PlanetMath: Cameron-Martin space

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Cameron-Martin space

(Definition)

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 $\int_0^\infty \vert\omega'(s)\vert^2\,ds < \infty$ . (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.

"Cameron-Martin space" is owned by neldredge.

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