Appendix A Compact Operators

In this Section, we collect miscellaneous results used in Sect. 2.2 for the description of the covariance operators used in this work, which are inverses of self-adjoint realizations of the differential operator \(- \Delta + m^2\) [154, Sec. 4.1]. These covariance operators are compact (at least when \(\Omega \subset \mathbb ^d\) is bounded) self-adjoint operators, and we begin this Section with a brief summary of the properties of such operators. We then state two results used in Sect. 2.2.

Let \(\mathcal \) be a real or complex separable Hilbert space with inner product \(\langle \rangle \) and norm \(\Vert .\Vert \). We can make some useful statements about the spectrum of self-adjoint compact operators. The Riesz-Schauder theorem [90, Thm. VI.15] states that the spectrum \(\sigma (T)\) of any compact operator T on \(\mathcal \) is a discrete set with either no limit points or \(\lambda = 0\) as its only limit point. Any non-zero \(\lambda \in \sigma (T)\) is an eigenvalue of T of finite multiplicity. If T is, in addition, self-adjoint, then, by the Hilbert–Schmidt theorem [90, Thm. VI.16], there exists a complete orthonormal basis \(\_^\infty \) of \(\mathcal \) such that \(T \phi _n = \lambda _n \phi _n\) and \(\_^\infty \) is a null sequence in \(\mathbb \). This also implies that a self-adjoint and compact T is strictly positive precisely when all its eigenvalues are positive. Furthermore, \(\min \sigma (T) = 0\) and \(\max \sigma (T) = \Vert T\Vert \), see [90, Thm. VI.6]. As such a T is both self-adjoint and injective, its inverse \(T^\) is an unbounded densely defined self-adjoint operator on \(\mathcal \), see [155, Prop. A.8.2].

Two important special cases of compact operators on \(\mathcal \) are operators of trace class and Hilbert–Schmidt operators. For any \(T \in \mathfrak ^+ (\mathcal )\), its trace is defined as \(\textrm (T) = \sum _^\infty \langle \rangle \), where \(\_^\infty \) is some orthonormal basis of \(\mathcal \). An operator \(T \in \mathfrak (\mathcal )\) is said to be of trace class if and only if \(\textrm (|T|) < + \infty \), where \(|T| = \sqrt\). We denote the family of all trace class operators on \(\mathcal \) by \(\mathfrak (\mathcal )\). If \(T \in \mathfrak (\mathcal )\) is self-adjoint, then \(\textrm (T) = \sum _^\infty \lambda _n\), where \(\_^\infty \) is the sequence of eigenvalues of T. An operator \(T \in \mathfrak (\mathcal )\) is called Hilbert–Schmidt if and only if \(\textrm (T^* T) < + \infty \). We denote the family of all Hilbert–Schmidt operators on \(\mathcal \) by \(\textrm (\mathcal )\). Hilbert–Schmidt operators on \(L^2\)-spaces have a special form. An operator \(T \in \mathfrak (L^2 (M, \mu ))\), where \((M, \mu )\) is a measure space, is Hilbert–Schmidt precisely when there exists a function \(K \in L^2 (M \times M, \mu \otimes \mu )\), called the kernel of T, such that \((Tf) (x) = \int K (x,y) f (y) \, \textrm\mu (y)\) \(\mu \)-a.e. for all \(f \in L^2 (M, \mu )\). In other words, an operator on \(L^2 (M, \mu )\) is Hilbert–Schmidt if and only if its kernel is square integrable in the sense that

$$\begin \iint _M \left| K (x,y) \right| ^2 \, \textrm\mu (x) \, \textrm\mu (y) < + \infty \; . \end$$

(A1)

For a useful collection of properties of bounded integral operators on \(L^2\)-spaces, see, e.g., [156].

As already stated, the covariance operators used in this work are the inverses of self-adjoint extensions of \((- \Delta + m^2)|_\). The following Proposition states that such an operator has an inverse that is defined on all of \(\mathcal \).

Proposition 14

Let \(A \,: \, \mathfrak (A) \rightarrow \mathcal \) be a self-adjoint operator that is bounded from below by \(c > 0\). Then, A is bijective.

