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

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

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

For an irreducible Gaussian probability $(\mathrm{\Omega },{ℱ},P,{\mathscr{H}})$ the fundamental algebra ${\mathbb{𝔸}}_{{\mathscr{H}}}$ 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 ${𝒢}=\{W(g):g\in G\}$.

See also

• Malliavin calculus
• Malliavin derivative

Literature

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