L p pD0q ď C}φ} L q pDq . (6.2)
Note also that we have the discrete integration by parts formula since h is suitably chosen:

ż D δ α h upxqφpxqdx " ż D upxqpδ α h q ˚φpxqdx. (6.3)
Therefore, ˇˇˇż D upxqpδ α h q ˚φpxqdx ˇˇˇď C}φ} L q pDq . (6.4)

The Lebesgue dominated convergence theorem yields that the left hand side converges to ˇˇş D upxqB α φpxqdx ˇˇ. We therefore have piiq. piq ùñ piiiq: We will use recursively the fact that if O Ă R d is an open set and if f P W 1,p pOq X C 8 pOq, then for all open set O 0 Ă O and y P R d such that O 0 `ty Ă O for all t P r0, 1s, we have

}∆ y f } p L p pO0q " }τ y f ´f } p L p pO0q ď |y| p }∇f } p L p pOq " |y| p d ÿ j"1 }B xj f } p L p pOq . (6.5)
This is a slight generalization of equation ( 4) p. 268 in [START_REF] Brézis | Functional analysis, Sobolev spaces and partial differential equations[END_REF], found in the proof of Proposition 9.3 in [START_REF] Brézis | Functional analysis, Sobolev spaces and partial differential equations[END_REF]. The proof of equation (6.5) is an exact copy of the proof of equation ( 4) p. 268 in [START_REF] Brézis | Functional analysis, Sobolev spaces and partial differential equations[END_REF], which relies on writing the quantity upx `hq ´upxq as the integral of the derivative of the map t Þ Ñ upx `thq, t P r0, 1s. We first show equation (2.14) under the assumption that u P W m,p pDq X C 8 pDq. The Meyers-Serrin theorem ( [1], Theorem 3.17), which asserts the density of W m,p pDq X C 8 pDq in W m,p pDq for arbitrary open sets D Ă R d , will imply that equation (2.14) holds for general u P W m,p pDq. Let D 0 Ť D be an open set, ℓ ď m and py 1 , ..., y ℓ q P pR d q ℓ be such that ř ℓ i"1 |y i | ă distpD 0 , BDq. We begin by constructing a sequence of open sets pD k q 0ďkďℓ starting from D 0 such that D 0 Ă D 1 Ă ... Ă D ℓ Ť D, which additionally verify D k´1 `ty k Ă D k for all t P r0, 1s and all k P t1, ..., ℓu, with ř ℓ i"k |y i | ă distpD k´1 , BDq for all k P t1, ..., ℓu. This will enable us to use equation (6.5) recursively over k P t1, ..., ℓu, to obtain the desired finite difference control (2.14). We detail below this construction. We define this sequence recursively as follow, for all k P t1, ..., ℓu and starting from the open set D 0 , ´|z ´y| ´t|y k | ¯(6.7)

D k :" ď tPr0,1s pD k´1 `ty k q. ( 6
ě inf zPD k´1 ,yPBD |z ´y| ´|y k | " distpD k´1 , BDq ´|y k | ą ℓ ÿ i"k`1 |y i |. (6.8)
In particular, distpD ℓ , BDq ą 0. In equation (6.8), we used the assumption that ř ℓ i"k |y i | ă distpD k´1 , BDq. One also checks that D k Ť D, as follow. Given x P D k , write x " z `ty k for some z P D k´1 and t P r0, 1s. This yields

|x ´z| ď |y k | ď ℓ ÿ i"k |y i | ": r k .
(6.9)

Hence, x P Bpz, r k q. Moreover, Bpz, r k q Ă D, since z P D k´1 and r k ă distpD k´1 , BDq (indeed, one checks that Bpz, rq Ă D for all z P D and r ą 0 such that r ă distpz, BDq, e.g. by observing that BD Ă ty : |z ´y| ą ru, thus (taking the complement) Bpz, rq Ă D YIntpD c q, that z P D and that Bpz, rq is path connected). From equation (6.9), we obtain that D k Ă Ť xPD k´1 Bpx, r k q Ă D, with distp Ť xPD k´1 Bpx, r k q, BDq ě distpD k´1 , BDq ´rk ą 0. This finally yields that D k Ť D. In particular, D ℓ Ť D.

Given the sequence pD k q 0ďkďℓ , we can now prove the finite difference control (2.14). First, one easily checks that classical partial derivatives and finite difference operators all commute together, as long as both are well-defined. Recall that ∆ py1,...,y ℓ q " ∆ y1 ˝∆py2,...,y ℓ q .

(6.10)

Note now that ∆ py2,...,y ℓ q u lies in W 1,p pD 1 q X C 8 pD 1 q, since ř ℓ i"2 |y i | ă distpD 1 , BDq. Using equation (6.5) with O " D 1 , O 0 " D 0 and f " ∆ py2,...,y ℓ q u, }∆ py1,...,y ℓ q u} p L p pD0q "

› › ∆ y1 p∆ py2,...,y ℓ q uq › › p L p pD0q ď |y 1 | p › › ∇p∆ py2,...,y ℓ q uq › › p L p pD1q (6.11) ď |y 1 | p d ÿ j"1 › › B xj p∆ py2,...,y ℓ q uq › › p L p pD1q ď |y 1 | p d ÿ j"1 › › ∆ py2,...,y ℓ q `Bxj u ˘› › p L p pD1q . (6.12)
We used equation (6.5) in equation (6.11). We also commuted finite difference operators and partial derivatives in equation (6.12). If m " 1, then we have proved equation (2.14) for u P W 1,p pDq X C 8 pDq. If m ě 2, note that for all j, B xj u P W m´1,p pDq X C 8 pDq. In particular, B xj u P W 1,p pDq X C 8 pDq. One can then proceed by induction and perform the above step sequentially over k P t1, ..., ℓu (recall that ℓ ď m), successively using equation (6.5) for O " D k , O 0 " D k´1 and f " ∆ py k`1 ,...,y ℓ q u P W 1,p pD k q X C 8 pD k q (the latter holds because ř ℓ i"k`1 |y i | ă distpD k , BDq). This yields }∆ py1,...,y ℓ q u} p L p pD0q ď |y This shows equation (2.14) with C " }u} W m,p pDq (in fact, the equations above show that finite differences ∆ py1,...,y ℓ q u of order ℓ ď m are more accurately controlled by derivatives of order ℓ, taking C ℓ " p ř |β|"ℓ }B β u} p L p pDq q 1{p ). The general case where u P W m,p pDq is settled by equation (6.13) conjoined with the Meyers-Serrin theorem. We finally show that }B α u} L p pDq ď C, given any C which verifies equation (2.14). For this, copy the previous steps of piiiq ùñ piiq, which prove that for all φ P C 8 c pDq, the control from equation (2.12) holds for this C. Using the extremal equality case of Hölder's inequality in equation (2.12) indeed yields }B α u} L p pDq " sup This finishes the proof.

