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Revision difference : Cameron-Martin space |
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| \begin{definition} |
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.) This can be thought of as the set of paths with ``finite energy.'' |
| 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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| \end{definition} |
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| This can be thought of as the set of paths with ``finite energy.'' |
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| 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. |
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. |
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