Wiener measure

Definition 1.

The Wiener space W() is just the set of all continuousMathworldPlanetmathPlanetmath paths ω:[0,) satisfying ω(0)=0. It may be made into a measurable spaceMathworldPlanetmathPlanetmath by equipping it with the σ-algebra generated by all projection maps ωω(t) (or the completionPlanetmathPlanetmath of this under Wiener measure, see below).

Thus, an -valued continuous-time stochastic process Xt with continuous sample paths can be thought of as a random variableMathworldPlanetmath taking its values in W().

Definition 2.

In the case where Xt=Wt is Brownian motionMathworldPlanetmath, the distributionPlanetmathPlanetmathPlanetmath measureMathworldPlanetmath P induced on W() is called the Wiener measure. That is, P is the unique probability measure on W() such that for any finite sequencePlanetmathPlanetmath of times 0<t1<<tn and Borel sets A1,,An

P({ω:ω(t1)A1,,ω(tn)An}) = A1Anp(t1,0,x1)p(t2-t1,x1,x2) (2)

where p(t,x,y)=12πtexp(-(x-y)22t) defined for any x,y 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() which are nowhere differentiableMathworldPlanetmathPlanetmath is of P-measure 1.

The Wiener space W(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