Proof. (Lemma 2.4) We begin by explicitly constructing the family pΦ q n q. First, use the fact that L q pDq is a separable Banach space ( [1], Theorem 2.21) : let pf n q nPN Ă L q pDq be a dense countable subset of L q pDq. For all n P N, let pϕ nm q mPN Ă C 8 c pDq be such that ϕ nm ÝÑ f n for the L q pDq topology (recall that C 8 c pDq is dense in L q pDq, [1], Corollary 2.30). We relabel the countable family pϕ nm q n,mPN as pφ n q nPN , which is thus dense in L q pDq. Second, let ph n q nPN Ă C 8 c pDq be a dense subset of C 8 c pDq for its LF-space topology (see Lemma 2.3). We then define E q to be the set of all finite linear combinations of elements of pφ n q and ph n q with rational coefficients :

E q " Span Q tφ n , n P Nu `Span Q th m , m P Nu (6.15) " ď n,mPN ! n ÿ i"1 q i φ i `m ÿ j"1
r j h j , pq 1 , ..., q n , r 1 , ..., r m q P Q n`m

) . (6.16)

Note that E q is countable, as a countable union of countable sets. We then define the family pΦ q n q to be an enumeration of E q : E q " tΦ q n , n P Nu. Proof of piq: Suppose that T " T v for some v P L p pDq. Then the control (2.18) is obviously true. Now, suppose that this countable control holds : let us show that T " T v for some v P L p pDq.

We begin by showing that the map T |Eq (restriction of T to the set E q ) can be uniquely extended to a continuous linear form T over L q pDq. Begin with the fact that for all f, g P E q , then f ´g P E q and from equation (2.23), |T pf q ´T pgq| " |T pf ´gq| ď C}f ´g} q .

(6.17) Equation (6.17) shows that T |Eq is Lipschitz over E q and therefore uniformly continuous on E q . Since R is complete and E q is dense in L q pDq, T |Eq can be uniquely extended by a map T defined over L q pDq, which is itself uniformly continuous ( [START_REF] Royden | Real analysis[END_REF], Problem 44,p. 196). We briefly recall the construction procedure of T over L q pDq. Given f P L q pDq and pf n q Ă E q any sequence such that }f n ´f } L q Ñ 0, one shows that the sequence pT pf n qq nPN is Cauchy, thus convergent and one sets T pf q :" lim n T pf n q. One proves that the value T pf q does not depend on the sequence pf n q, which implies that T is well defined and coincides with T on E q . We now check that T remains linear. Let f, g P L q pDq and λ P R. Let pf n q, pg n q Ă E q and pλ n q Ă Q be sequences such that f n Ñ f, g n Ñ g both in L q pDq and λ n Ñ λ. Then λ n f n `gn Ñ λf `g in L q pDq, and the sequence pλ n f n `gn q is contained in E q . Since T is well defined, we have that T pλf `gq " lim nÑ8 T pλ n f n `gn q " lim nÑ8 λ n T pf n q `T pg n q " λ T pf q `T pgq.

(6.18) Thus, T is a (uniformly) continuous linear form over L q pDq. Riesz' representation lemma yields a function v P L p pDq such that

@f P L q pDq, T pf q " ż D f pxqvpxqdx. (6.19)
We now need to check that in fact T pφq " T pφq if φ P C 8 c pDq, to show that T is indeed an extension of T . For this, notice that T and T both define continuous linear forms over C 8 c pDq, w.r.t. its LF-topology (v lies in L 1 loc pDq). Note also that T and T coincide on E q , by construction of T : @n P N, T pΦ n q ´T pΦ n q " 0. (6.20)

But E q is chosen so that it contains ph n q, which is a dense subset of C 8 c pDq. Given φ P C 8 c pDq, consider pj n q a subsequence of ph n q such that j n ÝÑ φ for the topology of C 8 c pDq. Then, pT ´T qpφq " lim nÑ8 pT ´T qpj n q " lim nÑ8 0 " 0, (6.21) which shows that in fact, T pφq " T pφq.

Proof of piiq: if b can be extended to a continuous linear form over L q pDq, then the estimate (2.20) is obviously true, by continuity over L q pDq of the said extension. Suppose now that (2.20) holds. Let φ P E q . Then L φ , the continuous linear form over C 8 c pDq defined by

@ψ P C 8 c pDq, L φ pψq " bpφ, ψq, (6.22)
verifies @ψ P E q , |L φ pψq| ď C}φ} q }ψ} q . (6.23)

From the point piq, L φ is a regular distribution with a representer v φ P L p pDq which is unique in L p pDq. Define the map B : E q Ñ L p pDq by Bφ " v φ . Then B verifies @φ P E q , @ψ P L q pDq, |xBφ, ψy L p ,L q | " |L φ pψq| ď C}φ} q }ψ} q . (6.24)

Taking the supremum w.r.t. ψ P L q pDq yields @φ P E q , }Bφ} p ď C}φ} q . (6.25)

Observe now that the bilinearity of b yields Bpφ `λψq " Bφ `λBψ if φ, ψ P E q and λ P Q.

