## Surface Measures on Paths in an Embedded Riemannian Manifold

- We construct and study two surface measures on the space C([0,1],M) of paths in a compact Riemannian manifold M embedded into the Euclidean space R^n. The first one is induced by conditioning the usual Wiener measure on C([0,T],R^n) to the event that the Brownian particle does not leave the tubular epsilon-neighborhood of M up to time T, and passing to the limit. The second one is defined as the limit of the laws of reflected Brownian motions with reflection on the boundaries of the tubular epsilon-neighborhoods of M. We prove that the both surface measures exist and compare them with the Wiener measure W_M on C([0,T],M). We show that the first one is equivalent to W_M and compute the corresponding density explicitly in terms of the scalar curvature and the mean curvature vector of M. Further, we show that the second surface measure coincides with W_M. Finally, we study the limit behavior of the both surface measures as T tends to infinity.
- Oberflächenmaße auf Räumen von mannigfaltigkeitswertigen Pfaden

Author: | Nadja Sidorova |
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URN (permanent link): | urn:nbn:de:bsz:386-kluedo-16268 |

Advisor: | Günter Trautmann |

Document Type: | Doctoral Thesis |

Language of publication: | English |

Year of Completion: | 2003 |

Year of Publication: | 2003 |

Publishing Institute: | Technische Universität Kaiserslautern |

Granting Institute: | Technische Universität Kaiserslautern |

Acceptance Date of the Thesis: | 2003/07/22 |

Tag: | Brownsche Bewegung; Oberflächenmaße; Riemannsche Mannigfaltigkeiten; Stochastische ProcesseBrownian motion; Riemannian manifolds; stochastic processes; surface measures |

Faculties / Organisational entities: | Fachbereich Mathematik |

DDC-Cassification: | 510 Mathematik |