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Some problems on stochastic analysis on path and loop spaces
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Chen, Xin, Ph.D. (2011) Some problems on stochastic analysis on path and loop spaces. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2569141~S1
Abstract
In the thesis, some problems on the stochastic analysis on path and loop space over manifold are studied. In particular, in the 2nd Chapter, the Poincaré inequality for O-U Dirichlet form and pinned Wiener measure on loop space over hyperbolic space is proved and some weighted inequalities for other reference measure on the same space are also derived. In the 3rd Chapter, we give a concrete estimate for the rate function of weak Poincaré inequality (for O-U Dirichlet form and pinned Wiener measure) on loop space over compact simply connected manifold with strictly positive Ricci curvature and use that to prove F-Sobolev inequality for the reference measure induced by the ground state of a Schrodinger operator. In the 4th Chapter, an integration by parts formula for free loop space over noncompact manifold are derived under some curvature and heat kernel estimate condition. In the 5th Chapter, for several kinds of SDE with non-Lipschitz coefficients, the approximation results for their derivative processes are obtained and are used to prove differential formula and some integration by parts formula on path space endowed with corresponding probability measure as the distribution of such SDE. In the 6th chapter, the results in Chapter 5 are generalized to SDE with non-Lipschitz drift coefficients on complete compact Riemannian manifolds.
| Item Type: | Thesis or Dissertation (PhD) |
|---|---|
| Subjects: | Q Science > QA Mathematics |
| Library of Congress Subject Headings (LCSH): | Stochastic analysis, Loop spaces, Manifolds (Mathematics) |
| Date: | June 2011 |
| Institution: | University of Warwick |
| Theses Department: | Mathematics Institute |
| Thesis Type: | PhD |
| Publication Status: | Unpublished |
| Supervisor(s)/Advisor: | Li, X-M. (Xue-Mei), 1964- |
| Extent: | v, 187 leaves |
| Language: | eng |
| URI: | http://wrap.warwick.ac.uk/id/eprint/47194 |
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