Taking the exact same steps as for the proof of point piq and using equation (6.25), B : E q Ñ L p pDq is Lipschitz continuous over E q , and can thus be uniquely extended as a uniformly continuous map B : L q pDq Ñ L p pDq. This relies on the fact that E q is dense in L q pDq and that L q pDq is complete. As previously, one checks that B is linear. Being uniformly continuous, it is then a bounded operator from L q pDq to L p pDq (its adjoint B˚i s then automatically bounded).

Denote by b the continuous bilinear form over L q pDq defined by bpf, gq " x Bf, gy L p ,L q , @f, g P L q pDq. (6.26)

We now need to check that b indeed coincides with b over C 8 c pDq, so that it is indeed an extension of b. For this, let φ, ψ P C 8 c pDq and pφ n q, pψ n q two sequences of elements of E q that converge to φ and ψ respectively, in the LF topology. Then b and b coincide on E q : bpφ n , ψ m q " bpφ n , ψ m q.

(6.27)

Observe that the following chain of equalities holds. It relies on the sequential continuity (for the LF topology of C 8 c pDq) of the linear forms φ Þ Ñ bpφ, ψq, ψ Þ Ñ bpφ, ψq and T v : φ Þ Ñ T v pφq " xv, φy L q ,L p for any v P L q pDq, as well equation (6.27). x Bφ n , ψy L p ,L q " lim nÑ8 xφ n , B˚ψ y L q ,L p " lim nÑ8 T B˚ψ pφ n q " T B˚ψ pφq " xφ, B˚ψ y L q ,L p " x Bφ, ψy L p ,L q " bpφ, ψq.

(6.28)

The uniqueness of b follows from the uniqueness of B as an extension of B.

Proof. (Lemma 2.5) Let pK n q be an increasing sequence of compact subsets of D such that Ť n K n " D. From the measurability of U and Tonelli's theorem, ω Þ Ñ ş Kn |U ω pxq|dx is measurable and we have that From equation (6.29), ω Þ Ñ ş Kn |U ω pxq|dx is finite almost surely. Since the family pK n q is countable, one obtains a set Ω 0 Ă Ω of probability one such that for all ω P Ω 0 and for all n P N, ş Kn |U ω pxq|dx ă `8. Given now any compact subset K of D, there exists N P N such that K Ă K N and thus for all ω P Ω 0 , ş K |U ω pxq|dx ă `8. Therefore, the sample paths of U lie in L 1 loc pDq almost surely. From this fact and Fubini's theorem, we next obtain that given any φ P C 8 c and |α| ď m, the following map

U α φ : Ω Q ω Þ ÝÑ ż D U ω pxqB α φpxqdx (6.30)
is a well defined random variable (i.e. it is measurable; see e.g. [START_REF] Doob | Stochastic processes[END_REF], Theorem 2.7, p. 62). Moreover, one can show that it is a limit in probability of suitably chosen Riemann sums of the integrand ( [20], Theorem 2.8, p. 65). But here, those Riemann sums are all Gaussian random variables because U is a Gaussian process. Thus U α φ is a Gaussian random variable. a a limit in probability of Gaussian random variables. This also shows that tU α φ , φ P C 8 c pDqu is in fact a Gaussian process, since the linearity of B α yields

n ÿ i"1 a i U α φi " U α p ř n i"1 aiφiq , (6.31)
and thus ř n i"1 a i U α φi is a Gaussian random variable. An alternative proof of the Gaussianity of U α φ is found in [START_REF] Bogachev | Gaussian measures[END_REF], Example 2.3.16. p. 58-59.

Proof. (Proposition 3.5) Let γ ą 0, set s n :" n ´γ , m n :" ps n `sn`1 q{2 and consider the functions defined on p0, 1q by ϕ n pxq :" 1 rsn`1,mns pxq ´1rmn,snq pxq, ψ n pxq " ż x 0 ϕ n psqds. (6.32)

The functions ψ n are nonnegative hat functions supported on rs n`1 , s n s, with slope ˘1. Consider now the following covariance functions defined for all px, yq P p0, 1q 2 by kpx, yq " The infinite sums above have in fact only one non zero term given any fixed px, yq P p0, 1q 2 , hence the functions g and k are well-defined. We now prove the announced properties on k. piq : since for fixed px, yq P D ˆD, the sums in equation (6.33) only involve one basis function at a time, it is clear that for all x, y ‰ s n or m n , B x B y kpx, yq " gpx, yq. It is also clear that g and k are the covariance functions of the measurable Gaussian processes given by U pxq " ř `8 n"1 ξ i ψ n pxq and V pxq " ř `8 n"1 ξ i ϕ n pxq respectively, where pξ i q is a sequence of independent standard Gaussian random variables (again, these Gaussian processes are well-defined as only one basis function is activated at a time, given x P p0, 1q). piiq : observe that gpx, xq " 1 for all x P p0, 1q (except possibly for x " s n or m n for some n P N). Likewise, kpx, xq " x p dx " 2 ps n ´mn q p`1 p `1 , }ψ n } 2 p " ˆ2 p `1 ˙2{p ps n ´mn q 2`2{p . (6.36)

Hence, since |s n ´mn | ď C{n γ`1 for some C ą 0, ps n ´mn q 2`2{p ď C 1 {n p2`2{pqpγ`1q for some dx " s n ´sn`1 " C{n γ`1 , (

C
for some C ą 0. Thus, }ϕ n } 2 p " C 1 {n 2pγ`1q{p and ř n }ϕ n } 2 p converges only if γ ą p{2 ´1. Therefore, our conterexample is found by taking any γ P p0, p{2 ´1s (observe that when p ď 2, this interval becomes empty!) Proof. (Lemma 4.7) First, the map k is measurable over D ˆD. Then, given a compact set K Ă D ˆD, there exists a compact set K 0 Ă D such that K Ă K 0 ˆK0 (see e.g. the text before equation (3.11)). Then, using the Cauchy-Schwarz inequality for k, xF α k ϕ j , ϕ j y L 2 " TrpF α k q. (6.49)

ż K |kpx,
Taking the limit when n goes to infinity yields ř `8 i"1 λ i }B α ϕ i } 2 L 2 ď TrpF α k q. This shows that if ř `8 i"1 λ i }B α ϕ i } 2 L 2 " `8, then TrpF α k q " `8. Suppose now that TrpF α k q ă `8. Equation (6.49) shows that the series of functions ř i λ i B α ϕ i b B α ϕ i converges in norm in L 2 pD ˆDq. Moreover, we check that it is equal to B α,α k : taking φ P C

Lemma 2 . 6 (

 26 [START_REF] Adler | Random Fields and Geometry[END_REF], Theorem 2.1.2). Let pU n q nPN be a Gaussian sequence and set |U | :" sup n |U n |. Suppose that Pp|U | ă `8q " 1. Then there exists ε ą 0 such that Erexppε|U | 2 qs ă `8.

