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Let $\mathcal{S}(\mathbb{R}^d)$ be the Frechet space of Schwartz functions on $\mathbb{R}^n$. Its dual space $\mathcal{S}'(\mathbb{R}^d)$ is the space of tempered distributions.

Minlos Theorem as stated in ME then provide examples of Borel probability measures on $\mathcal{S}'(\mathbb{R}^d)$.

As for $L^2(\mathbb{R}^d)$, any symmetric, positive definite trace-class operator (a.k.a $S$-operator) yields a unique Borel Gaussian probability measure on $L^2(\mathbb{R}^d)$.

Now, I wonder if there are corresponding examples of Borel probability measures on the Schwartz space $\mathcal{S}(\mathbb{R}^d)$ as opposed to $\mathcal{S}'(\mathbb{R}^d)$ or $L^2(\mathbb{R}^d)$.

I searched for relevant references myself, but have not been successful. Could anyone please help me?

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    $\begingroup$ Could you please show a reference for existence of Gaussian measures on $L^2$? $\endgroup$
    – tsnao
    Commented Dec 21, 2023 at 7:17
  • $\begingroup$ @tsnao Gaussian Measures in Hilbert Space - Alexander Kukush $\endgroup$
    – Isaac
    Commented Dec 21, 2023 at 8:11

1 Answer 1

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$\newcommand\S{\mathcal S}\newcommand\R{\mathbb R}$$\S(\R^d)$ is a complete nuclear separable metrizable locally convex vector space. So, by part (i) of the Corollary on p. 19 and part (a) of the Corollary on p. 2 in the paper by Vakhania and Tarieladze, the class of the covariance operators of Borel Gaussian measures over $\S(\R^d)$ coincides with the class of all symmetric positive operators mapping $\S'(\R^d)$ into $\S(\R^d)$.

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