In physics literature, the covariance of a Gaussian measure $\mu$ on a function space is denoted as $C(x,y)$. Moreover, they say that if the covariance is translation-invariant, then actually $C(x,y)=\widetilde{C}(x-y)$.

I am extremely confused about exact meaning of such statement. For example, let us consider the following Hilbert space \begin{equation} V:=\{ f \in L^2(S^1) \mid f(x)=-f(-x) \text{ and } f \text{ is real-valued}\} \end{equation} where $S^1$ is the circle.

Then, by Minlos theorem, there exists a centered Gaussian measure $d\mu$ on $V$ such that \begin{equation} \exp\Bigl(-\frac{1}{2}\bigl\langle f, (-\Delta)^{-1} f \bigr\rangle \Bigr)=\int_{V} e^{i\langle f, g\rangle}d\mu(g) \end{equation} for all $f \in V$. Here $\langle, \rangle$ is the $L^2$ inner product.

$(-\Delta)^{-1} : V \to V$ clearly commutes with the translation operation. Also, the $2-$point moment of $d\mu$ is defined as the bilinear map $C : V \times V \to \mathbb{R}$ \begin{equation} C(f,g):=\int_{V} \langle f, h \rangle \langle g,h\rangle d\mu(h). \end{equation}

I suspect that what physics literature mean by "covariance" is $C(f,g)$ here. But in what sense is $C(f,g)$ translation-invariant?

Such things are so frustatingly confusing...Could anyone please clarify?

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