Classical Wiener space

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.

Consider ${\displaystyle E\subseteq \mathbb {R} ^{n}}$ and a metric space ${\displaystyle (M,d)}$. The classical Wiener space ${\displaystyle C(E,M)}$ is the space of all continuous functions ${\displaystyle f:E\to M.}$ That is, for every fixed ${\displaystyle t\in E,}$

In almost all applications, one takes ${\displaystyle E=[0,T]}$ or ${\displaystyle E=\mathbb {R} _{+}=[0,+\infty )}$ and ${\displaystyle M=\mathbb {R} ^{n}}$ for some ${\displaystyle n\in \mathbb {N} .}$ For brevity, write ${\displaystyle C}$ for ${\displaystyle C([0,T]);}$ this is a vector space. Write ${\displaystyle C_{0}}$ for the linear subspace consisting only of those functions that take the value zero at the infimum of the set ${\displaystyle E.}$ Many authors refer to ${\displaystyle C_{0}}$ as "classical Wiener space".

The vector space ${\displaystyle C}$ can be equipped with the uniform norm

turning it into a normed vector space (in fact a Banach space since ${\displaystyle [0,T]}$ is compact). This norm induces a metric on ${\displaystyle C}$ 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 ${\displaystyle [0,T],}$ or the uniform topology.

Thinking of the domain ${\displaystyle [0,T]}$ as "time" and the range ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle f}$ to lie on top of the graph of ${\displaystyle g}$, while leaving time fixed. Contrast this with the Skorokhod topology, which allows us to "wiggle" both space and time.

If one looks at the more general domain ${\displaystyle \mathbb {R} _{+}}$ with

then the Wiener space is no longer a Banach space, however it can be made into one if the Wiener space is defined under the additional constraint