Proposition 2 . 9 (

 29 [START_REF] Bogachev | Gaussian measures[END_REF], Proposition 3.11.15 and Example 2.3.16).

Proposition 3 . 1 (

 31 Sample path Banach-Sobolev regularity for Gaussian processes). Let D Ă R d be an open set. Let pU pxqq xPD " GP p0, kq be a measurable centered Gaussian process, defined on a probability set pΩ, F, Pq, such that its standard deviation function σ lies in L 1 loc pDq. Let p P p1, `8q. Then the following statements are equivalent.(i) (Sample path regularity) The sample paths of pU pxqq xPD lie in W m,p pDq almost surely.

  2)-(6.3)-(6.4) in the proof of Lemma 2.1. Let φ P C 8 c pD ˆDq. Since it is compactly supported in D ˆD, find an open set D 0 Ť D such that Supppφq Ă D 0 . Use Hölder's inequality and equation (3.15):

1 |C r | 2 ż

 12 then introduces the Hardy-Littlewood maximal functions of f and g, as M pdq f pxq :" sup rą0 1 |C r | ż Crpxq |f ptq|dt, M p2dq gpx, yq :" sup rą0 Crpxq ż Crpxq |gps, tq|dsdt.

pdq r f

  pxq Ñ f pxq a.e.; but we also have the pointwise domination |A pdq r f pxq ´f pxq| ď |A pdq r f pxq| `|f pxq| ď M pdq f pxq `|f pxq| a.e.. (3.53) From equation (3.52), the function on the right-hand side of equation (3.53) lies in L p pR d q and

  equations (3.62), (3.63), (3.64) and Lebesgue's dominated convergence in L p pPq, Er}A pdq r U ´U } p p s Ý ÝÝ Ñ

pdq r , pA pdq r U

 r pxqq xPD is a centered measurable Gaussian process whose covariance function is given by CovpA pdq r U pxq, A pdq r U pyqq " `Apdq r b A pdq r ˘kpx, yq " A p2dq r kpx, yq, @px, yq P R d ˆRd (3.67) (Note then that A p2dq r kpx, xq " VarpA pdq r U pxqq ě 0, which also shows that the limit in equation (3.58) is nonnegative a.e.) The proof of the Gaussianity of pA pdq r U pxqq xPD is carried out similarly

  (4.4), the random variable is defined as sup xPK |V x | :" sup xPA |V x | where A is any countable subset of K, dense in K. Different choices of A only modify sup xPK |V x | on a set of probability 0 ([START_REF] Dudley | The sizes of compact subsets of Hilbert space and continuity of Gaussian processes[END_REF], p. 291), which leaves equation (4.3) unchanged. We will use the two following results below, taken from[START_REF] Dudley | The sizes of compact subsets of Hilbert space and continuity of Gaussian processes[END_REF]. Proposition 4.2 ( [21], p. 293 and [21], Proposition 3.4). We have the two following facts.

Proposition 4 . 3 (

 43 [START_REF] Dudley | The sizes of compact subsets of Hilbert space and continuity of Gaussian processes[END_REF], Proposition 6.3). Suppose that T is compact and self-adjoint. Then T pBq is a GB-set if and only if ř nPN λ 2 n ă 8, i.e. T pBq is a "Schmidt ellipsoid".We can now state our result pertaining to the H m pDq-regularity of Gaussian processes, given an arbitrary open set D Ă R d .

Proposition 4 . 4 (

 44 Sample path Hilbert-Sobolev regularity for Gaussian processes). Let D Ă R d be an open set. Let pU pxqq xPD " GP p0, kq be a measurable centered Gaussian process, defined on a probability set pΩ, F, Pq, such that its standard deviation function σ lies in L 1 loc pDq. The following statements are equivalent:(i) (Sample path regularity) The sample paths of U lie in H m pDq almost surely.

Lemma 4 . 7 .

 47 Let k be a measurable positive definite function defined on an open set D. Suppose that σ P L 1 loc pDq. Then k P L 1 loc pD ˆDq. Given a multi-index α, its distributional derivative D α,α k exists and we can introduce the associated continuous bilinear form over C 8 c pDq, b α pφ, ψq :" D α,α kpφ b ψq " ż DˆD kpx, yqB α φpxqB α ψpyqdxdy. (4.15)

C 8 c

 8 pDq, D α,α kpφ b ψq " ż DˆD kpx, yqB α φpxqB α ψpyqdxdy (4.34) " ż DˆD k α px, yqφpxqψpyqdxdy " T kα pφ b ψq. (4.35) Equation (4.35) shows that the distributional derivative D α,α k and the regular distribution T kα coincide on the set DpDq b DpDq. From the Schwartz kernel theorem ( [48], Theorem 51.7), D α,α k " T kα in D 1 pD ˆDq, which shows that B α,α k exists in L 2 pD ˆDq and that B α,α k " k α .

