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Since the product of real numbers is commutative, for any permutation $\sigma$, $$ \int_{S'}\phi(f_{\sigma(1)})\cdots\phi(f_{\sigma(n)})\ d\mu(\phi)= \int_{S'}\phi(f_{1})\cdots\phi(f_{n})\ d\mu(\phi) …

I am not aware of a theorem which has UV stability as a hypothesis and delivers the construction of a continuum limit, even modulo taking subsequences. UV stability is just a combination of suitable u …

My formula in the comment for $\|z\|=1$, was obtained as follows. First note that the integral is well defined in $[0,\infty]$, and by monotone convergence, is given by $$ \int_{S^{2n-1}}\frac{1}{|1-\ …

There is a choice to be made here: working with Banach spaces or with spaces of distributions like $\mathscr{S}'(\mathbb{R}^d)$. There are pros and cons for each of these two settings. Let me stick to …

This is easy to do using a graphical calculus for contractions of old-fashioned tensors. See this recent article for an example of application of such techniques and hopefully useful references. Here …

This is easy to compute using techniques from perturbative quantum field theory, i.e., Wick's Theorem (due to Isserlis) for moments of Gaussian measures. Consider $$ (2\pi)^{-\frac{p}{2}}\int_{\mathbb …

I could add a couple things to the already existing good answers. Continuous time processes are not that bad/that much more difficult to define or to simulate numerically in practice. For a Markov ch …

The result is $$ I(k)=\frac{2\pi^4\ \Gamma\left(\frac{k}{2}+1\right)\Gamma\left(\frac{k}{2}+2\right)}{\Gamma(k+4)}\ , $$ which can be simplified a bit more using the Legendre Duplication Formula. More …

While waiting for more context around the question, one can already mention the main definitions and tools for this topic. I will assume the space on which these probability measures live is the space …

The axioms don't tell you what theory you constructed. For that you need to go beyond the construction of correlation functions of the elementary field $\phi$ (the basic chapter on renormalization in …

Essentially, what is asked is the continuation of my previous MO answer Reformulation - Construction of thermodynamic limit for GFF and the solution of the exercise I mentioned at the end of that answ …

For $x\in\mathbb{Z}^d$ I will denote by $\bar{x}$ the corresponding equivalence class in the discrete finite torus $\Lambda_{L}=\mathbb{Z}^d/L\mathbb{Z}^d$. I will view a field $\phi\in\mathbb{R}^{\La …

The source of the confusion is not saying explicitly what are the sets and $\sigma$-algebras the measures are supposed to be on. For example, a sentence like ''By Kolmogorov's Extension Theorem, there …

Not quite an answer (because I still don't understand the question) but too long for a comment. Take the following very simple example where the $\varphi_{x}$ are iid $\mathscr{N}(0,1)$ random variab …

Just a quick answer for now. I would need to read carefully the definitions in the paper to be more precise. In general you need the Bochner-Minlos Theorem which says there is a unique probability me …