In this article, we consider multiparameter processes, that is, our time is multidimensional. Throughout the paper, the dimension of time n≥1 is arbitrary but fixed.

We use the following notation throughout this article: t,s,u∈Rn are n-dimensional multiparameters of time: t=(t1,…,tn) , s=(s1,…,sn) , u=(u1,…,un ); 0 is an n-dimensional vector of 0s, and 1 is an n-dimensional vector of 1s. We denote s≤t if sk≤tk for all k≤n . For s≤t , the set [s,t]⊂Rn is the n-dimensional rectangle {u∈Rn;s≤u≤t} .

Let X=(Xt)t∈[0,1] be a real-valued centered Gaussian multiparameter process or field defined on some complete probability space (Ω,F,P) . We assume that the Gaussian field X is separable, that is, its linear space, or the 1st chaos, H1=cl(span{Xt;t∈[0,1]}) is separable. Here cl means closure in L2(Ω,F,P) .

Our main result, Theorem 1, shows when the Gaussian field X can be represented in terms of the Brownian sheet. Recall that the Brownian sheet W=(Wt)t∈[0,1] is the centered Gaussian field with the covariance E[WtWs]= ∏k=1nmin(tk,sk). The Brownian sheet can also be considered as the Gaussian white noise on [0,1] with the Lebesgue control measure. This means that dW is a random measure on ([0,1],B([0,1]),Leb([0,1])) characterized by the following properties: 1.

∫AdWt∼N(0,Leb(A)) ,

In this article, we show the Fredholm representation for Gaussian fields satisfying the trace condition (3) in Section 2, Theorem 1. In Section 3, we apply the Fredholm representation to give a representation for Gaussian fields that are equivalent in law, and in Section 4, we show how to generate series expansions for Gaussian fields by using the Fredholm representation. The Fredholm representation of Theorem 1 can also be used to provide a transfer principle that builds stochastic analysis and Malliavin calculus for Gaussian fields from the corresponding well-known theory for the Brownian sheet. We do not do that in this article, although it would be quite straightforward given the results for the one-dimensional case provided in [9].

Recall that X is a separable centered Gaussian field with covariance function R and W is a Brownian sheet. Suppose that X is defined on a complete probability space (Ω,F,P) that is rich enough to carry Brownian sheets.

The following theorem states that the field X can be realized as a Wiener integral with respect to a Brownian sheet. Let us note that it is not always possible to construct the Brownian sheet W directly from the field X. Indeed, consider the trivial field X≡0 to see this. As a consequence, the Karhunen representation theorem (see, e.g., [2, Thm. 41]) cannot be applied here. Consequently, the Brownian sheet in representation (2) is not guaranteed to exist on the same probability space with X.

In any case, representation (2) holds in law. This means that for a given Brownian sheet W, the field given by (2) is a Gaussian field with the same law as X.

The reproducing kernel Hilbert space (RKHS) of the Gaussian field X is the Hilbert space H that is isometric to the linear space H1 , and the defining isometry is R(t,·)↦Xt . In other words, the RKHS is the Hilbert space of functions over [0,1] extended and closed linearly by the relation ⟨R(t,·),R(s,·)⟩H=R(t,s). The RKHS is of paramount importance in the analysis of Gaussian processes. In this respect, the Fredholm representation (2) is also very important. Indeed, if the kernel K of Theorem 1 is known, then the RKHS is also known as the following reformulation of Lifshits [6, Prop. 4.1] states. Proposition 1.

Let X admit representation (2). Then H={f;f(t)=∫[0,1]f˜(s)K(t,s)ds,f˜∈L2([0,1])}. Moreover, the inner product in H is given by ⟨f,g⟩H=inff˜,g˜∫[0,1]f˜(t)g˜(t)dt, where the infimum is taken over all such f˜ and g˜ that f(t)=∫[0,1]f˜(s)K(t,s)ds,g(t)=∫[0,1]g˜(t)K(t,s)ds.

Two random objects ξ and ζ are equivalent in law if, their distributions satisfy P[ξ∈B]>0 0$]]> if and only if P[ζ∈B]>0 0$]]> for all measurable sets B. On the contrary, the random objects ξ and ζ are singular in law if there exists a measurable set B such that P[ξ∈B]=1 but P[ζ∈B]=0 . For centered Gaussian random objects there is the well-known dichotomy that two centered Gaussian objects are either equivalent or singular in law; see [4, Thm. 6.1].

There is a complete characterization of the equivalence by any two Gaussian processes due to Kallianpur and Oodaira; see [5, Thms. 9.2.1 and 9.2.2]. It is possible to extend this to Gaussian fields and formulate it in terms of the operator K. The result would remain quite abstract, though. Therefore, we due not pursue in that direction. Instead, the following Proposition 2 gives a partial solution to the problem what do Gaussian fields equivalent to a given Gaussian field X look like. Proposition 2 uses only the Hitsuda representation theorem, which is, unlike the Kallianpur–Oodaira theorem, quite concrete.

Let X˜=(X˜t)t∈[0,1] be a centered Gaussian field with covariance function  R˜ , and let X=(Xt)t∈[0,1] be a centered Gaussian field with covariance function R. Proposition 2

Suppose that X has representation (2) with kernel K and Brownian sheet W. If (8) X˜t=∫[0,1]K(t,s)dWs−∫[0,1]∫[s,1]K(t,s)L(s,u)dWuds for some L∈L2([0,1]) , then X˜ is equivalent in law to X.

The Mercer square root (7) can be used to build the Karhunen–Loève expansion for the Gaussian process X. But the Mercer form (7) is seldom known. However, if we can somehow find any kernel K such that representation (2) holds, then we can construct a series expansion for X by using the Fredholm representation of Theorem 1 as follows.

The proof below uses reproducing kernel Hilbert space technique. For more details on this, we refer to [3], where the series expansion is constructed for fractional Brownian motion by using the transfer principle.