Cameron-Martin space

Definition 1.

Let W(d) be Wiener space. The Cameron-Martin space H(d) is the subspace of W(d) consisting of all paths ω such that ω is absolutely continuousMathworldPlanetmath and 0|ω(s)|2𝑑s<. (Note that if ω is absolutely continuous, then it is almost everywhere differentiableMathworldPlanetmathPlanetmath, so the integral makes sense.)

This can be thought of as the set of paths with “finite energy.”

Note that H(d) has Wiener measure 0, since sample paths of Brownian motionMathworldPlanetmath are nowhere differentiable, whereas a path from H(d) is almost everywhere differentiable.

Title Cameron-Martin space
Canonical name CameronMartinSpace
Date of creation 2013-03-22 15:55:56
Last modified on 2013-03-22 15:55:56
Owner neldredge (4974)
Last modified by neldredge (4974)
Numerical id 6
Author neldredge (4974)
Entry type Definition
Classification msc 60H99
Related topic WienerMeasure
Defines Cameron-Martin space