# Wiener measure

###### Definition 1.

The *Wiener space* $W(\mathbb{R})$ is just the set of all continuous^{} paths $\omega :[0,\mathrm{\infty})\to \mathbb{R}$ 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 $$ and Borel sets ${A}_{1},\mathrm{\dots},{A}_{n}\subset \mathbb{R}$

$P(\{\omega :\omega ({t}_{1})\in {A}_{1},\mathrm{\dots},\omega ({t}_{n})\in {A}_{n}\})$ | $=$ | ${\int}_{{A}_{1}}}\mathrm{\cdots}{\displaystyle {\int}_{{A}_{n}}}p({t}_{1},0,{x}_{1})p({t}_{2}-{t}_{1},{x}_{1},{x}_{2})\mathrm{\cdots$ | (2) | ||

$\mathrm{\cdots}p({t}_{n}-{t}_{n-1},{x}_{n-1},{x}_{n})d{x}_{1}\mathrm{\cdots}d{x}_{n},$ |

where $p(t,x,y)=\frac{1}{\sqrt{2\pi t}}\mathrm{exp}(-\frac{{(x-y)}^{2}}{2t})$ 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 |
---|---|

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 |