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In [[Probability theory|probability theory]] particularly in the [[Malliavin calculus]], a '''Gaussian probability space''' is a [[Probability space|probability space]] together with a [[Hilbert space]] of mean zero, real-valued Gaussian random variables. Important examples include the [[Classical Wiener space|classical]] or [[Abstract Wiener space|abstract Wiener space]] with some suitable collection of Gaussian random variables.<ref>{{cite book|first1=Paul |last1=Malliavin |publisher=Springer |title=Stochastic analysis |place=Berlin, Heidelberg |date=1997 |isbn=3-540-57024-1 |doi=10.1007/978-3-642-15074-6}}</ref><ref>{{cite book|first1=David |last1=Nualart |publisher=Springer |title= The Malliavin calculus and related topics |place=New York |date=2013|page=3 |doi=10.1007/978-1-4757-2437-0}}</ref>
==Definition==
A '''Gaussian probability space''' <math>(\Omega,\mathcal{F},P,\mathcal{H},\mathcal{F}^{\perp}_{\mathcal{H}})</math> consists of
* a ([[Probability space#Complete probability space|complete]]) probability space <math>(\Omega,\mathcal{F},P)</math>,
* a closed subspace <math>\mathcal{H}\subset L^2(\Omega,\mathcal{F},P)</math> called the ''Gaussian space'' such that all <math>X\in \mathcal{H}</math> are mean zero Gaussian variables. Their [[Σ-algebra|σ-algebra]] is denoted as <math>\mathcal{F}_{\mathcal{H}}</math>.
* a [[Σ-algebra|σ-algebra]] <math>\mathcal{F}^{\perp}_{\mathcal{H}}</math> called the ''transverse σ-algebra'' which is defined through
:: <math>\mathcal{F}=\mathcal{F}_{\mathcal{H}} \otimes \mathcal{F}^{\perp}_{\mathcal{H}}.</math><ref name="Malliavin">{{cite book|first1=Paul |last1=Malliavin |publisher=Springer |title=Stochastic analysis |place=Berlin, Heidelberg |date=1997 |isbn=3-540-57024-1| pages=4–5 |doi=10.1007/978-3-642-15074-6}}</ref>

===Irreducibility===
A Gaussian probability space is called ''irreducible'' if <math>\mathcal{F}=\mathcal{F}_{\mathcal{H}}</math>. Such spaces are denoted as <math>(\Omega,\mathcal{F},P,\mathcal{H})</math>. Non-irreducible spaces are used to work on subspaces or to extend a given probability space.<ref name="Malliavin" /> Irreducible Gaussian probability spaces are classified by the dimension of the Gaussian space <math>\mathcal{H}</math>.<ref name="Malliavin2">{{cite book|first1=Paul |last1=Malliavin |publisher=Springer |title=Stochastic analysis |place=Berlin, Heidelberg |date=1997 |isbn=3-540-57024-1| pages=13–14 |doi=10.1007/978-3-642-15074-6}}</ref>

===Subspaces===
A ''subspace'' <math>(\Omega,\mathcal{F},P,\mathcal{H}_1,\mathcal{A}^{\perp}_{\mathcal{H}_1})</math> of a Gaussian probability space <math>(\Omega,\mathcal{F},P,\mathcal{H},\mathcal{F}^{\perp}_{\mathcal{H}})</math> consists of
* a closed subspace <math>\mathcal{H}_1\subset \mathcal{H}</math>,
* a [[Σ-algebra#Sub σ-algebras|sub σ-algebra]] <math>\mathcal{A}^{\perp}_{\mathcal{H}_1}\subset \mathcal{F}</math> of ''transverse random variables'' such that <math>\mathcal{A}^{\perp}_{\mathcal{H}_1}</math> and <math>\mathcal{A}_{\mathcal{H}_1}</math> are independent, <math>\mathcal{A}=\mathcal{A}_{\mathcal{H}_1}\otimes \mathcal{A}^{\perp}_{\mathcal{H}_1}</math> and <math>\mathcal{A}\cap\mathcal{F}^{\perp}_{\mathcal{H}}=\mathcal{A}^{\perp}_{\mathcal{H}_1}</math>.<ref name="Malliavin" />