µ j ψ j pxqψ j pyq ¯2dydx ( 4 µ j ψ j pxqψ j pyq ¯2dydx ( 4

 44

  }φ} L q pDq dx ˇˇˇˇ. piiiq ùñ piiq: suppose piiiq, let us show piiq. Let |α| ď m and let φ P C 8 c pDq. Note K :" Supppφq its compact support and consider an open set D 0 such that K Ă D 0 Ť D. Let α P N d 0

xT

  Bφ n , ψ m y L p ,L q " lim nÑ8 lim mÑ8 Bφn pψ m q " lim nÑ8 T Bφn pψq " lim nÑ8

`8 ÿ n" 1 ψ

 1 n pxqψ n pyq, gpx, yq "

  1{2 ds ˙2 " x 2 .

  adjoint and nonnegative p@x P H 1 , xx, T ˚T xy H1 ě 0q. If H 1 is separable, T ˚T can then be diagonalized in an orthonormal basis pe n q of H 1 . Denote pλ n q the nonnegative eigenvalues of T

	roots s n :"	?	˚T , their square λ n are called the singular values of T . If H 1 is separable, T is said to be Hilbert-
	Schmidt if	ř	nPN }T f n } 2 H2

  e n y " ˚is trace-class, in which case TrpT ˚T q " }T } 2 HS " }T ˚}2 HS . If H 1 " H 2 " L 2 pDq, if T is trace class with kernel k and if k is sufficiently smooth (say continuous), then the trace of T " E k is given by TrpT q " ş D kpx, xqdx. Extensions of this formula to general Hilbert-Schmidt kernels k P L 2 pD ˆDq of trace class operators is studied in

	ÿ
	µ n .	(2.7)
	nPN
	Any trace-class operator is Hilbert-Schmidt, and T is Hilbert-Schmidt if and only if either T	˚T
	or T T

  In equation(3.34), we used equation(3.33) and pushforward integration to write the integral w.r.t. P as an integral w.r.t. µ " P U . Likewise in equation(3.35) where we write the integral w.r.t. µ as an integral w.r.t. µ 0 using the pushforward identity µ " µ 0A . In equation(3.36), To prove the remaining inequality (inf E k "ASA ˚}A} 2 νpSq ď }σ} 2 p ), we use an explicit decomposition E k " ASA ˚by first setting

	we used Jensen's inequality for concave functions (0 ă p{2 ă 1). In equation (3.37), we used
	the trace identity for Gaussian measures over Hilbert spaces from [6], equation 2.3.2 and the
	one following p. 49. Moreover, from the nuclear norm estimate of [48], Proposition 47.1 pp.
	479-480,
		νpE k q " νpASA ˚q ď }A}νpSq}A ˚} ď }A} 2 νpSq	(3.38)
	. In equations (3.37) and (3.38), taking the infimum over all representations E k " ASA ˚yields
	νpE k q ď	inf A,S s.t.		2 νpSq, }σ} 2 p ď C ´2 p p	inf A,S s.t.	2 νpSq.	(3.39)
			E k "ASA ˚}A}	E k "ASA ˚}A}
							Af pxq " f pxqσpxq 1´p{2 .	(3.40)
	Using Hölder's inequality with a " 2{p, 1{a `1{b " 1 (notice that a ą 1), we obtain
					ż
	}Af } p p "	|f pxq| p σpxq pp1´p{2q dx	(3.41)
					D
					ˆżD	˙p{2 ˆżD	˙1{b
				ď		|f pxq| 2 dx	σpxq bpp1´p{2q dx	.	(3.42)
	But b " a a´1 " 2{p 2{p´1 " 1 1´p{2 and bp1 ´p{2q " 1, which together with equation (3.41) yields
							}Af } p p ď }f } p 2 }σ} pp1´p{2q p	.	(3.43)
								34)
		ż						ż
	"	L 2 pDq	}Ag} p p µ 0 pdgq ď }A} p	L 2 pDq	}g} p 2 µ 0 pdgq	(3.35)
	ď }A} p	ż L 2 pDq	xg, gy p{2 L 2 µ 0 pdgq ď }A} p	ˆżL 2 pDq	xg, gy L 2 µ 0 pdgq	˙p{2	(3.36)

ď }A} p TrpSq p{2 " }A} p νpSq p{2 .

(3.37)

Thus A : L 2 pDq Ñ L p pDq is bounded and }A} ď }σ} 1´p{2 p

  For this, first observe that for all x P R d and n P N 0 , the following domination holds:|µ n | ˆ|A pdq r ψ n pxqA pdq r ϕ n pxq| ď |µ n | ˆM pdq ψ n pxqM pdq ϕ n pxq. (3.59)But the series obtained by summing the right-hand side term of equation (3.59) is an a.e. finite function of x, as Minkowski's inequality in L p{2 pR d q and equation (3.52) yield:

						(3.56)
	With equations (3.54), (3.55) and (3.56), we now prove the desired result. We first focus on
	the decomposition k "	ř	n µ n ψ n b ϕ n , for which the following pointwise equality holds ( [10],
	Corollary 2.2 and Lemma 2.3, or equation 3.6 from [10])
	A p2dq r	kpx, yq "	`8 ÿ	µ n A pdq r ψ n pxqA pdq r ϕ n pyq, @px, yq P R d ˆRd .	(3.57)
						n"0
	We now prove that from this decomposition, we can deduce a first important fact, which is
			lim rÑ0	A p2dq r	kpx, xq "	`8 ÿ n"0

b φqpx, yq ď M pdq ψpxqM pdq φpyq, (3.55) M pdq pψ `φqpxq ď M pdq ψpxq `M pdq φpxq. µ n ψ n pxqϕ n pxq a.e.. (3.58)

  . For this, observe that equation (3.51) together with the sublinear properties of M pdq (equations (3.55) and (3.56)) lead to the domination |µ n |M pdq ψ n pxqM pdq ϕ n pxq @x P R d ,

		|A p2dq r	kpx, xq| ď M p2dq kpx, xq ď	`8 ÿ	(3.69)
							n"0
	and the right-hand side of equation (3.69) indeed lies in L p{2 pDq, from equation (3.61). We
	finally conclude from Lebesgue's dominated convergence theorem that
	lim rÑ0	Er}A pdq r U } p p s " C p	ż R d	lim rÑ0	´Ap2dq r	kpx, xq

