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
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
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
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
.
For
we have
.
Fundamental algebra
Given a Gaussian probability space
one defines the algebra of cylindrical random variables
![{\displaystyle \mathbb {A} _{\mathcal {H}}=\{F=P(X_{1},\dots ,X_{n}):X_{i}\in {\mathcal {H}}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16b448dead7acd920680e669c1edd36bb351b82f)
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
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
- Malliavin, Paul (1997). Stochastic analysis. Berlin, Heidelberg: Springer. doi:10.1007/978-3-642-15074-6. ISBN 3-540-57024-1.
References
- ^ 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.