Proof

As A is bounded from below by a positive number, it is strictly positive and hence possesses a unique strictly positive self-adjoint square root [157, 158], which we denote by B. Then, for all \(f \in \mathfrak (A)\), \(0 \le \langle \rangle = \Vert B f \Vert ^2\), where equality holds if and only if \(f = 0\). This implies that \(B f = 0\) if and only if \(f = 0\), which means B is injective. As the composition of two injective maps is injective, \(A = B^2\) is injective.

Since \(\textrm (T)^\perp = \textrm (T^*)\) for any densely defined T [93, Prop. 1.6], the fact that A is injective implies

$$\begin \overline (A)} = \^\perp = \mathcal \; , \end$$

(A2)

i.e., the range of A is dense in \(\mathcal \). As A is bounded from below by some \(c > 0\), we have \(c \Vert f\Vert ^2 \le \langle \rangle \le \Vert f\Vert \, \Vert A f\Vert \), i.e., \(\Vert A f \Vert \ge c \, \Vert f \Vert \) for all \(f \in \mathfrak (A)\). Fix \(\varphi \in \mathcal \). Then, there exists \(\_^\infty \subset \textrm (A)\) such that \(\varphi _n \rightarrow \varphi \) in \(\mathcal \). Let \(\_^\infty \subset \mathfrak (A)\) be such that \(\varphi _n = A f_n\) for all \(n \in \mathbb \). By the semi-boundedness of A,

$$\begin \Vert f_n - f_m \Vert \le c^ \Vert A (f_n - f_m) \Vert = c^ \Vert \varphi _n - \varphi _m \Vert \; , \end$$

(A3)

which implies that \(\_^\infty \) is Cauchy. We denote the limit of \(\_^\infty \) by f. As A is self-adjoint, it is closed. Thus, its graph \(\Gamma (A) = \ (A) \}\) is closed in \(\mathcal \oplus \mathcal \) (and hence complete) and \(\lim _ (f_n, A f_n) = (f, \varphi ) = (f, Af) \in \Gamma (A)\), which implies that \(\varphi \in \textrm (A)\). Therefore, \(\textrm (A) = \mathcal \). \(\square \)

Furthermore, this inverse is bounded.

Corollary 15

Let \(A \,: \, \mathfrak (A) \rightarrow \mathcal \) be a self-adjoint operator that is bounded from below by \(c > 0\). Then, the inverse of A is a strictly positive bounded operator on \(\mathcal \) with \(\Vert A^\Vert \le c^\).

Proof

As A is self-adjoint, \(A^: \mathcal \rightarrow \mathfrak (A)\) is also self-adjoint. Then, the Hellinger-Toeplitz Theorem [90, p. 84] implies that \(A^\) is bounded. Fix \(\psi \in \mathcal \), which can be written as \(\psi = A \varphi \) as \(\mathcal = \textrm (A)\). Then,

$$\begin \langle \psi }\rangle = \langle \rangle \ge c \Vert \varphi \Vert ^2 \ge 0 \; , \end$$

(A4)

where equality holds if and only if \(\varphi = 0\). As A is injective, \(\varphi = 0\) if and only if \(\psi = 0\). Thus, \(A^\) is strictly positive. Lastly,

$$\begin \Vert A^\Vert = \sup _ \frac \psi \Vert } = \sup _ \frac = \sup _ \left( \Vert A \varphi \Vert \right) ^ \le c^ \; . \end$$

(A5)

\(\square \)

Appendix B Sobolev Spaces

In this Section, we define the Hilbert–Sobolev spaces needed in the main part of this work. For the most part, we follow the definitions and notations of [98, Ch. 3].

In the following, let \(s \in \mathbb \), \(d \in \mathbb \) and \(m > 0\). We denote by \(\mathscr (\mathbb ^d)\) the Schwartz space of functions of rapid decrease. This space equipped with its usual Fréchet topology is denoted by \(\mathcal (\mathbb ^d)\), see, e.g., [159]. The operator \(\mathcal ^s: \mathcal (\mathbb ^d) \rightarrow \mathcal (\mathbb ^d)\), called the Bessel potential of order s and mass m, defined by

$$\begin (\mathcal ^s f) (x) = \int _^d} (|\varvec|^2 + m^2)^} \hat (\varvec) \; \textrm^\varvec \cdot \varvec} \; \frac^d p} \; , \end$$

