---
tag: 5Z00K7K1HY
kind: object
name: Classical Wiener space
msc: [60J65, 28C20]
instance_of: [06GZ956657]
sources:
  - url: https://en.wikipedia.org/wiki/Classical_Wiener_space
same_as:
  wikidata: Q5128316
  wikipedia: Classical Wiener space
---

## Description

Let $T > 0$ and $n \ge 1$.
The classical Wiener space is the set
$$C_0([0, T]; \R^n) = \set{w \colon [0, T] \to \R^n : w \text{ continuous}, \ w(0) = 0},$$
with the norm $\norm{w} = \sup_{0 \le t \le T} \abs{w(t)}$ and the Wiener measure $\mu$.
The Wiener measure is the unique Borel probability measure on this space under which the coordinate process $W_t(w) = w(t)$ is an $n$-dimensional [[Wiener process|552AWX3QXN]] on $[0, T]$.

Check that $C_0([0, T]; \R^n)$ is a [[Banach space|06GZ956657]]:

1. It is a vector space, and $\norm{\cdot}$ is a norm on it, since a continuous function on a compact interval is bounded.
2. A Cauchy sequence for $\norm{\cdot}$ converges uniformly to a function $w$; a uniform limit of continuous functions is continuous, and $w(0) = 0$.

It is also separable: piecewise linear paths with rational vertices are dense.

## Remarks

The Wiener measure is the law of Brownian motion, seen as a random element of the path space.
The Cameron–Martin space is the subspace of absolutely continuous paths $h$ with $h(0) = 0$ and $\int_0^T \abs{h'(t)}^2 \, dt < \infty$.
Translations of $\mu$ by elements of the Cameron–Martin space give measures equivalent to $\mu$, but the Cameron–Martin space itself has $\mu$-measure zero.
The space is the standard setting of [[Malliavin calculus|1EC7VPHFAY]].

## Explanations

### L1 human

The classical Wiener space is the set of all possible paths of a Brownian particle that starts at the origin.
Each point of the space is a whole continuous path, and two paths are close when they stay close at every time.
The Wiener measure tells how likely each set of paths is, so that "pick a random path" means "run a Brownian motion".