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 $(\Omega ,{\mathcal {F}},P,{\mathcal {H}},{\mathcal {F}}_{\mathcal {H}}^{\perp })$ 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 ${\mathcal {F}}_{\mathcal {H}}^{\perp }$ called the transverse σ-algebra which is defined through

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

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 $(\Omega ,{\mathcal {F}},P,{\mathcal {H}}_{1},{\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp })$ of a Gaussian probability space $(\Omega ,{\mathcal {F}},P,{\mathcal {H}},{\mathcal {F}}_{\mathcal {H}}^{\perp })$ consists of

• a closed subspace ${\mathcal {H}}_{1}\subset {\mathcal {H}}$,
• a sub σ-algebra ${\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp }\subset {\mathcal {F}}$ of transverse random variables such that ${\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp }$ and ${\mathcal {A}}_{{\mathcal {H}}_{1}}$ are independent, ${\mathcal {A}}={\mathcal {A}}_{{\mathcal {H}}_{1}}\otimes {\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp }$ and ${\mathcal {A}}\cap {\mathcal {F}}_{\mathcal {H}}^{\perp }={\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp }$.

Example:

Let $(\Omega ,{\mathcal {F}},P,{\mathcal {H}},{\mathcal {F}}_{\mathcal {H}}^{\perp })$ 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 ${\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp }$ via ${\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp }:=\sigma ({\mathcal {A}}_{V},{\mathcal {F}}_{\mathcal {H}}^{\perp })={\mathcal {A}}_{V}\vee {\mathcal {F}}_{\mathcal {H}}^{\perp }$.

Remark

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

Fundamental algebra

Given a Gaussian probability space $(\Omega ,{\mathcal {F}},P,{\mathcal {H}},{\mathcal {F}}_{\mathcal {H}}^{\perp })$ 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 $\mathbb {R} [X_{n},\dots ,X_{n}]$ 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 (named after Irving Segal) with ${\mathcal {G}}$ as a Gaussian space. In this setting, one usually writes for an element $g\in {\mathcal {G}}$ 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 ${\mathcal {G}}=\{W(g):g\in G\}$.

Literature

• Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.