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Classical Wiener space

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Inmathematics, 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.

Definition[edit]

Consider E ⊆ Rⁿ and a metric space (M, d). The classical Wiener space C(E; M) is the space of all continuous functions f : E → M. I.e. for every fixed tinE,

${\displaystyle d(f($s),f(t))\to 0} as${\displaystyle |s-t|\to 0.}$

In almost all applications, one takes E = [0, T ] or [0, +∞) and M = Rⁿ for some ninN. For brevity, write C for C([0, T ]; Rⁿ); this is a vector space. Write C₀ for the linear subspace consisting only of those functions that take the value zero at the infimum of the set E. Many authors refer to C₀ as "classical Wiener space".

For a stochastic process ${\displaystyle \{X_{t},t\in T\}:(\Omega ,{\mathcal {F}},P)\to (E,{\mathcal {B}})}$ and the space ${\displaystyle {\mathcal {F}}(T,E)}$ of all functions from ${\displaystyle T}$to${\displaystyle E}$, one looks at the map ${\displaystyle \varphi :\Omega \to {\mathcal {F}}(T,E)\cong E^{T}}$. One can then define the coordinate mapsorcanonical versions ${\displaystyle Y_{t}:E^{T}\to E}$ defined by ${\displaystyle Y_{t}(\omega )=\omega ($t)} . The ${\displaystyle \{Y_{t},t\in T\}}$ form another process. The Wiener measure is then the unique measure on ${\displaystyle C_{0}(\mathbb {R} _{+},\mathbb {R} )}$ such that the coordinate process is a Brownian motion.^[1]

Properties of classical Wiener space[edit]

Uniform topology[edit]

The vector space C can be equipped with the uniform norm

${\displaystyle \|f\|:=\sup _{t\in [0,\,T]}|f($t)|}

turning it into a normed vector space (in fact a Banach space). This norm induces a metriconC in the usual way: ${\displaystyle d(f,g):=\|f-g\|}$. The topology generated by the open sets in this metric is the topology of uniform convergence on [0, T ], or the uniform topology.

Thinking of the domain [0, T ] as "time" and the range Rⁿ as "space", an intuitive view of the uniform topology is that two functions are "close" if we can "wiggle space slightly" and get the graph of f to lie on top of the graph of g, while leaving time fixed. Contrast this with the Skorokhod topology, which allows us to "wiggle" both space and time.

Separability and completeness[edit]

With respect to the uniform metric, 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 is a Polish space.

Tightness in classical Wiener space[edit]

Recall that the modulus of continuity for a function f : [0, T ] → Rⁿ is defined by

${\displaystyle \omega _{f}(\delta ):=\sup \left\{|f($s)-f(t)|:s,t\in [0,T],\,|s-t|\leq \delta \right\}.}

This definition makes sense even if f is not continuous, and it can be shown that f is continuous if and only if its modulus of continuity tends to zero as δ → 0:

${\displaystyle f\in C\iff \omega _{f}(\delta )\to 0{\text{ as }}\delta \to 0}$.

By an application of the Arzelà-Ascoli theorem, one can show that a sequence ${\displaystyle (\mu _{n})_{n=1}^{\infty }}$ofprobability measures on classical Wiener space Cistight if and only if both the following conditions are met:

${\displaystyle \lim _{a\to \infty }\limsup _{n\to \infty }\mu _{n}\{f\in C\mid |f(0)|\geq a\}=0,}$ and

${\displaystyle \lim _{\delta \to 0}\limsup _{n\to \infty }\mu _{n}\{f\in C\mid \omega _{f}(\delta )\geq \varepsilon \}=0}$ for all ε > 0.

Classical Wiener measure[edit]

There is a "standard" measure on C₀, 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 B : [0, T ] × Ω → Rⁿ, starting at the origin, with almost surely continuous paths and independent increments

${\displaystyle B_{t}-B_{s}\sim \,\mathrm {Normal} \left(0,|t-s|\right),}$

then classical Wiener measure γ is the law of the process B.

Alternatively, one may use the abstract Wiener space construction, in which classical Wiener measure γ is the radonification of the canonical Gaussian cylinder set measure on the Cameron-Martin Hilbert space corresponding to C₀.

Classical Wiener measure is a Gaussian measure: in particular, it is a strictly positive probability measure.

Given classical Wiener measure γ on C₀, the product measure γⁿ × γ is a probability measure on C, where γⁿ denotes the standard Gaussian measureonRⁿ.

See also[edit]

• Abstract Wiener space
• Skorokhod space, a generalization of classical Wiener space, which allows functions to be discontinuous
• Wiener process

References[edit]