I am currently reading a paper about fractional Gaussian fields and try to find a proof of the Bochner-Minlos theorem for the Schwartz space. The version I consider is the following:
A complex valued function $\Psi:\mathcal{S}(\mathbb{R}^d)\rightarrow\mathbb{C}$ is the characteristic function of a probability measure $\mu$ on $\mathcal{S}'(\mathbb{R}^d)$, i.e.
$\Psi(\phi)=\displaystyle\int_{\mathcal{S}(\mathbb{R}^d)}e^{i(f,\phi)}d\mu(f)$ for all $\phi\in\mathcal{S}(\mathbb{R}^d)$,
if and only if $\Psi(0)=1$, $\Psi$ is continuous and positive definite, i.e.
$\displaystyle\sum_{i,j=1}^nz_i\bar{z_j}\Psi(\phi_i-\phi_j)\ge0$ for all $z_1,…,z_n\in\mathbb{C}$ and $\phi_1,…,\phi_n\in\mathcal{S}(\mathbb{R})$.
Does anyone know where I can find a proof of that theorem? Any help would be highly appreciated.
Best Answer
Chances are, if you need this you will also need other results like the Lévy continuity theorem on $\mathcal{S}'(\mathbb{R}^d)$. In this case, the best way to proceed is to prove the isomorphism with the space of sequences of at most polynomial growth. It's an investment, but it pays off in considerably simplifying the proofs of Bochner-Minlos, Lévy continuity, the kernel theorem, Fubini's theorem for distributions, etc., etc.
For Bochner-Minlos see, Section 1.4.2 of the thesis by Ajay Chandra "Construction and Analysis of a Hierarchical Massless Quantum Field Theory". See also, Ch. 8 of the notes by Antti Kupiainen "Introduction to the Renormalization Group".
For the Lévy continuity theorem see "Generalized random fields and Lévy's continuity theorem on the space of tempered distributions" by Biermé et al.