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Let us consider the real Hilbert space $H:=L^2\bigl(\mathbb{R}^n, \mathbb{R}^n\bigr)$ and "any" centered Gaussian measure $d\mu$ on it.

Next, denote a generic element of $H$ by the column vector $f(x)=\bigl[f_1(x), \cdots, f_n(x) \bigr]^T$.

Then, is it true that any $(ij)$-entry of the $2$-point correlation function for $\mu$, given by \begin{equation} C_{ij}(x,y):=\int_{H} f_i(x) f_j(y) d\mu(f) \end{equation} in facts depends on $x-y$ only?

That is, do we have \begin{equation} C_{ij}(x,y)=C_{ij}(x-y) \end{equation} for all $i,j=1,2,\cdots, n$?

I encountered such statements in the context of free scalar theories in QFT. However, I wonder if the property holds true in general.

Could anyone please help me?

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    $\begingroup$ the covariance matrix should be translationally invariant for this to hold; just saying that the measure is Gaussian is not enough, the translational invariance is an additional assumption $\endgroup$ Commented Sep 20, 2023 at 7:39
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    $\begingroup$ @CarloBeenakker I see. Could you provide an example of a Gaussian measure on above $H$ wth translation-invariant covarince? For Schwartz space, I can set $-\Delta+1$, but I cannot come up with one on $H$. $\endgroup$
    – Isaac
    Commented Sep 20, 2023 at 7:43

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$\newcommand\R{\mathbb R}$No, of course not. E.g., let $n=1$. Let $(B_t)_{t\in\R}$ be a standard Brownian motion. Let $$\R\ni x\mapsto F(x):=B_x\,1(x\in[0,1]),$$ so that the random function $F$ is a centered Gaussian random vector in $L^2(\R,\R)$. Let $\mu$ be the distribution of $F$ in $L^2(\R,\R)$.

Then $$C_{11}(x,y)=EF(x)F(y)=EB_x B_y\,1(x\in[0,1])\,1(y\in[0,1])\\ =\min(x,y)\,1(x\in[0,1])\,1(y\in[0,1]),$$ which is not a function of $x-y$.


More generally, if $G$ is any centered nonzero Gaussian random vector in $H:=L^2(\R^n,\R^n)$ with translation-invariant covariance, then $c:=\sum_{j=1}^n EG_j(x)^2>0$ does not depend on $x\in\R^n$. So, $$E\|G\|_H^2=E\int_{\R^n}dx \sum_{j=1}^n G_j(x)^2= \int_{\R^n}dx\, c =\infty,$$ which contradicts Fernique's theorem. So, there is no centered nonzero Gaussian random vector in $H:=L^2(\R^n,\R^n)$ with translation-invariant covariance.

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  • $\begingroup$ Thank you for your concrete example. Could you provide a Gaussian measure on some $L^2$ space whose covariance is translation-invariant? Perhaps, is there a way to restrict the one on the space of distributions with covariance $-\Delta+1$? $\endgroup$
    – Isaac
    Commented Sep 20, 2023 at 14:22
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    $\begingroup$ @Isaac : I have added an answer to this question as well. $\endgroup$ Commented Sep 20, 2023 at 14:53

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