Wiener measure

Definition 1.

The Wiener space $W(ℝ)$ is just the set of all continuous paths $\omega :[0,\mathrm{\infty })\to ℝ$ satisfying $\omega (0)=0$. It may be made into a measurable space by equipping it with the $\sigma$-algebra $ℱ$ generated by all projection maps $\omega \mapsto \omega (t)$ (or the completion of this under Wiener measure, see below).

Thus, an $ℝ$-valued continuous-time stochastic process $X_t$ with continuous sample paths can be thought of as a random variable taking its values in $W(ℝ)$.

Definition 2.

In the case where $X_t=W_t$ is Brownian motion, the distribution measure $P$ induced on $W(ℝ)$ is called the Wiener measure. That is, $P$ is the unique probability measure on $W(ℝ)$ such that for any finite sequence of times and Borel sets $A_1,\mathrm{\dots },A_n\subset ℝ$

$P(\{\omega :\omega (t_1)\in A_1,\mathrm{\dots },\omega (t_n)\in A_n\})$ $=$ ${\displaystyle {\int }_{A_1}}\mathrm{\cdots }{\displaystyle {\int }_{A_n}}p(t_1,0,x_1)p(t_2-t_1,x_1,x_2)\mathrm{\cdots }$ (2)

$\mathrm{\cdots }p(t_n-t_{n-1},x_{n-1},x_n)d{x}_1\mathrm{\cdots }d{x}_n,$

where $p(t,x,y)=\frac{1}{\sqrt{2\pi t}}\exp (-\frac{{(x-y)}^2}{2t})$ defined for any $x,y\in ℝ$ and $t>0$.

This of course corresponds to the defining property of Brownian motion. The other properties carry over as well; for instance, the set of paths in $W(ℝ)$ which are nowhere differentiable is of $P$-measure $1$.

The Wiener space $W({ℝ}^d)$ and corresponding Wiener measure are defined similarly, in which case $P$ is the distribution of a $d$-dimensional Brownian motion.

Title	Wiener measure
Canonical name	WienerMeasure
Date of creation	2013-03-22 15:55:53
Last modified on	2013-03-22 15:55:53
Owner	neldredge (4974)
Last modified by	neldredge (4974)
Numerical id	7
Author	neldredge (4974)
Entry type	Definition
Classification	msc 60G15
Related topic	BrownianMotion
Related topic	CameronMartinSpace
Defines	Wiener space
Defines	Wiener measure