(B6)

where \(\hat\) denotes the Fourier transform of f, is continuous and can be continuously extended (with respect to the weak-* topology on \(\mathcal ^* (\mathbb ^d))\) to an operator acting on tempered distributions, \(\mathcal ^s: \mathcal ^* (\mathbb ^d) \rightarrow \mathcal ^* (\mathbb ^d)\). The Hilbert–Sobolev space \(H^s (\mathbb ^d)\) of order s on \(\mathbb ^d\) is defined as the completion of \(\mathscr (\mathbb ^d)\) with respect to the norm induced by the inner product

$$\begin \langle \rangle _s \langle ^s f, \mathcal ^s g}\rangle _^d)} \; . \end$$

(B7)

We have the following scale of Hilbert spaces,

$$\begin \ldots \subset H^ (\mathbb ^d) \subset H^ (\mathbb ^d) \subset H^0 (\mathbb ^d) = L^2 (\mathbb ^d) \subset H^ (\mathbb ^d) \subset H^ (\mathbb ^d) \subset \ldots \; , \end$$

(B8)

where each inclusion \(H^s (\mathbb ^d) \hookrightarrow H^ (\mathbb ^d)\) is continuous with dense image. We thus see that \(H^s (\mathbb ^d)\) for \(s > 0\) is a subset of \(L^2 (\mathbb ^d)\) and consists precisely of those square-integrable functions that fulfill additional regularity requirements, see the discussion of the spaces \(H^s (\Omega )\), \(\Omega \subset \mathbb ^d\) open, below. On the other hand, for \(s < 0\), elements in \(H^s (\mathbb ^d)\) need not be square integrable and furthermore need not be functions at all. For example, the Dirac delta distribution is an element of \(H^s (\mathbb ^d)\) if (and only if) \(s < -d / 2\), as can be seen from the definition of the norm in \(H^s (\mathbb ^d)\) and the Fourier transform of the Dirac delta.

Let K be a closed subset of \(\mathbb ^d\). The linear space of distributions with support in K,

$$\begin H^s_K = \left\^d) \; : \; \textrm \, \varphi \subseteq K \right\} \; , \end$$

(B9)

is a closed subspace of \(H^s (\mathbb ^d)\) and a Hilbert space when equipped with the inner product of \(H^s (\mathbb ^d)\). We denote by \(e_K\) the orthogonal projection from \(H^s (\mathbb ^d)\) onto \(H^s_K\).

Let \(\Omega \subset \mathbb ^d\) be open. The set of restrictions to \(\Omega \) (in the sense of distributions) of elements in \(H^s (\mathbb ^d)\),

$$\begin H^s (\Omega ) = \left\^* (\Omega ) \, : \, f = F|_\Omega \, , \; F \in H^s (\mathbb ^d) \right\} \; , \end$$

(B10)

is a Hilbert space when equipped with the inner product

$$\begin \langle \rangle _ \langle \rangle _s \; , \end$$

(B11)

where \(F,G \in H^s (\mathbb ^d)\) such that \(f = F|_\Omega \) and \(g = G|_\Omega \) and \(p_\Omega I - e_^d \Omega }\). The space \(H^s (\Omega )\) is furthermore the completion of \(C^\infty (\overline)\), the space of restrictions to \(\Omega \) of functions in \(C^\infty (\mathbb ^d)\), in the norm induced by the inner product (B11).

Let \(\alpha \in \mathbb _0^d\) be a multi-index. We define \(\textsf^\alpha \partial _^ \ldots \partial _^\), \(\partial _^ (\partial / \partial x_j)^\), a partial differential operator of order \(|\alpha | = \alpha _1 + \ldots + \alpha _d\). If \(s \in \mathbb _0\) and \(\Omega \) is a Lipschitz domain (see, e.g., [98, Def. 3.28]), an equivalent norm on \(H^s (\Omega )\) arises from the inner product

$$\begin \sum _ \langle ^\alpha f, \textsf^\alpha g}\rangle _ \; , \end$$

(B12)

cf. [98, Thm. 3.30] and [70, Sec. 1.1.1]. In particular, in this case a function f is in \(H^s (\Omega )\) if and only if it and all its weak derivatives up to order s are square integrable.

