Re: Gaussian distribution define in infinite-dimensional Hilbert space?

In article
<b5373cde-78a4-481b-90e8-eaa8894b18c8@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
istillshine@xxxxxxxxx wrote:

I know multivariate Gaussian distribution. But what is infinite-
dimensional Gaussian distribution? Is there any definition for this?

There is a measure associated with an infinite-dimensional Hilbert space
that is analogous to a multivariate Gaussian distribution. But there are
difficult technical details.

Let the starting Hilbert space be H; and let there be a one-to-one
mapping m of H into a Hilbert space K, such that the image of H is
dense in K, while m satisfies a condition that I don't quite remember.
(I think that having m be Hilbert-Schmidt is sufficient, but it's been a
long time since I worked on this stuff.) Then the Gaussian measure
associated with H can be defined as a countably additive measure on K.

Here is a reference :
Equivalence and Radon-Nikodym Derivatives of Gaussian Measures
C. H. Henrich
Journal of Mathematical Analysis and Applications, vol. 17 (1972)
255-270.

References in that paper will lead you to the basics of Gaussian
measures on infinite-dimensional Hilbert spaces.

There must be some good textbook exposition of this that is more recent,
but I just don't know, or don't remember.

--
Christopher J. Henrich
chenrich@xxxxxxxxxxxx
htp://www.mathinteract.com
.

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