  α kpx, yqf pyqdy (4.6) is trace class. Equivalently, there exists a representative k α of B α,α k in L 2 pD ˆDq which is the covariance function of a measurable Gaussian process. For all such k α , denoting σ α pxq :" k α px, xq 1{2 , we have

	ż
	TrpE α k q "	k α px, xqdx ă `8.	(4.7)
	D
	(iii) (Mercer decomposition) The kernel k has the following Mercer decomposition
	`8 ÿ
	kpx, yq "
	n"0

  .26) As in the proof of Proposition 3.1 (e.g. equation (3.27)), we deduce that the two random variables sup φPF2 |U α φ | and sup φPF2 |V Aαpφq | have the same law, and from equation (4.24), we obtain that Since equation (4.27) holds for all |α| ď m, this provides a set of probability 1 on which all the sample paths of U lie in H m pDq, which proves piq. piq ùñ piiq : From Lemma 2.4 and the assumption from piq, DˆD kpx, yqB α φpxqB α ψpyqdxdy " b α pφ, ψq " xE α k φ, ψy L 2 .

	From Proposition 2.6, we have that
			C α :" E "	sup φPF2	|U α φ | 2 ‰	ă `8.	(4.29)
	Introduce b α , the continuous bilinear form over C 8 c pDq given by
					ż
	b α pφ, ψq "				kpx, yqB α φpxqB α ψpyqdxdy.	(4.30)
					DˆD
	Consider now φ and ψ in F 2 . Then,
	|b α pφ, ψq| " ˇˇˇż	kpx, yqB α φpxqB α ψpyqdxdy ˇˇˇ" |ErU α φ U α ψ s|
	DˆD
	ď Er|U α φ0 U α ψ0 |s ď	1 2	E "	pU α φ0 q 2 `pU α ψ0 q 2 ‰	ď E "	sup φ0PF	pU α φ0 q 2 ‰	" C α .	(4.31)
	From Lemma 4.7, b α can be extended to a continuous bilinear form over L 2 pDq and there exists
	a unique bounded, self-adjoint and nonnegative operator E α k which verifies
	ż
	@φ, ψ P C 8 c pDq,									(4.32)
	Pp sup φPF2	|U α φ | ă `8q " Pp sup φPF2	|V Aαpφq | ă `8q " 1.	(4.27)
				Pp sup φPF2	|U α φ | ă `8q " 1.	(4.28)

  , p. 150. piiiq ùñ pivq: we first study how finite difference operators behave on elements of H k in order to use Lemma 2.1piiiq. First, using the reproducing formula (4.2), observe that for suitable x and y P D,∆ y f pxq " f px `yq ´f pxq " xf, kpx `y, ¨q ´kpx, ¨qy H k " xf, ∆ y kpx, ¨qy H k . (4.36)More generally, for any finite difference operator ∆ py1,...,y ℓ q of order ℓ ď m, and any open setD 0 Ť D such that ř ℓ i"1 |y i | ă distpD 0 , BDq,∆ py1,...,y ℓ q f pxq " xf, ∆ py1,...,y ℓ q kpx, ¨qy H k .(4.37)The Cauchy-Schwarz inequality in H k yields ∆ py1,...,y ℓ q f pxq 2 ď }f } 2 H k }∆ py1,...,y ℓ q kpx, ¨q} 2

	H k .	(4.38)

  .[START_REF] Serre | Systems of conservation laws[END_REF] We used equations (4.38) and (4.39) to obtain equation(4.40). In equation (4.41), we distributed ∆ py1,...,y ℓ q b ∆ py1,...,y ℓ q over the Mercer decomposition of k (which exists by the assumption piiiq). In equation (4.42), we used the fact that ϕ i P H m pDq (see Lemma 4.9piq) conjointly with the finite difference control of Lemma 2.1piiiq. In equation (4.43), the we used the trace equality from Lemma 4.9piiq. From equation (4.43) and Lemma 2.1piiiq again, we obtain that f lies in H m pDq. Consider now any open set D 0 Ť D. Equation (4.43) applied to δ α h , the finite difference approximation of B α from equation (2.11) with h " ph 1 , ..., h d q P pR ˚qd and |α| ď m such that ř d i"1 α i h i ă distpD 0 , BDq, yields that

  .6) This sequence clearly verifies D k´1 `ty k Ă D k for all t P r0, 1s. If D k´1 is open, then D k´1 `ty k is also open, hence D k is open. We now check that the property that ř ℓ i"k |y i | ă distpD k´1 , BDq is inherited recursively (it is true for k " 1). With the assumption that ř ℓ i"k |y i | ă distpD k´1 , BDq, we indeed have that distpD k , BDq " inf

		xPD k ,yPBD	|x ´y| "	inf zPD k´1 ,tPr0,1s	|z `ty k ´y|
					yPBD
	ě	inf zPD k´1 ,tPr0,1s	´|z ´y| ´t|y k | ¯"	inf zPD k´1 ,yPBD	inf tPr0,1s
		yPBD