More generally, Hilbert–Sobolev spaces on an open subset \(\Omega \) can be defined either à la Slobodeckij-Gagliardo (with the corresponding space denoted by \(W_2^s (\Omega )\)) or à la Bessel-Fourier (with the corresponding space denoted by \(H^s (\Omega )\)). Whether or not these two definitions lead to the same space depends on s and the regularity of \(\Omega \), see [70, Sec. 1.1.1] and references therein. We principally work with spaces \(H^s (\Omega )\) but occasionally (when \(H^s (\Omega ) = W^s_2 (\Omega )\)) make use of results from the theory of \(W^s_2 (\Omega )\) spaces. Notice that for the case \(\Omega = \mathbb ^d\), \(H^s (\mathbb ^d) = W^s_2 (\mathbb ^d)\) for all \(s \ge 0\) (in the sense that \(H^s (\mathbb ^d)\) and \(W^s_2 (\mathbb ^d)\) are norm-equivalent Hilbert spaces), see [98, Thm. 3.16].

Finally, we define, for every non-empty open \(\Omega \subset \mathbb ^d\) and \(s \in \mathbb \), the space \(\widetilde^s (\Omega )\) as the closure of \(C^\infty _0 (\Omega )\) in \(H^s (\mathbb ^d)\) and the space \(H^s_0 (\Omega )\) as the closure of \(C^\infty _0 (\Omega )\) in \(H^s (\Omega )\). The spaces \(\widetilde^s (\Omega )\) and \(H^s_0 (\Omega )\) are Hilbert spaces when equipped with the inner products of \(H^s (\mathbb ^d)\) and \(H^s (\Omega )\), respectively.

We conclude this Section with a brief discussion of the trace operator. Let \(\Omega \) be a Lipschitz domain with boundary \(\partial \Omega \). For every \(f \in C^\infty (\overline)\) we define the operator \(\widetilde\) to be the restriction of f to the boundary, i.e., \(\widetilde f = f|_\). This operator can be continuously extended to an operator \(\gamma \,: \, H^ (\Omega ) \rightarrow H^} (\partial \Omega )\), see [98, Thm. 3.37]. The operator \(\gamma \) is called the trace operator. For definitions of the trace spaces \(H^s (\partial \Omega )\) see, e.g., [117, Sec. 4.2].

Appendix C Quadratic Forms

In this Section we provide a brief summary of quadratic forms on Hilbert spaces. We closely follow [90, 91, 94, 95]. Let \(\mathcal \) be a complex separable Hilbert space with inner product \(\langle \rangle \) and norm \(\Vert .\Vert \) and \(\mathcal \) a dense linear subspace of \(\mathcal \). Let \(\mathfrak : \mathcal \times \mathcal \rightarrow \mathbb \) be a (not necessarily bounded) sesquilinear form. Then \(\mathcal \) is called the form domain of \(\mathfrak \), and we write \(\mathcal (\mathfrak ) = \mathcal \). The sesquilinear form \(\mathfrak \) defines a quadratic form from \(\mathcal (\mathfrak )\) into \(\mathbb \) via \(f \mapsto \mathfrak (f) \mathfrak (f,f)\). On the other hand, a quadratic form on a complex Hilbert space defines a sesquilinear form via polarization. We call a sesquilinear form \(\mathfrak \) symmetric if \(\mathfrak (f,g) = \overline(g,f)}\) for all \(f,g \in \mathcal (\mathfrak )\). Clearly, the quadratic form associated with a symmetric form is real-valued. A symmetric form \(\mathfrak \) is called bounded from below if there exists a real number c such that \(\mathfrak (f) \ge c \Vert f\Vert ^2\) for all \(f \in \mathcal (\mathfrak )\), in which case we simply write \(\mathfrak \ge c\). The largest such number c is called the lower bound of \(\mathfrak \). The symmetric form \(\mathfrak \) is called positive if \(\mathfrak \ge 0\).

A form \(\mathfrak \), bounded from below by \(c \in \mathbb \), is called closed if the form domain \(\mathcal (\mathfrak )\) is complete with respect to the norm

$$\begin |\!|\!|f|\!|\!|^2 = \mathfrak (f) + (1-c) \Vert f\Vert ^2 \; . \end$$

(C13)

If \(\mathfrak \) is closed and \(S \subset \mathcal (\mathfrak )\) is \(|\!|\!|.|\!|\!|\)-dense in \(\mathcal (\mathfrak )\), we say that S is a form core of \(\mathfrak \). A form \(\mathfrak \) is called closable if it has a closed extension. The following Theorem captures an important connection between closed forms that are bounded from below and self-adjoint operators.