Example:

Let <math>(\Omega,\mathcal{F},P,\mathcal{H},\mathcal{F}^{\perp}_{\mathcal{H}})</math> be a Gaussian probability space with a closed subspace <math>\mathcal{H}_1\subset \mathcal{H}</math>. Let <math>V</math> be the orthogonal complement of <math>\mathcal{H}_1</math> in <math>\mathcal{H}</math>. Since orthogonality implies independence between <math>V</math> and <math>\mathcal{H}_1</math>, we have that <math>\mathcal{A}_V</math> is independent of <math>\mathcal{A}_{\mathcal{H}_1}</math>. Define <math>\mathcal{A}^{\perp}_{\mathcal{H}_1}</math> via <math>\mathcal{A}^{\perp}_{\mathcal{H}_1}:=\sigma(\mathcal{A}_V,\mathcal{F}^{\perp}_{\mathcal{H}})=\mathcal{A}_V \vee \mathcal{F}^{\perp}_{\mathcal{H}}</math>.

===Remark===
For <math>G=L^2(\Omega,\mathcal{F}^{\perp}_{\mathcal{H}},P)</math> we have <math>L^2(\Omega,\mathcal{F},P)=L^2((\Omega,\mathcal{F}_{\mathcal{H}},P);G)</math>.

===Fundamental algebra===
Given a Gaussian probability space <math>(\Omega,\mathcal{F},P,\mathcal{H},\mathcal{F}^{\perp}_{\mathcal{H}})</math> one defines the [[Algebra|algebra]] of cylindrical random variables
:<math>\mathbb{A}_{\mathcal{H}}=\{F=P(X_1,\dots,X_n):X_i\in \mathcal{H}\}</math>
where <math>P</math> is a [[Polynomial|polynomial]] in <math>\R[X_n,\dots,X_n]</math> and calls <math>\mathbb{A}_{\mathcal{H}}</math> the ''fundamental algebra''. For any <math>p<\infty</math> it is true that <math>\mathbb{A}_{\mathcal{H}}\subset L^p(\Omega,\mathcal{F},P)</math>.

For an irreducible Gaussian probability <math>(\Omega,\mathcal{F},P,\mathcal{H})</math> the fundamental algebra <math>\mathbb{A}_{\mathcal{H}}</math> is a [[Dense set|dense set]] in <math>L^p(\Omega,\mathcal{F},P)</math> for all <math>p\in[1,\infty[</math>.<ref name="Malliavin2" />

===Numerical and Segal model===
An irreducible Gaussian probability <math>(\Omega,\mathcal{F},P,\mathcal{H})</math> where a basis was chosen for <math>\mathcal{H}</math> is called a ''numerical model''. Two numerical models are [[Isomorphism|isomorphic]] if their Gaussian spaces have the same dimension.<ref name="Malliavin2" />

Given a [[Separable space|separable]] Hilbert space <math>\mathcal{G}</math>, there exists always a canoncial irreducible Gaussian probability space <math>\operatorname{Seg}(\mathcal{G})</math> called the ''Segal model'' with <math>\mathcal{G}</math> as a Gaussian space.<ref>{{cite book|first1=Paul |last1=Malliavin |publisher=Springer |title=Stochastic analysis |place=Berlin, Heidelberg |date=1997 |isbn=3-540-57024-1|page=16 |doi=10.1007/978-3-642-15074-6}}</ref>

==Literature==
*{{cite book|first1=Paul |last1=Malliavin |publisher=Springer |title=Stochastic analysis |place=Berlin, Heidelberg |date=1997 |isbn=3-540-57024-1 |doi=10.1007/978-3-642-15074-6}}

==References==
<references />

[[Category:Probability theory]]

{{Sourceattribution|Gaussian probability space}}