This is related to my previous question. I'm having some problems understanding the local to global program discussed in Brydge's lecture notes. We are assuming $C=C_{1}+\cdots+C_{N}$ is a covariance matrix on $\mathbb{R}^{|\Lambda|}$. Besides, $\varphi$ is a random vector with distribution $\mu$ being Gaussian with covariance $C$; In addition, each $\xi_{j}$ is a random vector with distribution $\mu_{j}$ being Gaussian with covariance $C_{j}$, $j=1,...,N$. Now, on page 26, Brydges defines, for a given $X\subset \Lambda$, the set $\mathcal{N}_{j}(X)$ which is an algebra of functions measurable with respect to the $\sigma$-algebra generated by $\{\varphi_{j}(x), x\in X\}$. Here, $\varphi_{j}(x) = \sum_{k>j}\xi_{k}$ are random vectors. In my understanding, an element of $\mathcal{N}_{j}(X)$ is a real valued random variable defined on an underlying probability space, say $(\Omega, \mathcal{F},P)$. On page 27, Brydges defined $F^{X}=\prod_{B\in \mathcal{B}_{j}(X)}F(B)$. Again, $F^{X}$ seems to be a real valued function on $(\Omega, \mathcal{F},P)$. But equation (2.21) (Lemma 2.9) states that: \begin{eqnarray} \mathbb{E}_{j+1}F^{\Lambda} = (F')^{\Lambda} \end{eqnarray} and $\mathbb{E}_{j+1}$ is defined on page 24 as an integral with respect to $\mu_{j}$ which is a measure on $\mathbb{R}^{|\Lambda|}$. How can $F^{\Lambda}$ be viewed as a function on $\mathbb{R}^{|\Lambda|}$?
1 Answer
I don't have time to go back and read exactly the definitions, but I suspect the issue here is that Brydges explicitly writes the dependence on the supporting sets, but not the dependence on the fields. I recommend that you review the definitions by adding/restoring in your notation this dependence in the fields, with care: namely don't blur the difference between $\sum_{k>j}\xi_k$ and $\sum_{k\ge j}\xi_k$ for example. I know it sounds silly, but if you do this carefully, you should see that some field remains which has not been integrated over. So the result $(F')^{\Lambda}$ should be a function of that remaining field. The latter could a priori be a function of blocks of some scale (coarse field), but you can see it as a function of unit blocks (fine field) in a trivial way: the value of the fine field for unit box is the value of the coarse field on the big block that contains the unit box.
This is just a quick generic answer.
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$\begingroup$ Thanks for your answer! I followed your advice but I'm still a little confused. As you said, $(F')^{\Lambda}$ should be a function of the remaining fields but this is actually a composite function, as it seems. For instance, the function $F^{\Lambda}$ should be defined over $\Omega$ but it can be written as $F^{\Lambda} = f(\zeta_{j+1},\psi_{j+1})$ for some function $f:\mathbb{R}^{2\Lambda}\to \mathbb{R}$, so that $F^{\Lambda}(\omega) = f(\zeta_{j+1}(\omega),\psi_{j+1}(\omega))$. But Brydges integrates $F^{\Lambda$ over $\mathbb{R}^{\Lambda}$ as some sort of functional integral. I'm lost here. $\endgroup$ Commented Feb 4, 2020 at 21:34