Theorem 16

([90, Thm. VIII.15], [94, Thm. VI.2.1 & Thm. VI.2.6]). Let \(\mathfrak \) be a densely defined closed symmetric form bounded from below with form domain \(\mathcal (\mathfrak )\). Then, there exists a unique densely defined self-adjoint operator T, bounded from below with the same lower bound as \(\mathfrak \), with domain \(\mathfrak (T) \subset \mathcal \) such that

\(\mathfrak (T) \subset \mathcal (\mathfrak )\) and \(\mathfrak (f,g) = \langle \rangle \) for all \(f \in \mathfrak (T)\) and \(g \in \mathcal (\mathfrak )\),

\(\mathfrak (T)\) is a form core of \(\mathfrak \),

if \(f \in \mathcal (\mathfrak )\), \(h \in \mathcal \) and \(\mathfrak (f,g) = \langle \rangle \) for every g belonging to a core of \(\mathfrak \), then \(f \in \mathfrak (T)\) and \(T f = h\).

We call T the operator associated with the form \(\mathfrak \) and call \(\mathcal (T) \mathcal (\mathfrak )\) the form domain of the operator T. The case where \(\mathfrak \) is positive is of special importance in this work, namely when we consider self-adjoint realizations of the differential operator \(- \Delta + m^2\) in Sect. 2.2.

Theorem 17

([94, Thm. VI.2.23]). Let \(\mathfrak \) be a densely defined, closed, symmetric and positive quadratic form and let T be the positive self-adjoint operator associated with \(\mathfrak \). Then, \(\mathcal (T) = \mathfrak (T^)\) and \(\mathfrak (f,g) = \langle f, T^ g}\rangle \) for all \(f,g \in \mathcal (\mathfrak )\).

Appendix D Gaussian Functional Integrals

This Appendix contains details of the discussion of functional integrals omitted in Sect. 2. We discuss Gaussian measures on infinite dimensional spaces, particularly on Hilbert and locally convex spaces and recall Minlos’ Theorem [160], which ensures the existence of Gaussian measures on spaces of distributions with covariance operators relevant for this work.

A free scalar field theory can be defined by the formal expression

$$\begin \textrm\mu = \frac \, \exp \left[ - S_\textrm [\varphi ] \right] \, \mathcal \varphi \; , \end$$

(D14)

where \(S_\textrm [\varphi ]\) is the Euclidean action functional. As this measure is Gaussian, it is completely specified by a covariance operator and a mean. In statistical field theory, the covariance operator is the inverse of the elliptic differential operator appearing in the free Euclidean action, and its integral kernel is usually referred to as the Green’s function, propagator or two-point function. A detailed discussion of the covariance operators needed in this work can be found in Sect. 2.2. The mean of the Gaussian measure is interpreted as the expected field configuration. The exposition in this Section is taken primarily from [86, 87, 118]. We start with some very basic results from probability theory.

Before we continue, we note that, throughout this Section, every time we consider a Gaussian measure on a locally convex space X it is useful to keep the specific choice employed in the main part in mind, namely a Gaussian measure on the space of distributions \(\mathcal ^*_\beta (\Omega )\), the strong topological dual of \(\mathcal (\Omega )\), the space of test functions equipped with the usual LF-topology. For definitions of these spaces see, e.g., [159] as well as [90, Ch. V]. A similar remark holds for the discussion of nuclear spaces and Minlos’ Theorem at the end of this Section. More precisely, the choices \(X = \mathcal ^*_\beta (\Omega )\), \(X^* = \mathcal (\Omega )\), \(\mathfrak = \mathcal (\Omega )\) and \(\mathfrak ^*_\beta = \mathcal ^*_\beta (\Omega )\) correspond to the specific spaces used in the main body of this work.

A measure space is a triple \((X, \mathcal , \mu )\), where \(\mathcal \) is a \(\sigma \)-algebra of subsets of a set X and \(\mu \) is a measure on \(\mathcal \). In case that \(\mu \) is a probability measure, i.e., \(\mu : \mathcal \rightarrow [0,1]\) with \(\mu (X) = 1\), the triple \((X, \mathcal , \mu )\) is called a probability space. Let \((X, \mathcal , \mu )\) be a measure space and \((Y,\mathcal ')\) a measurable space. A function \(f: X \rightarrow Y\) is called measurable with respect to the pair \((\mathcal , \mathcal ')\) if, for all \(A \in \mathcal '\), \(f^ (A) \in \mathcal \). It is furthermore called \(\mu \)-measurable if it is measurable with respect to the pair \((\mathcal , \mathcal ')\) and \(\mathcal \) is complete with respect to \(\mu \). A \(\mu \)-measurable function induces a measure on Y, called the push-forward measure, \(f_* \mu \), defined by

$$\begin (f_* \mu ) (A) = (\mu \circ f^) (A) = \mu (f^(A)) \end$$

(D15)

for all \(A \in \mathcal '\). Measurable functions from a probability space \((X, \mathcal , \mu )\) to some measurable space \((Y,\mathcal ')\) are called random variables.

