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I've came across an identity once (I don't remember where) concerning convolutions of Gaussian measures. … He mentions that this identity is related to Wick's theorem for Gaussian measures. Note, however, that Brydges states (\ref{1}) in the case $f$ is a polynomial. …

In particular, the authors state: "We construct a Gaussian measure $d\mu_{0}(\phi)$ on a measure space of continuous functions $\phi(x), x\in \Lambda \subset \mathbb{R}^{3}$ with covariance $u$: \begin … straightforward to show that: \begin{eqnarray} e^{-\beta U} = \int d\mu_{0}(\phi) e^{i\sqrt{\beta}\sum_{\alpha}e_{i(\alpha)}\phi(x_{\alpha})}" \tag{2}\label{2} \end{eqnarray} First of all, how to construct such a Gaussian …