2005 Fiscal Year Final Research Report Summary
Stochastic analysis and its application to analysis of differential operators
| Project/Area Number |
15540116
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Nagoya University |
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Principal Investigator |
MATSUMOTO Hiroyuki Nagoya University, Graduate School of Information Science, Professor, 大学院・情報科学研究科, 教授 (00190538)
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| Co-Investigator(Kenkyū-buntansha) |
OSADA Hirohumi Kyushu University, Gradiate School of of Mathematics, Professor, 大学院・数理学研究院, 教授 (20177207)
SATO Junya Nagoya University, Graduate School of Information Science, Associate Professor, 大学院・情報科学研究科, 助教授 (20235352)
UEMURA Hideaki Aichi University of Education, faculty of Education, Associate professor, 教育学部, 助教授 (30203483)
LIANG Song Tohoku University, Graduate School of Information Science, Associate Professor, 大学院・情報科学研究科, 助教授 (60324399)
KAISE Hidehiro Nagoya University, Graduate School of Information Science, Assistant, 大学院・情報科学研究科, 助手 (60377778)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Poncare upper half plane / Laplacian / Brownian motions / Selberg trace formula / exponential Wiener functionals |
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| Research Abstract |
The stochastic differential equation for the Brownian motion on the Poincare upper half plane (the hyperbolic plane), a diffusion process generated by a half of the Laplacian, is explicitly solved and we have an concrete representation for the Brownian motion as a Wiener functional. In this research, as an extension of this known fact, we showed that the horizontal life of the Brownian motion to the bundle of orthonormal frames also has an expression as a Wiener functional. Based on this representation, we may show a probabilistic representation for the heat kernel of the Laplacian acting on the differential forms and give a proof of the Selberg trace formula for the differential 1-forms on a compact Riemannian surface, which may be given as a quotient space of the upper half plane by a hyperbolic discrete subgroup of the isometry group. This is an analytic and/or geometric proof and we do not need the harmonic analysis. Moreover we have obtained the Selberg trace formula in a very exp
… More
licit form, since we have restrict ourselves to the two-dimensional case.
The research for an extension to the general dimension case has been continued. In the two dimensional case, we can represent the rotation part of the horizontal lift by using an auxiliary one-dimensional Brownian motion and this representation plays a crucial role. We have not found a corresponding representation and this should be the next task. If we find such a representation, we will be able to give a proof for the Selberg trace formula following the idea of McKean which has been the basis of this research.
When we apply probability theory to the analysis on the upper half plane, the exponential Wiener functionals which is an integral of a geometric Brownian motion appear. Some studies on these functionals has been continued since the Wiener functionals of the same type also appear in the theory of Mathematical Finance and a study for some diffusion processes in random environments. In a joint project with Professot YOR, who is a foreign co-worker in thie research, we gathered some results and applications of these exponential Wiener functionals and gave an insight from analytic point of view. The results have been published in a journal. Less
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Research Products
(12 results)