We are interested in \(\sigma \)-algebras generated by families of sets or functions on X. If S is some family of subsets of X, then there exists a smallest \(\sigma \)-algebra in X, denoted \(\sigma (S)\), containing S. Given a family F of functions on X, there exists a smallest \(\sigma \)-algebra of subsets of X, denoted \(\mathcal (X,F)\), with respect to which all \(f \in F\) are measurable.

We now give some important examples of \(\sigma \)-algebras which we will need in this work. Suppose X is a topological space and let \(\mathscr _X\) denote the topology of X. Then \(\sigma (\mathscr _X) \mathcal (X)\) is called the Borel \(\sigma \)-algebra of X. A measure that is defined on \(\mathcal (X)\) is called a Borel measure on X. Now suppose that X is a locally convex space. The cylindrical \(\sigma \)-algebra generated by the topological dual \(X^*\), denoted \(\mathcal (X) \mathcal (X, X^*)\), is the smallest \(\sigma \)-field, with respect to which all continuous linear functionals on X are measurable. As \(\mathcal (X)\) is generated by continuous functions, \(\mathcal (X) \subseteq \mathcal (X)\) and in some important cases equality holds, see [87, Thm. A.3.7 & Thm. A.3.8].

Given a topological space X, we can generate a \(\sigma \)-algebra on it via its topology, which yields the Borel \(\sigma \)-algebra \(\mathcal (X)\). Radon measures are Borel measures that satisfy a collection of natural properties one expects from a measure on a topological space. For the definition of a Radon measure used in this work, see [87, Def. A.3.10]. A Radon measure on a locally convex space X is uniquely determined by its restriction to \(\mathcal (X)\) [87, Prop. A.3.12]. All measures considered in this work will be Radon measures on locally convex spaces, hence many theorems in the remainder of this section will be formulated in terms of Radon measures.

We now consider a special class of probability measures, namely Gaussian measures. We start by recalling some basic facts of Gaussian measures on \(\mathbb ^n\). A Gaussian measure on \(\mathbb \) is a Borel probability measure such that it is either the Dirac measure or its Radon-Nikodym derivative (or density) with respect to the Lebesgue measure is given by

$$\begin p (x) = \frac} \exp \left[ - \frac \right] \; , \end$$

(D16)

with \(a \in \mathbb \) the mean and \(\sigma ^2 > 0\) the variance. The case \(\sigma ^2 = 0\) corresponds formally to the Dirac measure, in which case the measure is also called a degenerate Gaussian measure. If the mean vanishes, we call the Gaussian measure centred. Similarly, a (non-degenerate) Gaussian measure on \(\mathbb ^n\) is defined as a Borel measure on \(\mathbb ^n\) with density

$$\begin p (\varvec) = \frac} \exp \left[ - \frac (\varvec-\varvec)^\textsf A^ (\varvec-\varvec) \right] \; , \end$$

(D17)

where A is a strictly positive symmetric matrix, called the covariance matrix of the Gaussian measure, and \(\varvec \in \mathbb ^n\) is the mean vector.

So far, we defined Gaussian measures as Borel measures on \(\mathbb ^n\) whose densities with respect to the Lebesgue measure are (normalized) Gaussian distributions. We can also use the following equivalent definition of Gaussian measures on \(\mathbb ^n\), which is particularly suitable for generalizations to infinite dimensional spaces. We define the family of Gaussian measures on \(\mathbb ^n\) to be precisely the family of measures on \(\mathbb ^n\) such that, for every linear functionalFootnote 28f on \(\mathbb ^n\), the push-forward measure \(f_* \mu \) is a Gaussian measure on \(\mathbb \) [87, Def. 1.2.1]. A (real-valued) random variable \(\xi \) on a probability space \((X,\mathcal ,\mu )\) is called Gaussian if the push-forward measure \(\xi _* \mu \) is a Gaussian measure on \(\mathcal (\mathbb )\). This leads to the following

Definition 3

A Borel probability measure \(\mu \) on \(\mathbb ^n\) is called Gaussian if every linear functional on \(\mathbb ^n\) is a Gaussian random variable with respect to \(\mu \).

