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The vanishing of harmonic one-forms on based path spaces
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Elworthy, K. D. and Yang, Y. (2013) The vanishing of harmonic one-forms on based path spaces. Journal of Functional Analysis, Volume 264 (Number 5). pp. 1168-1196. doi:10.1016/j.jfa.2012.12.008 ISSN 0022-1236.
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Official URL: http://dx.doi.org/10.1016/j.jfa.2012.12.008
Abstract
We prove the triviality of the first L-2 cohomology class of based path spaces of Riemannian manifolds furnished with Brownian motion measure, and the consequent vanishing of L-2 harmonic one-forms. We give explicit formulae for closed and co-closed one-forms expressed as differentials of functions and co-differentials of L-2 two-forms, respectively; these are considered as extended Clark-Ocone formulae. A feature of the proof is the use of the temporal structure of path spaces to relate a rough exterior derivative operator on one-forms to the exterior differentiation operator used to construct the de Rham complex and the self-adjoint Laplacian on L-2 one-forms. This Laplacian is shown to have a spectral gap.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Journal of Functional Analysis | ||||
Publisher: | Academic Press | ||||
ISSN: | 0022-1236 | ||||
Official Date: | 1 March 2013 | ||||
Dates: |
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Volume: | Volume 264 | ||||
Number: | Number 5 | ||||
Page Range: | pp. 1168-1196 | ||||
DOI: | 10.1016/j.jfa.2012.12.008 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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