# Gaussian probability space

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In [probability theory](/source/Probability_theory) particularly in the [Malliavin calculus](/source/Malliavin_calculus), a **Gaussian probability space** is a [probability space](/source/Probability_space) together with a [Hilbert space](/source/Hilbert_space) of mean zero, real-valued [Gaussian random variables](/source/Gaussian_random_variable). Important examples include the [classical](/source/Classical_Wiener_space) or [abstract Wiener space](/source/Abstract_Wiener_space) with some suitable collection of Gaussian random variables.[1][2]

## Definition

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

- a ([complete](/source/Probability_space#Complete_probability_space)) probability space (\Omega,\mathcal{F},P),
- a closed [linear subspace](/source/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](/source/%CE%A3-algebra) is denoted as \mathcal{F}_{\mathcal{H}}.
- a [σ-algebra](/source/%CE%A3-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}}.[3]

### 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.[3] Irreducible Gaussian probability spaces are classified by the dimension of the Gaussian space \mathcal{H}.[4]

### 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](/source/%CE%A3-algebra#Sub_σ-algebras) \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}.[3]

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](/source/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](/source/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](/source/Dense_set) in L^p(\Omega,\mathcal{F},P) for all p\in[1,\infty[.[4]

### 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](/source/Isomorphism) if their Gaussian spaces have the same dimension.[4]

Given a [separable](/source/Separable_space) 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](/source/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](/source/Isometry) 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\}.[5]

## See also

- [Malliavin calculus](/source/Malliavin_calculus)
- [Malliavin derivative](/source/Malliavin_derivative)

## Literature

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

## References

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

1. Nualart, David (2013). *The Malliavin calculus and related topics*. New York: Springer. p. 3. [doi:10.1007/978-1-4757-2437-0](https://doi.org/10.1007/978-1-4757-2437-0)

1. Malliavin, Paul (1997). *Stochastic analysis*. Berlin, Heidelberg: Springer. pp. 4–5. [doi:10.1007/978-3-642-15074-6](https://doi.org/10.1007/978-3-642-15074-6). ISBN 3-540-57024-1.

1. Malliavin, Paul (1997). *Stochastic analysis*. Berlin, Heidelberg: Springer. pp. 13–14. [doi:10.1007/978-3-642-15074-6](https://doi.org/10.1007/978-3-642-15074-6). ISBN 3-540-57024-1.

1. Malliavin, Paul (1997). *Stochastic analysis*. Berlin, Heidelberg: Springer. p. 16. [doi:10.1007/978-3-642-15074-6](https://doi.org/10.1007/978-3-642-15074-6). ISBN 3-540-57024-1.

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