Structure theorem for Gaussian measures: Difference between revisions

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{{Short description|Mathematical theorem}}

In [[mathematics]], the '''structure theorem for Gaussian measures''' shows that the [[abstract Wiener space]] construction is essentially the only way to obtain a [[Strictly positive measure|strictly positive]] [[Gaussian measure]] on a [[separable space|separable]] [[Banach space]]. It was proved in the 1970s by [[Gopinath Kallianpur|Kallianpur]]&ndash;SatoSatô&ndash;Stefan and [[Richard M. Dudley|Dudley]]&ndash;[[Jacob Feldman|Feldman]]&ndash;[[Lucien le Cam|le Cam]].<!-- I don't see those papers cited here. -->

There is the earlier result due to H. Satô (1969) <ref>[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1118797795 H. Satô, Gaussian Measure on a Banach Space and Abstract Wiener Measure], 1969.</ref> which proves that "any Gaussian measure on a separable Banach space is an [[abstract Wiener space | abstract Wiener measure]] in the sense of [[Leonard Gross | L. Gross]]". The result by Dudley et al. generalizes this result to the setting of Gaussian measures on a general [[topological vector space]].

==Statement of the theorem==

Let ''&gamma;'' be a strictly positive Gaussian measure on a separable Banach space (''E'',&nbsp;||&nbsp;||). Then there exists a separable [[Hilbert space]] (''H'',&nbsp;&lang;&nbsp;,&nbsp;&rang;) and a map ''i''&nbsp;:&nbsp;''H''&nbsp;&rarr;&nbsp;''E'' such that ''i''&nbsp;:&nbsp;''H''&nbsp;&rarr;&nbsp;''E'' is an abstract Wiener space with ''&gamma;''&nbsp;=&nbsp;''i''_&lowast;(''&gamma;''^''H''), where ''&gamma;''^''H'' is the [[canonical form|canonical]] Gaussian [[cylinder set measure]] on ''H''.

Let ''&gamma;γ'' be a strictly positive Gaussian measure on a separable Banach space (''E'',&nbsp;||&nbsp;||). Then there exists a [[separable [[Hilbert space]] (''H'',&nbsp;&lang;&nbsp;,&nbsp;&rang;) and a map ''i''&nbsp;:&nbsp;''H''&nbsp;&rarr;→&nbsp;''E'' such that ''i''&nbsp;:&nbsp;''H''&nbsp;&rarr;→&nbsp;''E'' is an abstract Wiener space with ''&gamma;γ''&nbsp;=&nbsp;''i''_&lowast;∗(''&gamma;γ''^''H''), where ''&gamma;γ''^''H'' is the [[canonical form|canonical]] Gaussian [[cylinder set measure]] on ''H''.

{{Reflist}}

==References==

{{reflist}}

* {{cite journal

Line 17 ⟶ 19:

| volume = 93

| year = 1971

| issue = 2 | pages = 390&ndash;408

| issn = 0003-486X

| doi=10.2307/1970780

| jstor = 1970780 | mr=0279272}}

{{Analysis in topological vector spaces}}

{{Measure theory}}

{{Banach spaces}}

[[Category:ProbabilityBanach theoremsspaces]]

[[Category:Theorems in probability theory]]

[[Category:Theorems in measure theory]]

[[Category:Probability theorems]]