An
abstract Wiener space is a
mathematical object in
measure theory, used to construct a "decent"
(
strictly positive and
locally finite) measure on an
infinite-
dimensional vector
space.
It is named after the American mathematician Norbert Wiener. Wiener's original
construction only applied to the space of real-valued
continuous paths on the
unit interval, known as
classical Wiener space;
Leonard Gross provided the generalization to
the case of a general
separable
Banach space.
Definition
Let
H be a separable
Hilbert
space. Let
E be a separable Banach space. Let
i :
H →
E be an
injective continuous linear map with
dense image
(i.e., the
closure of
i(
H) in
E is
E itself) that
radonifies the
canonical
Gaussian cylinder set measure γ^{H}
on
H. Then the triple
(
i,
H,
E) (or simply
i :
H →
E) is called
an
abstract Wiener space. The measure
γ
induced on
E is called the
abstract Wiener
measure of
i :
H →
E.
The Hilbert space
H is sometimes called the
Cameron–Martin space or
reproducing kernel
Hilbert space.
Some sources (e.g. Bell (2006)) consider
H to be a densely
embedded Hilbert subspace of the Banach space
E, with
i simply the
inclusion of
H into
E. There is no loss of generality in
taking this "embedded spaces" viewpoint instead of the "different
spaces" viewpoint given above.
Properties
- γ is a Borel measure: it
is defined on the Borel σ-algebra
generated by the open subsets of
E.
- γ is a Gaussian
measure in the sense that f_{∗}(γ) is
a Gaussian measure on R for every linear functional
f ∈ E^{∗},
f ≠ 0.
- Hence, γ is strictly positive and locally finite.
- If E is a finite-dimensional Banach space, we may take
E to be isomorphic to
R^{n} for some
n ∈ N. Setting
H = R^{n} and
i : H → E to be the
canonical isomorphism gives the abstract Wiener measure
γ = γ^{n}, the
standard Gaussian measure on
R^{n}.
- The behaviour of γ under translation is described by the
Cameron–Martin
theorem.
- Given two abstract Wiener spaces
i_{1} : H_{1} → E_{1}
and
i_{2} : H_{2} → E_{2},
one can show that
γ_{12} = γ_{1} ⊗ γ_{2}.
In full:
- :(i_{1} \times i_{2})_{*} (\gamma^{H_{1} \times H_{2}}) =
(i_{1})_{*} \left( \gamma^{H_{1}} \right) \otimes (i_{2})_{*}
\left( \gamma^{H_{2}} \right),
- i.e., the abstract Wiener measure γ_{12} on
the Cartesian product
E_{1} × E_{2} is the
product of the abstract Wiener measures on the two factors
E_{1} and E_{2}.
Example: Classical Wiener space
Arguably the most frequently-used abstract Wiener space is the
space of continuous
paths, and is
known as
classical Wiener space. This is the
abstract Wiener space with
- H := L_{0}^{2, 1} ([0, T]; \mathbb{R}^{n}) := \{ \text{paths
starting at 0 with first derivative} \in L^{2} \}
with
inner product
- \langle \sigma_{1}, \sigma_{2} \rangle_{L_{0}^{2,1}} :=
\int_{0}^{T} \langle \dot{\sigma}_{1} (t), \dot{\sigma}_{2} (t)
\rangle_{\mathbb{R}^{n}} \, \mathrm{d} t,
E =
C_{0}([0,
T];
R^{n})
with
norm
- \| \sigma \|_{C_{0}} := \sup_{t \in [0, T]} \| \sigma (t)
\|_{\mathbb{R}^{n}},
and
i :
H →
E the
inclusion map. The measure
γ
is called
classical Wiener measure or simply
Wiener measure.
See also
References