Wiener measureWienerMeasure2006-05-31 02:26:572006-05-31 03:28:40DefinitionWiener spaceWiener measure% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{definition}{Definition}\begin{definition}
The \emph{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).
\end{definition}
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 $W(\mathbb{R})$.
\begin{definition}
In the case where $X_t = W_t$ is Brownian motion, the distribution measure $P$ induced on $W(\mathbb{R})$ is called the \emph{Wiener measure}. That is, $P$ is the unique probability measure on $W(\mathbb{R})$ such that for any finite sequence of times $0<t_1<\ldots<t_n$ and Borel sets $A_1,\ldots,A_n \subset \mathbb{R}$
\begin{eqnarray}
P(\{\omega : \omega(t_1)\in A_1,\ldots,\omega(t_n) \in A_n\}) &=& \int_{A_1}\cdots\int_{A_n} p(t_1,0,x_1)p(t_2-t_1,x_1,x_2)\cdots \\
&& \cdots p(t_n-t_{n-1},x_{n-1},x_n) \; dx_1 \cdots \; dx_n,
\end{eqnarray}
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 $t>0$.
\end{definition}
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.