The concept of an abstract Wiener space is a mathematical construction developed by Leonard Gross to understand the structure of Gaussian measures on infinite-dimensional spaces. The construction emphasizes the fundamental role played by the Cameron–Martin space. The classical Wiener space is the prototypical example.
The structure theorem for Gaussian measures states that all Gaussian measures can be represented by the abstract Wiener space construction.
Contents
Motivation
Let
H
{\displaystyle H}
be a real Hilbert space, assumed to be infinite dimensional and separable. In the physics literature, one frequently encounters integrals of the form
1
Z
∫
H
f
(
v
)
e
−
1
2
‖
v
‖
2
D
v
,
{\frac {1}{Z}}\int _{H}f(v)e^{-{\frac {1}{2}}\Vert v\Vert ^{2}}Dv,
where
Z
{\displaystyle Z}
is supposed to be a normalization constant and where
D
v
{\displaystyle Dv}
is supposed to be the non-existent Lebesgue measure on
H
{\displaystyle H}
. Such integrals arise, notably, in the context of the Euclidean path-integral formulation of quantum field theory. At a mathematical level, such an integral cannot be interpreted as integration against a measure on the original Hilbert space
H
{\displaystyle H}
. On the other hand, suppose
B
{\displaystyle B}
is a Banach space that contains
H
{\displaystyle H}
as a dense subspace. If
B
{\displaystyle B}
is "sufficiently larger" than
H
{\displaystyle H}
, then the above integral can be interpreted as integration against a well-defined (Gaussian) measure on
B
{\displaystyle B}
. In that case, the pair
(
H
,
B
)
{\displaystyle (H,B)}
is referred to as an abstract Wiener space.
The prototypical example is the classical Wiener space, in which
H
{\displaystyle H}
is the Hilbert space of real-valued functions
b
{\displaystyle b}
on an interval
[
0
,
T
]
{\displaystyle [0,T]}
having first derivative in
L
2
{\displaystyle L^{2}}
and satisfying
b
(
0
)
=
0
{\displaystyle b(0)=0}
, with the norm being given by
‖
b
‖
2
=
∫
0
T
b
′
(
t
)
2
d
t
.
{\displaystyle \left\Vert b\right\Vert ^{2}=\int _{0}^{T}b'(t)^{2}\,dt.}
In that case,
B
{\displaystyle B}
may be taken to be the Banach space of continuous functions on
[
0
,
T
]
{\displaystyle [0,T]}
with the supremum norm. In this case, the measure on
B
{\displaystyle B}
is the Wiener measure describing Brownian motion starting at the origin. The original subspace
H
⊂
B
{\displaystyle H\subset B}
is called the Cameron–Martin space, which forms a set of measure zero with respect to the Wiener measure.
What the preceding example means is that we have a formal expression for the Wiener measure given by
d
μ
(
b
)
=
1
Z
exp
{
−
1
2
∫
0
T
b
′
(
t
)
2
d
t
}
D
b
.
{\displaystyle d\mu (b)={\frac {1}{Z}}\exp \left\{-{\frac {1}{2}}\int _{0}^{T}b'(t)^{2}\,dt\right\}\,Db.}
Although this formal expression suggests that the Wiener measure should live on the space of paths for which
∫
0
T
b
′
(
t
)
2
d
t
<
∞
{\textstyle \int _{0}^{T}b'(t)^{2}\,dt<\infty }
, this is not actually the case, as sample Brownian paths are known to be almost surely nowhere differentiable, though it can be generalized to random measures as tempered distributions through the characteristic function as the white noise measure.
Gross's abstract Wiener space construction abstracts the situation for the classical Wiener space and provides a necessary and sufficient (if sometimes difficult to check) condition for the Gaussian measure to exist on
B
{\displaystyle B}
. Although the Gaussian measure
μ
{\displaystyle \mu }
lives on
B
{\displaystyle B}
rather than
H
{\displaystyle H}
, it is the geometry of
H
{\displaystyle H}
rather than
B
{\displaystyle B}
that controls the properties of
μ
{\displaystyle \mu }
. As Gross himself puts it (adapted to our notation), "However, it only became apparent with the work of I.E. Segal dealing with the normal distribution on a real Hilbert space, that the role of the Hilbert space
H
{\displaystyle H}
was indeed central, and that in so far as analysis on
B
{\displaystyle B}
is concerned, the role of
B
{\displaystyle B}
itself was auxiliary for many of Cameron and Martin's theorems, and in some instances even unnecessary." One of the appealing features of Gross's abstract Wiener space construction is that it takes
H
{\displaystyle H}
as the starting point and treats
B
{\displaystyle B}
as an auxiliary object.
