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For infinite-dimensional Gaussian measures, we often see the definition of Gaussian random variables like this:

Let $H(\Omega;\mathbb{R})$ be a separable Hilbert space. A random variable $u \in H$ is said to be a Gaussian random variable if $\langle u,v \rangle$ is Gaussian (i.e., $\langle u,v \rangle$ follows the normal distribution on $\mathbb{R}$) for all $v \in H$.

Even more generally, I found in this lecture note (Definition 2.1) that Gaussian r.v. in topological vector space is defined as follows

Let $W$ be a topological vector space, and $\mu$ a Borel probability measure on $W$. $\mu$ is Gaussian iff, for each continuous linear functional $f\in W_*$, the pushforward $\mu \circ f^{−1}$ is a Gaussian measure on $\mathbb{R}$.

My question might be a bit vague: I would like to know the intuition and motivation of such definition. It seems to me that the infinite-dimensional Gaussian measure is defined via its finite-dimensional analogy. I can think of the analogy from characteristic function $\mathbb{E}[e^{i \langle u,v \rangle}]$ or that linear combinations of Gaussians $\Leftrightarrow$ Gaussian. What benefits does this kind of definition bring? Are there other definitions of Gaussian r.v.s in infinite-dimensional spaces?


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Even in finite dimensions this definition is more convenient since it is independent of coordinates. If you are interested in geometric applications this is what you need.

This definition has the advantage that clarifies the nature of the the various invariants. Here are some more details.

A Gaussian measure on a a topological vector space is a Borel measure $\mu$ such that any continuous linear functional $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bE}{\mathbb{E}}$$\newcommand{\bC}{\mathbb{C}}$ $\xi: V\to\bR$, viewed as a random variable has a Gaussian distribution. The Gaussian measures on $\bR$ are characterized by their mean and variance. Denote by $m(\xi)$ and respectively $V(\xi)$ the mean and respectively variance of $\xi$. Observe that the map $$ m:V^*\to\bR,\;\; \xi \mapsto m(\xi)\in \bR $$ is linear so it naturally lives in $V^{**}$, the bidual of $V$. With a bit of luck $m$ lands in $V\subset V^{**}$. This explains why in many applications some form of reflexivity is assumed about $V$.

As for the variance $V(-)$, note that it is defined by the covariance form $$ C: V^*\times V^*\to \bR,\;\;C(\xi,\eta)=\bE\big[\big(\,\xi-m(\xi)\,\big)\big(\,\eta-m(\eta)\,\big)\big]. $$ The covariance form is a symmetric nonnegative definite form again on the dual $V^*$ and $V(\xi)=C(\xi,\xi)$.

The Fourier transform (a.k.a. the characteristic function) is then the function $\newcommand{\ii}{\boldsymbol{i}}$ $$ \widehat{\mu}:V^*\to\bC,\;\;\widehat{\mu}(\xi)= \bE_\mu\big[ e^{\ii \xi}\,\big]=\int_V e^{\ii \xi(v)} \mu[dv]. $$ The characteristic function $f_\xi$ of $\xi$ is $$ f_\xi(t)= \widehat{\mu}(t\xi),\;\;t\in\bR. $$ The book Gaussian measures by V. Bogachev adopts this point and it is worth consulting it. One source that I like very much is the fourth volume of the treatise on generalized functions by Gelfand

I. M. Gelfand, N.Ya. Vilenkin: Generalized Functions. Volume 4. Applications of harmonic Analysis.

If $V$ is finite dimensional then for any $m\in V^{**}$ and $C$ symmetric nonnegative definite form on $V^*$ there exists a unique Gaussian measure with mean $m$ and covariance form $C$. This is no longer universally true in infinite dimensions. This a rather subtle issue. Things work out nicely if $V$ is the dual of nuclear space, for example if $V=C^{-\infty}(\bR^n)$ the dual of $C^\infty_0( \bR^n)$.

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    $\begingroup$ Just for reference, the mean $m$ lands in $V$ as soon as $\mu$ is Radon and $V$ is locally convex, see Thm 3.2.3 in Bogachev's 'Gaussian Measures' book. $\endgroup$ Nov 1, 2021 at 22:02
  • $\begingroup$ Yes, indeed the Radon condition olays an imoortant role in this story. $\endgroup$ Nov 2, 2021 at 7:25
  • $\begingroup$ @LiviuNicolaescu Apologies for commenting on an older post but I came across this question as I am studying Gaussian measures (especially on infinite dimensional spaces and for applications to QFT). How is Gelfand's book for learning about this topic? Is it in anyway out of date or are there better resources? $\endgroup$
    – CBBAM
    Jan 28 at 9:25
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    $\begingroup$ The default reference is Bogachev's book on Gaussian measures. It covers a lot and ir is very dense. For Gaussian measures over Banach spaces see the recent book by Strook. Gelfand's book does not go into probabilistic details but it is an ejoyable read $\endgroup$ Jan 29 at 8:27
  • $\begingroup$ @LiviuNicolaescu Thank you. $\endgroup$
    – CBBAM
    Jan 29 at 22:11

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