Cameron-Martin space

Definition 1.

Let $W(\mathbb{R}^{d})$ be Wiener space. The Cameron-Martin space $H(\mathbb{R}^{d})$ is the subspace of $W(\mathbb{R}^{d})$ consisting of all paths $\omega$ such that $\omega$ is absolutely continuous and $\int_{0}^{\infty}|\omega^{\prime}(s)|^{2}\,ds<\infty$ . (Note that if $\omega$ is absolutely continuous, then it is almost everywhere differentiable, so the integral makes sense.)

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

Note that $H(\mathbb{R}^{d})$ has Wiener measure $0$ , since sample paths of Brownian motion are nowhere differentiable, whereas a path from $H(\mathbb{R}^{d})$ 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