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 $0<t_{1}<\\ldots<t_{n}$ 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)
			$\\displaystyle\\cdots p(t_{n}-t_{n-1},x_{n-1},x_{n})\\;dx_{1}\\cdots\\;dx_{n},$		(2)

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.