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



"Cameron-Martin space" is owned by neldredge.
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See Also: Wiener measure

Also defines:  Cameron-Martin space
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Cross-references: nowhere differentiable, Brownian motion, sample paths, Wiener measure, integral, differentiable, almost everywhere, absolutely continuous, paths, subspace, Wiener space

This is version 3 of Cameron-Martin space, born on 2006-05-31, modified 2006-05-31.
Object id is 7941, canonical name is CameronMartinSpace.
Accessed 2243 times total.

Classification:
AMS MSC: 60H99 (Probability theory and stochastic processes :: Stochastic analysis :: Miscellaneous)
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