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 $(\mathrm{\Omega },ℱ,P,\mathscr{H},{ℱ}_{\mathscr{H}}^{\perp })$ consists of

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

$ℱ={ℱ}_{\mathscr{H}}\otimes {ℱ}_{\mathscr{H}}^{\perp }.$

Irreducibility

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

Subspaces

A subspace $(\mathrm{\Omega },ℱ,P,{\mathscr{H}}_1,{𝒜}_{{\mathscr{H}}_1}^{\perp })$ of a Gaussian probability space $(\mathrm{\Omega },ℱ,P,\mathscr{H},{ℱ}_{\mathscr{H}}^{\perp })$ consists of

• a closed subspace ${\mathscr{H}}_1\subset \mathscr{H}$,
• a sub σ-algebra ${𝒜}_{{\mathscr{H}}_1}^{\perp }\subset ℱ$ of transverse random variables such that ${𝒜}_{{\mathscr{H}}_1}^{\perp }$ and ${𝒜}_{{\mathscr{H}}_1}$ are independent, $𝒜={𝒜}_{{\mathscr{H}}_1}\otimes {𝒜}_{{\mathscr{H}}_1}^{\perp }$ and $𝒜\cap {ℱ}_{\mathscr{H}}^{\perp }={𝒜}_{{\mathscr{H}}_1}^{\perp }$.

Example:

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

Remark

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

Fundamental algebra

Given a Gaussian probability space $(\mathrm{\Omega },ℱ,P,\mathscr{H},{ℱ}_{\mathscr{H}}^{\perp })$ one defines the algebra of cylindrical random variables

${𝔸}_{\mathscr{H}}=\{F=P(X_1,\dots ,X_n):X_i\in \mathscr{H}\}$

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

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

Numerical and Segal model

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

Given a separable Hilbert space $𝒢$, there exists always a canoncial irreducible Gaussian probability space $\mathrm{Seg}(𝒢)$ called the Segal model (named after Irving Segal) with $𝒢$ as a Gaussian space. In this setting, one usually writes for an element $g\in 𝒢$ 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 $𝒢=\{W(g):g\in G\}$.

Literature

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