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.

单词	CameronMartinSpace
释义	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.
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