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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

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Author:Nadja Sidorova
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
Date of the Publication (Server):2003/09/29
Tag:Brownsche Bewegung; Oberflächenmaße; Riemannsche Mannigfaltigkeiten; Stochastische Processe
Brownian motion; Riemannian manifolds; stochastic processes; surface measures
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011