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] A Gaussian probability space
consists of 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
of a Gaussian probability space
consists of Example: Let
be a Gaussian probability space with a closed subspace
be the orthogonal complement of
Since orthogonality implies independence between
Given a Gaussian probability space
one defines the algebra of cylindrical random variables where
the fundamental algebra.
For an irreducible Gaussian probability
the fundamental algebra
is a dense set in
[4] 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 (named after Irving Segal) with
as a Gaussian space.
In this setting, one usually writes for an element
the associated Gaussian random variable in the Segal model as
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