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.{{cite book|first1=Paul |last1=Malliavin |publisher=Springer |title=Stochastic analysis |place=Berlin, Heidelberg |date=1997 |isbn=3-540-57024-1 |doi=10.1007/978-3-642-15074-6}}{{cite book|first1=David |last1=Nualart |publisher=Springer |title= The Malliavin calculus and related topics |place=New York |date=2013|page=3 |doi=10.1007/978-1-4757-2437-0}}

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}}.{{cite book|first1=Paul |last1=Malliavin |publisher=Springer |title=Stochastic analysis |place=Berlin, Heidelberg |date=1997 |isbn=3-540-57024-1| pages=4–5 |doi=10.1007/978-3-642-15074-6}}

=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}.{{cite book|first1=Paul |last1=Malliavin |publisher=Springer |title=Stochastic analysis |place=Berlin, Heidelberg |date=1997 |isbn=3-540-57024-1| pages=13–14 |doi=10.1007/978-3-642-15074-6}}

=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 (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\}.{{cite book|first1=Paul |last1=Malliavin |publisher=Springer |title=Stochastic analysis |place=Berlin, Heidelberg |date=1997 |isbn=3-540-57024-1|page=16 |doi=10.1007/978-3-642-15074-6}}

See also

Literature

  • {{cite book|first1=Paul |last1=Malliavin |publisher=Springer |title=Stochastic analysis |place=Berlin, Heidelberg |date=1997 |isbn=3-540-57024-1 |doi=10.1007/978-3-642-15074-6}}

References