In mathematics, classical Wiener space is the collection of all continuous functions on a given domain, taking values in a metric space. Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions. It is named after the American mathematician Norbert Wiener.
Article
In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a subinterval of the real line), taking values in a metric space (usually n-dimensional Euclidean space). Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions. It is named after the American mathematician Norbert Wiener.
Contents
Definition
Consider
E
⊆
R
n
{\displaystyle E\subseteq \mathbb {R} ^{n}}
and a metric space
(
M
,
d
)
{\displaystyle (M,d)}
. The classical Wiener space
C
(
E
,
M
)
{\displaystyle C(E,M)}
is the space of all continuous functions
f
:
E
→
M
.
{\displaystyle f:E\to M.}
That is, for every fixed
t
∈
E
,
{\displaystyle t\in E,}
d
(
f
(
s
)
,
f
(
t
)
)
→
0
{\displaystyle d(f(s),f(t))\to 0}
as
|
s
−
t
|
→
0.
{\displaystyle |s-t|\to 0.}
In almost all applications, one takes
E
=
[
0
,
T
]
{\displaystyle E=[0,T]}
or
E
=
R
+
=
[
0
,
+
∞
)
{\displaystyle E=\mathbb {R} _{+}=[0,+\infty )}
and
M
=
R
n
{\displaystyle M=\mathbb {R} ^{n}}
for some
n
∈
N
.
{\displaystyle n\in \mathbb {N} .}
For brevity, write
C
{\displaystyle C}
for
C
(
[
0
,
T
]
)
;
{\displaystyle C([0,T]);}
this is a vector space. Write
C
0
{\displaystyle C_{0}}
for the linear subspace consisting only of those functions that take the value zero at the infimum of the set
E
.
{\displaystyle E.}
Many authors refer to
C
0
{\displaystyle C_{0}}
as "classical Wiener space".
Properties of classical Wiener space
Uniform topology
The vector space
C
{\displaystyle C}
can be equipped with the uniform norm
‖
f
‖
:=
sup
t
∈
[
0
,
T
]
|
f
(
t
)
|
{\displaystyle \|f\|:=\sup _{t\in [0,\,T]}|f(t)|}
turning it into a normed vector space (in fact a Banach space since
[
0
,
T
]
{\displaystyle [0,T]}
is compact). This norm induces a metric on
C
{\displaystyle C}
Separability and completeness
With respect to the uniform metric,
C
{\displaystyle C}
is both a separable and a complete space:
Separability is a consequence of the Stone–Weierstrass theorem;
Completeness is a consequence of the fact that the uniform limit of a sequence of continuous functions is itself continuous.
Since it is both separable and complete,
C
{\displaystyle C}
is a Polish space.
Tightness in classical Wiener space
Recall that the modulus of continuity for a function
f
:
[
0
,
T
]
→
R
n
{\displaystyle f:[0,T]\to \mathbb {R} ^{n}}
is defined by
ω
f
(
δ
)
:=
sup
{
|
f
(
s
)
−
f
(
t
)
|
:
s
,
t
Classical Wiener measure
There is a "standard" measure on
C
0
,
{\displaystyle C_{0},}
known as classical Wiener measure (or simply Wiener measure). Wiener measure has (at least) two equivalent characterizations:
If one defines Brownian motion to be a Markov stochastic process
