Cameron-Martin spaceCameronMartinSpace2006-05-31 02:51:362006-05-31 03:04:23DefinitionCameron-Martin space% this is the default PlanetMath preamble. as your knowledge
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Let $W(\mathbb{R}^d)$ be Wiener space. The \emph{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.)
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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.