# Is the $2$-point function translation invariant for general Gaussian meaures?

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 $$$$C_{ij}(x,y):=\int_{H} f_i(x) f_j(y) d\mu(f)$$$$ in facts depends on $$x-y$$ only?

That is, do we have $$$$C_{ij}(x,y)=C_{ij}(x-y)$$$$ for all $$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.

• 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 Commented Sep 20, 2023 at 7:39
• @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 $H$. Commented Sep 20, 2023 at 7:43

$$\newcommand\R{\mathbb R}$$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.
• 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 $-\Delta+1$? Commented Sep 20, 2023 at 14:22