Intuitively, we can think of Gaussian measures on \(\mathbb ^n\) to be Gaussian “in every direction”. We can now use the above definition of Gaussian measures on \(\mathbb ^n\) to generalize the concept of a Gaussian measure to arbitrary linear spaces, including infinite dimensional ones. This general definition does not require any topology on the linear space, see [86, 87]. Nevertheless, we will specify on the case of Gaussian measures on locally convex spaces.Footnote 29 Let X be a locally convex space and \(X^*\) its topological dual.

Definition 4

([87, Def. 2.2.1]). Let X be a locally convex space. A probability measure \(\mu \) on the measurable space \((X,\mathcal (X))\) is called Gaussian if every \(f \in X^*\) is a Gaussian random variable. It is called a centred Gaussian measure if all \(f \in X^*\) are centred Gaussian random variables.

This definition can be restated in the language of random processes, see, e.g., [104]. A probability measure \(\mu \) on \((X,\mathcal (X))\) is Gaussian if the random process on \((X, \mathcal (X), \mu )\) indexed by \(X^*\), \(\_\), where \(\varphi _f = f (\varphi )\), is Gaussian [86]. Recalling that a Radon measure is uniquely determined by its restriction to \(\mathcal (X)\), a Radon measure \(\mu \) on a locally convex space X is called Gaussian if its restriction to \(\mathcal (X)\) is Gaussian [87, Def. 3.1.1].

Definition 5

([87, Def. 2.2.7], see also [87, Thm. 3.2.3]). Let \(\mu \) be a Radon Gaussian measure on a locally convex space X. The mean of \(\mu \), denoted \(a_\mu \), is an element of X and is defined by

$$\begin a_\mu (f) = \mathbb [f] = \int _X f (\varphi ) \; \textrm\mu (\varphi ) \; , \qquad f \in X^* \; . \end$$

(D18)

The covariance operator of \(\mu \), denoted \(R_\mu \), is a linear map \(R_\mu : X^* \rightarrow X\), defined by

$$\begin R_\mu (f) (g)&= \mathbb [(f-a_\mu (f))(g-a_\mu (g))] \nonumber \\&= \int _X \left( f (\varphi ) - a_\mu (f) \right) \left( g (\varphi ) - a_\mu (g) \right) \textrm\mu (\varphi ) \; , \qquad f, g \in X^* \; . \end$$

(D19)

The covariance operator \(R_\mu \) induces the symmetric bilinear form \(\textrm\) on \(X^* \times X^*\) via \(\textrm (f,g) R_\mu (f) (g)\) for all \(f,g \in X^*\). The corresponding quadratic form, called the covariance of \(\mu \), is positive, i.e., \(\textrm (f,f) \ge 0\) for all \(f \in X^*\).

A Borel measure \(\mu \) on \(\mathbb ^n\) is Gaussian precisely when its Fourier transform is given by [87, Prop. 1.2.2]

$$\begin \hat (\varvec) \int _^n} \exp \left[ \textrm\; \varvec \cdot \varvec \right] \; \textrm\mu (\varvec) = \exp \left[ - \frac \, \varvec^\textsf A \varvec + \textrm\; \varvec \cdot \varvec \right] \; , \end$$

(D20)

with \(\varvec \in \mathbb ^n\) the mean and A the positive symmetric covariance matrix. An analogous statement also holds for measures on infinite dimensional spaces. First, however, we need to define the Fourier transform of such measures. Let X be a locally convex space and let \(\mu \) be a measure on \(\mathcal (X)\). Following [87, Def. A.3.17], the Fourier transform of \(\mu \), denoted \(\hat\), is a map \(\hat \,: \, X^* \rightarrow \mathbb \) defined by

$$\begin \hat (f) = \int _X \exp \left[ \textrmf (\varphi ) \right] \; \textrm\mu (\varphi ) \; , \qquad f \in X^* \; . \end$$

(D21)

A Gaussian measure on a locally convex space X is fully characterized by its Fourier transform.

Theorem 18

([87, Thm. 2.2.4]). A Radon measure