# 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 CameronMartinSpace 2013-03-22 15:55:56 2013-03-22 15:55:56 neldredge (4974) neldredge (4974) 6 neldredge (4974) Definition msc 60H99 WienerMeasure Cameron-Martin space