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

Irreducibility
A Gaussian probability space is called irreducible if $$\mathcal{F}=\mathcal{F}_{\mathcal{H}}$$. Such spaces are denoted as $$(\Omega,\mathcal{F},P,\mathcal{H})$$. 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 $$\mathcal{H}$$.

Subspaces
A subspace $$(\Omega,\mathcal{F},P,\mathcal{H}_1,\mathcal{A}^{\perp}_{\mathcal{H}_1})$$ of a Gaussian probability space $$(\Omega,\mathcal{F},P,\mathcal{H},\mathcal{F}^{\perp}_{\mathcal{H}})$$ consists of
 * a closed subspace $$\mathcal{H}_1\subset \mathcal{H}$$,
 * a sub σ-algebra $$\mathcal{A}^{\perp}_{\mathcal{H}_1}\subset \mathcal{F}$$ of transverse random variables such that $$\mathcal{A}^{\perp}_{\mathcal{H}_1}$$ and $$\mathcal{A}_{\mathcal{H}_1}$$ are independent, $$\mathcal{A}=\mathcal{A}_{\mathcal{H}_1}\otimes \mathcal{A}^{\perp}_{\mathcal{H}_1}$$ and $$\mathcal{A}\cap\mathcal{F}^{\perp}_{\mathcal{H}}=\mathcal{A}^{\perp}_{\mathcal{H}_1}$$.

Example:

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

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

Fundamental algebra
Given a Gaussian probability space $$(\Omega,\mathcal{F},P,\mathcal{H},\mathcal{F}^{\perp}_{\mathcal{H}})$$ one defines the algebra of cylindrical random variables
 * $$\mathbb{A}_{\mathcal{H}}=\{F=P(X_1,\dots,X_n):X_i\in \mathcal{H}\}$$

where $$P$$ is a polynomial in $$\R[X_n,\dots,X_n]$$ and calls $$\mathbb{A}_{\mathcal{H}}$$ the fundamental algebra. For any $$p<\infty$$ it is true that $$\mathbb{A}_{\mathcal{H}}\subset L^p(\Omega,\mathcal{F},P)$$.

For an irreducible Gaussian probability $$(\Omega,\mathcal{F},P,\mathcal{H})$$ the fundamental algebra $$\mathbb{A}_{\mathcal{H}}$$ is a dense set in $$L^p(\Omega,\mathcal{F},P)$$ for all $$p\in[1,\infty[$$.

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

Given a separable Hilbert space $$\mathcal{G}$$, there exists always a canoncial irreducible Gaussian probability space $$\operatorname{Seg}(\mathcal{G})$$ called the Segal model with $$\mathcal{G}$$ as a Gaussian space.