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
editA 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
Irreducibility
editA 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
editA 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
editFor we have .
Fundamental algebra
editGiven 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
editAn 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
edit- Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.
References
edit- ^ Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.
- ^ Nualart, David (2013). The Malliavin calculus and related topics. New York: Springer. p. 3. doi:10.1007/978-1-4757-2437-0.
- ^ 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.
- ^ 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.
- ^ Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. p. 16. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.