  1 ą 0 and ř n }ψ n } 2 p ă `8. Next,

	ż sn
	}ψ 1 n } p p " }ϕ n } p p "
	sn`1

  yq|dxdy ď Therefore, k P L 1 loc pD ˆDq and for all mutli-index α, b α is a bilinear continuous form over C 8 c pDq. From Lemma 2.2, b α can be uniquely extended to a continuous bilinear form over L 2 pDq. Denote by F α k the associated bounded operator over L 2 pDq. We now need to show that F α k is self-adjoint and nonnegative. First note that for all φ, ψ P C 8 c pDq, xF α k φ, ψy L 2 " ż DˆD kpx, yqB α φpxqB α ψpyqdydx " xφ, F α k ψy L 2 . (6.39) Equation (6.39), conjoined with the density of C 8 c pDq in L 2 pDq and the continuity of the bilinear form pf, gq Þ Ñ xF α k f, gy L 2 yields that xF α k f, gy L 2 " xf, F α k gy L 2 for all f, g P L 2 pDq. Therefore F α k is self-adjoint. For the positivity, consider again φ P C 8 c pDq. Then from Fubini's theorem (justified below), Equation (6.41), conjoined with the density of C 8 c pDq in L 2 pDq and the continuity of the quadratic form f Þ Ñ xF α k f, f y L 2 yields that xF α k f, f y L 2 ě 0 for all f P L 2 pDq. Therefore F α k is nonnegative. Proof. (Lemma 4.8) From the definition of b α over C 8 c pDq, b α pφ, ψq " From Cauchy-Schwarz's inequality, it verifies @φ, ψ P C 8 c pDq, |b α pφ, ψq| ď }B α,α k} 2 }φ} 2 }ψ} 2 (6.43) From Lemma 4.7, there exists a unique bounded, self-adjoint and nonnegative operator F α k over L 2 pDq such that b α pφ, ψq " xF α k φ, ψy L 2 for all φ, ψ P C 8 c pDq. The uniqueness of F α k and equation (6.42) yield F α k " E α k , and thus E α k is self-adjoint and nonnegative. Proof. (Lemma 4.9) piq : Let n P N 0 be such that λ n ‰ 0. Let φ P C 8 c pDq. Then DˆD B α,α kpx, yqφpxqφpyqdxdy ď }B α,α k} L 2 pDˆDq }φ} 2 L 2 pDq . (6.44) Therefore, from Lemma 2.1, B α ϕ n P L 2 pDq. piiq : introduce the finite rank kernel k n defined by k n px, yq " Then its mixed derivative B α,α k n px, yq is equal to ř n i"1 λ i B α ϕ i pxqB α ϕ i pyq in L 2 pD ˆDq and the associated operator E α kn is trace class, with i xB α ϕ i , ϕ j y 2 Now, observe that E α kn ď F α k in the sense of the Loewner order. Indeed, let first φ P C 8 c pDq: xpF α k ´Eα kn qφ, φy L 2 " xpE k ´Ekn qB α φ, B α φy L 2 " i xϕ i , B α φy 2 L 2 ě 0. (6.48) The density of C 8 c pDq in L 2 pDq and the continuity of the quadratic form f Þ Ñ xpF α k ´Eα kn qf, f y L 2 over L 2 pDq yields indeed that E α kn ď F α k . Taking the trace : i }B α ϕ i } 2 L 2 " TrpE α kn q " xE α kn ϕ j , ϕ j y L 2 ď

	Indeed the following integrability condition holds, setting K " Supppφq :
	"	ż					ȷ	ż
	E	DˆD ż	|B α φpxqB α φpyqU pxqU pyq|dxdy TrpE α kn q " `8 ÿ xE α kn ϕ j , ϕ j y L 2 " " KˆK ˆżK `8 ÿ	|B α φpxqB α φpyq|Er|U pxqU pyq|sdxdy ˙2 n ÿ L 2 (6.46)
		ď			|B α φpxqB α φpyq|σpxqσpyqdxdy " j"1	j"1	|B α φpxq|σpxqdx i"1
		KˆK ď sup |B α φpxq| 2	ˆżK	σpxqdx ˙2 ă `8. i"1 j"1 " n ÿ λ i `8 ÿ xB α ϕ i , ϕ j y 2 L 2 "	i"1 n ÿ	λ i }B α ϕ i } 2 L 2 .	(6.41) (6.47)
		xPK
							`8 ÿ
							i"n`1
						ż	ż
							kpx, yqB α φpxqB α ψpyqdxdy "	B α,α kpx, yqφpxqψpyqdxdy
						DˆD	DˆD
					" xE α k φ, ψy L 2 .	`8 ÿ	`8 ÿ	(6.42)
							j"1	j"1
		λ n	ˆżD	ż K0ˆK0 ϕ n pxqB α φpxqdx ˙2 ď	σpxqσpyqdxdy " i"1 λ i ϕ i pxqB α φpxqdx ˆżK0 σpxqdx ˙2 ă `8. `8 ÿ ˆżD ˙2	(6.38)
							`8 ÿ	ż
							ď	λ i
							i"1
							n
							ÿ
	xF α k φ, φy "	ż DˆD	kpx, yqB α φpxqB α φpyqdydx " i"1 λ i ϕ i pxqϕ i pyq. ż DˆD ErU pxqU pyqsB α φpxqB α φpyqdydx (6.45)
					"ˆż	˙2ȷ
			" E		U pxqB α φpxqdx	ě 0.	(6.40)
						D

DˆD ϕ i pxqϕ i pyqB α φpxqB α φpyqdxdy ď ż DˆD kpx, yqB α φpxqB α φpyqdxdy ď ż λ λ n ÿ i"1 λ

  DˆD ˆÿ i λ i B α ϕ i pxqB α ϕ i pyq ˙φpx, yqdxdy.(6.52)Moreover, since we have shown that B α,α k P L 2 pD ˆDq, Lemma 4.8 implies that F α k " E α k . We can then write, following the steps of equation(6.46),xE α k ϕ j , ϕ j y L 2 " xB α ϕ i , ϕ j y 2 L 2 " }B α ϕ i } 2 L 2 ă `8.Then as observed before, the series of functions ř i λ i B α ϕ i b B α ϕ i converges in norm in L 2 pD ˆDq, one verifies that B α,α k exists in L 2 pDq and is in fact given by B α,α k " ÿ i λ i B α ϕ i b B α ϕ i in L 2 pD ˆDq.(6.54)

							8 c pD ˆDq, then
	ż				ż
	kpx, yqB α,α φpx, yqdxdy "	ÿ	λ i	ϕ i pxqϕ i pyqB α,α φpx, yqdxdy	(6.50)
	DˆD			i	DˆD
					ż
				ÿ
			"		λ i
				i
	TrpF α k q " TrpE α k q "	`8 ÿ		`8 ÿ	ÿ	`8 ÿ	λ i }B α ϕ i } 2 L 2 .	(6.53)
			j"1		j"1		i"1
	Suppose now that	ř `8 i"1 λ i

DˆD

B α ϕ i pxqB α ϕ i pyqφpx, yqdxdy (6.51)

" ż i λ i

Recall that a topological space X is separable if there exists a countable subset Y Ă X which is dense in X for the topology of X. As for continuity and sequential continuity, separability and sequential separability agree for metric spaces but not for general topological spaces (see e.g.[START_REF] Bella | Sequential + separable vs sequentially separable and another variation on selective separability[END_REF] for further discussions).

