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

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A Gaussian probability space   consists of

  • a (complete) probability space  ,
  • a closed linear subspace   called the Gaussian space such that all   are mean zero Gaussian variables. Their σ-algebra is denoted as  .
  • a σ-algebra   called the transverse σ-algebra which is defined through
 [3]

Irreducibility

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A Gaussian probability space is called irreducible if  . Such spaces are denoted as  . 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  .[4]

Subspaces

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A subspace   of a Gaussian probability space   consists of

  • a closed subspace  ,
  • a sub σ-algebra   of transverse random variables such that   and   are independent,   and  .[3]

Example:

Let   be a Gaussian probability space with a closed subspace  . Let   be the orthogonal complement of   in  . Since orthogonality implies independence between   and  , we have that   is independent of  . Define   via  .

Remark

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For   we have  .

Fundamental algebra

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Given a Gaussian probability space   one defines the algebra of cylindrical random variables

 

where   is a polynomial in   and calls   the fundamental algebra. For any   it is true that  .

For an irreducible Gaussian probability   the fundamental algebra   is a dense set in   for all  .[4]

Numerical and Segal model

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An irreducible Gaussian probability   where a basis was chosen for   is called a numerical model. Two numerical models are isomorphic if their Gaussian spaces have the same dimension.[4]

Given a separable Hilbert space  , there exists always a canoncial irreducible Gaussian probability space   called the Segal model with   as a Gaussian space.[5]

Literature

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

References

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  1. ^ Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.
  2. ^ Nualart, David (2013). The Malliavin calculus and related topics. New York: Springer. p. 3. doi:10.1007/978-1-4757-2437-0.
  3. ^ a b c Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. pp. 4–5. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.
  4. ^ a b c Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. pp. 13–14. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.
  5. ^ Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. p. 16. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.