Gaussian Probability Space

In probability theory particularly in the Malliavin calculus, a Gaussian probability space is a probability space together with a Hilbert space of mean zero, real-valued Gaussian random variables. Important examples include the classical or abstract Wiener space with some suitable collection of Gaussian random variables.

Definition

A Gaussian probability space $(\Omega,\mathcal,P,\mathcal,\mathcal^_)$ consists of
* a (complete) probability space $(\Omega,\mathcal,P)$,
* a closed linear subspace $\mathcal\subset L^2(\Omega,\mathcal,P)$ called the ''Gaussian space'' such that all $X\in \mathcal$ are mean zero Gaussian variables. Their σ-algebra is denoted as $\mathcal_$.
* a σ-algebra $\mathcal^_$ called the ''transverse σ-algebra'' which is defined through
:: $\mathcal=\mathcal_ \otimes \mathcal^_.$

Irreducibility

A Gaussian probability space is called ''irreducible'' if $\mathcal=\mathcal_$. Such spaces are denoted as $(\Omega,\mathcal,P,\mathcal)$. Non-irreducible spaces are used to work on subspaces or to extend a given probability space. Irreducible Gaussian probability spaces are classified by the dimension of the Gaussian space $\mathcal$.

Subspaces

A ''subspace'' $(\Omega,\mathcal,P,\mathcal_1,\mathcal^_)$ of a Gaussian probability space $(\Omega,\mathcal,P,\mathcal,\mathcal^_)$ consists of
* a closed subspace $\mathcal_1\subset \mathcal$,
* a sub σ-algebra $\mathcal^_\subset \mathcal$ of ''transverse random variables'' such that $\mathcal^_$ and $\mathcal_$ are independent, $\mathcal=\mathcal_\otimes \mathcal^_$ and $\mathcal\cap\mathcal^_=\mathcal^_$.

Example:

Let $(\Omega,\mathcal,P,\mathcal,\mathcal^_)$ be a Gaussian probability space with a closed subspace $\mathcal_1\subset \mathcal$. Let $V$ be the orthogonal complement of $\mathcal_1$ in $\mathcal$. Since orthogonality implies independence between $V$ and $\mathcal_1$, we have that $\mathcal_V$ is independent of $\mathcal_$. Define $\mathcal^_$ via $\mathcal^_:=\sigma(\mathcal_V,\mathcal^_)=\mathcal_V \vee \mathcal^_$.

Remark

For $G=L^2(\Omega,\mathcal^_,P)$ we have $L^2(\Omega,\mathcal,P)=L^2((\Omega,\mathcal_,P);G)$.

Fundamental algebra

Given a Gaussian probability space $(\Omega,\mathcal,P,\mathcal,\mathcal^_)$ one defines the algebra of cylindrical random variables
:$\mathbb_=\$
where $P$ is a polynomial in \R _n,\dots,X_n/math> and calls $\mathbb_$ the ''fundamental algebra''. For any $p<\infty$ it is true that $\mathbb_\subset L^p(\Omega,\mathcal,P)$.

For an irreducible Gaussian probability $(\Omega,\mathcal,P,\mathcal)$ the fundamental algebra $\mathbb_$ is a dense set in $L^p(\Omega,\mathcal,P)$ for all $p\in$

Numerical and Segal model

An irreducible Gaussian probability $(\Omega,\mathcal,P,\mathcal)$ where a basis was chosen for $\mathcal$ is called a ''numerical model''. Two numerical models are isomorphic if their Gaussian spaces have the same dimension.

Given a separable space">separable Hilbert space $\mathcal$, there exists always a canoncial irreducible Gaussian probability space $\operatorname(\mathcal)$ called the ''Segal model'' (named after Irving Segal) with $\mathcal$ as a Gaussian space. In this setting, one usually writes for an element $g\in \mathcal$ the associated Gaussian random variable in the Segal model as $W(g)$. The notation is that of an isornomal Gaussian process and typically the Gaussian space is defined through one. One can then easily choose an arbitrary Hilbert space $G$ and have the Gaussian space as $\mathcal=\$.

See also

* Malliavin calculus
* Malliavin derivative

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

Probability theory