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.^[1][2]

Definition

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

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

${\displaystyle {\mathcal {F}}={\mathcal {F}}_{\mathcal {H}}\otimes {\mathcal {F}}_{\mathcal {H}}^{\perp }.}$^[3]

Irreducibility

A Gaussian probability space is called irreducible if ${\displaystyle {\mathcal {F}}={\mathcal {F}}_{\mathcal {H}}}$. Such spaces are denoted as ${\displaystyle (\Omega ,{\mathcal {F}},P,{\mathcal {H}})}$. Non-irreducible spaces are used to work on subspaces or to extend a given probability space.^[3] Irreducible Gaussian probability spaces are classified by the dimension of the Gaussian space ${\displaystyle {\mathcal {H}}}$.^[4]

Subspaces

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

• a closed subspace ${\displaystyle {\mathcal {H}}_{1}\subset {\mathcal {H}}}$,
• a sub σ-algebra ${\displaystyle {\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp }\subset {\mathcal {F}}}$ of transverse random variables such that ${\displaystyle {\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp }}$ and ${\displaystyle {\mathcal {A}}_{{\mathcal {H}}_{1}}}$ are independent, ${\displaystyle {\mathcal {A}}={\mathcal {A}}_{{\mathcal {H}}_{1}}\otimes {\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp }}$ and ${\displaystyle {\mathcal {A}}\cap {\mathcal {F}}_{\mathcal {H}}^{\perp }={\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp }}$.^[3]

Example:

Let ${\displaystyle (\Omega ,{\mathcal {F}},P,{\mathcal {H}},{\mathcal {F}}_{\mathcal {H}}^{\perp })}$ be a Gaussian probability space with a closed subspace ${\displaystyle {\mathcal {H}}_{1}\subset {\mathcal {H}}}$. Let ${\displaystyle V}$ be the orthogonal complement of ${\displaystyle {\mathcal {H}}_{1}}$ in ${\displaystyle {\mathcal {H}}}$. Since orthogonality implies independence between ${\displaystyle V}$ and ${\displaystyle {\mathcal {H}}_{1}}$, we have that ${\displaystyle {\mathcal {A}}_{V}}$ is independent of ${\displaystyle {\mathcal {A}}_{{\mathcal {H}}_{1}}}$. Define ${\displaystyle {\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp }}$ via ${\displaystyle {\mathcal {A}}_{{\mathcal {H}}_{1}}^{\perp }:=\sigma ({\mathcal {A}}_{V},{\mathcal {F}}_{\mathcal {H}}^{\perp })={\mathcal {A}}_{V}\vee {\mathcal {F}}_{\mathcal {H}}^{\perp }}$.

Remark

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

Fundamental algebra

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

${\displaystyle \mathbb {A} _{\mathcal {H}}=\{F=P(X_{1},\dots ,X_{n}):X_{i}\in {\mathcal {H}}\}}$

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

For an irreducible Gaussian probability ${\displaystyle (\Omega ,{\mathcal {F}},P,{\mathcal {H}})}$ the fundamental algebra ${\displaystyle \mathbb {A} _{\mathcal {H}}}$ is a dense set in ${\displaystyle L^{p}(\Omega ,{\mathcal {F}},P)}$ for all ${\displaystyle p\in [1,\infty [}$.^[4]

Numerical and Segal model

An irreducible Gaussian probability ${\displaystyle (\Omega ,{\mathcal {F}},P,{\mathcal {H}})}$ where a basis was chosen for ${\displaystyle {\mathcal {H}}}$ is called a numerical model. Two numerical models are isomorphic if their Gaussian spaces have the same dimension.^[4]

Given a separable Hilbert space ${\displaystyle {\mathcal {G}}}$, there exists always a canoncial irreducible Gaussian probability space ${\displaystyle \operatorname {Seg} ({\mathcal {G}})}$ called the Segal model with ${\displaystyle {\mathcal {G}}}$ as a Gaussian space.^[5]

Literature

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

References