# 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 $$. (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.

Title | Cameron-Martin space |
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Canonical name | CameronMartinSpace |

Date of creation | 2013-03-22 15:55:56 |

Last modified on | 2013-03-22 15:55:56 |

Owner | neldredge (4974) |

Last modified by | neldredge (4974) |

Numerical id | 6 |

Author | neldredge (4974) |

Entry type | Definition |

Classification | msc 60H99 |

Related topic | WienerMeasure |

Defines | Cameron-Martin space |