Although the formal expressions for
μ
{\displaystyle \mu }
appearing earlier in this section are purely formal, physics-style expressions, they are very useful in helping to understand properties of
μ
{\displaystyle \mu }
. Notably, one can easily use these expressions to derive the (correct!) formula for the density of the translated measure
d
μ
(
b
+
h
)
{\displaystyle d\mu (b+h)}
relative to
d
μ
(
b
)
{\displaystyle d\mu (b)}
, for
h
∈
H
{\displaystyle h\in H}
. (See the Cameron–Martin theorem.)
Mathematical description
Cylinder set measure on H
Let
H
{\displaystyle H}
be a Hilbert space defined over the real numbers, assumed to be infinite dimensional and separable. A cylinder set in
H
{\displaystyle H}
is a set defined in terms of the values of a finite collection of linear functionals on
H
{\displaystyle H}
. Specifically, suppose
ϕ
1
,
…
,
ϕ
n
{\displaystyle \phi _{1},\ldots ,\phi _{n}}
are continuous linear functionals on
H
{\displaystyle H}
and
E
{\displaystyle E}
is a Borel set in
R
n
{\displaystyle \mathbb {R} ^{n}}
Nonexistence of the measure on H
The set functional
μ
{\displaystyle \mu }
is called the standard Gaussian cylinder set measure on
H
{\displaystyle H}
. Assuming (as we do) that
H
{\displaystyle H}
is infinite dimensional,
μ
{\displaystyle \mu }
does not extend to a countably additive measure on the
σ
{\displaystyle \sigma }
-algebra generated by the collection of cylinder sets in
H
{\displaystyle H}
(that is, it does not extend to the cylindrical σ-algebra generated by the cylinder algebra.) One can understand the difficulty by considering the behavior of the standard Gaussian measure on
R
n
,
{\displaystyle \mathbb {R} ^{n},}
given by
(
2
Existence of the measure on B
Now suppose that
B
{\displaystyle B}
is a separable Banach space and that
i
:
H
→
B
{\displaystyle i:H\rightarrow B}
is an injective continuous linear map whose image is dense in
B
{\displaystyle B}
. It is then harmless (and convenient) to identify
H
{\displaystyle H}
with its image inside
B
{\displaystyle B}
and thus regard
H
{\displaystyle H}
as a dense subset of
B
{\displaystyle B}
. We may then construct a cylinder set measure on
B
{\displaystyle B}
Universality of the construction
The abstract Wiener space construction is not simply one method of building Gaussian measures. Rather, every Gaussian measure on an infinite-dimensional Banach space occurs in this way. (See the structure theorem for Gaussian measures.) That is, given a Gaussian measure
μ
{\displaystyle \mu }
on an infinite-dimensional, separable Banach space (over
R
{\displaystyle \mathbb {R} }
), one can identify a Cameron–Martin subspace
H
⊂
B
{\displaystyle H\subset B}
, at which point the pair
(
H
,
B
)
{\displaystyle (H,B)}
becomes an abstract Wiener space and
μ
{\displaystyle \mu }
is the associated Gaussian measure.
Properties
μ
{\displaystyle \mu }
is a Borel measure: it is defined on the Borel σ-algebra generated by the open subsets of B.
μ
{\displaystyle \mu }
is a Gaussian measure in the sense that f∗(
μ
{\displaystyle \mu }
) is a Gaussian measure on R for every linear functional f ∈ B∗, f ≠ 0.
Hence,
μ
{\displaystyle \mu }
is strictly positive and locally finite.
The behaviour of
μ
{\displaystyle \mu }
under translation is described by the Cameron–Martin theorem.
Given two abstract Wiener spaces i1 : H1 → B1 and i2 : H2 → B2, one can show that
γ
12
=
γ
1
⊗
γ
2
Example: Classical Wiener space
The prototypical example of an abstract Wiener space takes the space
B
{\displaystyle B}
to be classical Wiener space, the space of continuous paths. The subspace
H
{\displaystyle H}
is given by
H
:=
L
0
2
,
1
(
[
0
,
T
]
;
R
n
)
:=
{
Absolutely continuous paths starting at 0 with square-integrable first derivative
}
{\displaystyle {\begin{aligned}H&:=L_{0}^{2,1}([0,T];\mathbb {R} ^{n})\\&:=\{{\text{Absolutely continuous paths starting at 0 with square-integrable first derivative}}\}\end{aligned}}}