From the constant estimate "}B α f } 2 ď C" from Lemma 2.1piiiq, we then obtain that

Summing the inequality (4.45) for all |α| ď m, we obtain that

with C " `N ř |α|ďm TrpE α k q ˘1{2 and N is the number of multi-indexes α such that |α| ď m. Therefore H k Ă H m pDq and the corresponding imbedding i : H k Ñ H m pDq is continuous. Using the reproducing formula (4.2), its transpose i ˚: H m pDq Ñ H k is given by

Above, B α y denotes differentation w.r.t. the y coordinate (note that i ˚pf q is indeed defined pointwise, since i ˚pf q P H k ). Let pψ j q be an orthonormal basis of H m pDq and k " ř i λ i ψ i b ψ i be the Mercer decomposition of k provided by the assumption piiiq. The trace of the nonnegative self-adjoint operator ii ˚is given by (explanation below)

In equation (4.48), we used the fact that i ˚pψ j q given by equation (4.47) is a representative of ii ˚pψ j q in H m pDq. In equation (4.49), we used the fact that the series of functions

) to distribute the partial derivatives over to the Mercer decomposition of k. We also used Fubini's and Tonelli's theorems ad libitum, as all the series

by assumption, equation (4.50) finishes to prove pivq when H k is separable.

When H k is not separable, observe that kerpiq is closed in H k since i is continuous. Therefore H k " kerpiq'kerpiq K and kerpiq K endowed with the topology of H k is a Hilbert space. Moreover, i ˚: H m pDq Ñ H k is compact since ii ˚is trace class. Thus its closed range impi ˚q is separable ( [START_REF] Conway | A course in functional analysis[END_REF], Exercise 3 p. 176). Finally, observe that impi ˚q " kerpiq K ( [16], Theorem 4.12) so that kerpiq K is a separable Hilbert space. Consider now j :" i | kerpiq K , the restriction of i to kerpiq K . Then ii ˚" jj ˚, so that equation (4.50) indeed yields that j is Hilbert-Schmidt.

Above, we used Tonelli's theorem in equation (4.58). We imbedded k x in H m pDq in equation (4.59). We used equations (4.54) and (4.55) in equation (4.60). We used Jensen's discrete inequality on the squaring function p¨q 2 with the weights µ j {S (µ j {S ě 0, ř j µ j {S " 1) in equation (4.61). We imbedded i ˚pψ j q in H m pDq and used Tonelli's theorem in equation (4.62). We used equation (4.56) in equation (4.63).

Therefore we have proved that equation (4.53) holds. Since, for |α| ď m,

. By the assumption that ii ˚is trace class and using the trace equalities from Lemma 4.9piiq (TrpE α k q " TrpF α k q "

Therefore, Lemma 4.9piiq implies that every E α k is indeed trace-class, which shows piiiq.

Concluding remarks and perspectives

Given p P p1, `8q and m P N 0 , we showed that the W m,p -Sobolev regularity of integer order of a measurable Gaussian process ppU pxqq xPD " GP p0, kq is fully equivalent to the fact that B α,α k lies in L p pD ˆDq combined with the integrability in L p pDq of the standard deviation associated to B α,α k, provided we use a suitable representative of B α,α k in L p pD ˆDq. Using general results on Gaussian measures over Banach spaces of type 2 and cotype 2, we translated this criteria as the existence of suitable nuclear decompositions of the covariance. These can be understood as generalizations to Banach spaces of the eigenfunction expansion of symmetric, nonnegative and trace class operators. In the Hilbert space case p " 2, we linked this property with the Hilbert-Schmidt nature of the imbedding of the RKHS in H m pDq, and gave explicit formulas for the traces of the involved integral operators in terms of the Mercer decomposition of the kernel. The results presented in this article provide a theoretical background w.r.t. the use of Gaussian processes for solving physics-related machine learning problems, in particular when modeling solutions of PDEs as sample paths of some Gaussian process. These results also come along with suitable quantities for controlling the Sobolev norm of the corresponding sample paths (see Remark 3.7). The application of the Gaussian process principles identified here to PDE-related machine learning problems, e.g. following the approach of [START_REF] Chen | Solving and learning nonlinear PDEs with Gaussian processes[END_REF], is certainly an interesting continuation of the results of this article. Controlling the small ball probability (see e.g. [START_REF] Mas | Representation of small ball probabilities in hilbert space and lower bound in regression for functional data[END_REF] for further details) of the Sobolev norm of a Gaussian process, perhaps in terms of some nuclear norm, is also a relevant question for further applications of Gaussian process techniques in such machine learning problems. Finally, the following question (which was implicit in this article) is interesting for probability theory: are all Gaussian measures over W m,p pDq induced by some Gaussian process? Proposition 2.9 states that this is true for m " 0, i.e. L p pDq.

Finally, since B α,α k P L 2 pD ˆDq, E α k is bounded over L 2 pDq and from equation (6.54), `8 ÿ i"1

To see that this also finishes to prove equation (4.20) in the infinite case, observe that if TrpF α k q " `8, then the previous computation implies that the series ř i λ i }B α ϕ i } 2 2 " `8: if this were not the case, Lemma 4.8 would apply again and we would have E α k " F α k , which would then be trace class. For asymmetric derivatives, simply observe that for all |α|, |β| ď m,

Therefore the norm convergence of the series ř iPN λ i }B α ϕ i b B α ϕ i } L 2 for all |α| ď m implies that of all the series of the form ř iPN λ i }B α ϕ i b B β ϕ i } L 2 converge, provided that |α| ď m and |β| ď m. As previously, one then deduces that B α,β k " ř 8 i"0 λ i B α ϕ i b B β ϕ i .