Wiener measure

Definition 1.

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

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

Definition 2.

In the case where $X_{t}=W_{t}$ is Brownian motion, the distribution measure $P$ induced on $W(\mathbb{R})$ is called the Wiener measure. That is, $P$ is the unique probability measure on $W(\mathbb{R})$ such that for any finite sequence of times $0 and Borel sets $A_{1},\ldots,A_{n}\subset\mathbb{R}$

 $\displaystyle P(\{\omega:\omega(t_{1})\in A_{1},\ldots,\omega(t_{n})\in A_{n}\})$ $\displaystyle=$ $\displaystyle\int_{A_{1}}\cdots\int_{A_{n}}p(t_{1},0,x_{1})p(t_{2}-t_{1},x_{1}% ,x_{2})\cdots$ (2) $\displaystyle\cdots p(t_{n}-t_{n-1},x_{n-1},x_{n})\;dx_{1}\cdots\;dx_{n},$

where $p(t,x,y)=\frac{1}{\sqrt{2\pi t}}\exp(-\frac{(x-y)^{2}}{2t})$ defined for any $x,y\in\mathbb{R}$ 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 $W(\mathbb{R})$ which are nowhere differentiable is of $P$-measure $1$.

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

Title Wiener measure WienerMeasure 2013-03-22 15:55:53 2013-03-22 15:55:53 neldredge (4974) neldredge (4974) 7 neldredge (4974) Definition msc 60G15 BrownianMotion CameronMartinSpace Wiener space Wiener measure