PlanetMath: Wiener measure

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Wiener measure

(Definition)

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 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

is Brownian motion, the distribution measure

induced on $W(\mathbb{R})$ is called the Wiener measure. That is,

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 $A_1,\ldots,A_n \subset \mathbb{R}$

$\displaystyle P(\{\omega : \omega(t_1)\in A_1,\ldots,\omega(t_n) \in A_n\})$	$\displaystyle =$	$\displaystyle \int_{A_1}\cdots\int_{A_n} p(t_1,0,x_1)p(t_2-t_1,x_1,x_2)\cdots$	(1)
		$\displaystyle \cdots p(t_n-t_{n-1},x_{n-1},x_n) \; dx_1 \cdots \; dx_n,$	(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

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 -measure .

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

"Wiener measure" is owned by neldredge.

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Also defines:

Wiener space, Wiener measure

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Cross-references: nowhere differentiable, properties, defining property, Borel sets, finite sequence, probability measure, induced, measure, distribution, Brownian motion, random variable, sample paths, stochastic process, completion, projection maps, generated by, measurable space, paths, continuous
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This is version 4 of Wiener measure, born on 2006-05-31, modified 2006-05-31.
Object id is 7940, canonical name is WienerMeasure.
Accessed 3514 times total.

Classification:

AMS MSC:

60G15 (Probability theory and stochastic processes :: Stochastic processes :: Gaussian processes)

Pending Errata and Addenda

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