2002 Fiscal Year Final Research Report Summary
Stochastic Analysis and its application to differential operators
| Project/Area Number |
12640117
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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 |
MATUMOTO Hiroyuki Nagoya University, Graduate School of Human Informatics., Professor, 大学院・人間情報学研究科, 教授 (00190538)
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| Co-Investigator(Kenkyū-buntansha) |
UEMURA Hideaki Aichi Univ.of Education, Dept.Math., Ass.Prof., 教育学部, 助教授 (30203483)
IHARA Shunsuke Nagoya Univ., Dept.Info.Sci., Professor, 情報文化学部, 教授 (00023200)
ITO Masayuki Nagoya Univ., Dept.Info.Sci., Proferssor, 情報文化学部, 教授 (60022638)
UEKI Naomasa Kyoto Univ., Graduate School of Human Eiviroments, Ass.Prof., 大学院・人間環境学研究科, 助教授 (80211069)
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| Project Period (FY) |
2000 – 2002
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| Keywords | Brownian motion / Wiener functionals / Mathematical finance / Hyperbolic spaces / Pitman's theorem / Levy's theorem / Laplacian |
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| Research Abstract |
If we consider a standard Brownian motion and its maximum process, the difference has a same probability law as reflecting Brownian motion and the difference between twice the maximun process and the original Brownian motion has the same law as a three dimensional Bessel process. These results are well known as Levy's and Pitman's theorems. The head investigator Matsumoto, in joint work with Yor, showed analogus or extensions of these theorems for an exponential Brownian functionals by using the classical Laplace method. The exponential Brownian functional appears also in the studies of mathematical finance and in stochastic analysis on hyperbolic spaces. Various aspects of this functional and their applications have been given under the support of this grant. Among others, some explicit expressions for the heat kernels of the semigroups generated by the Laplacians on non-compact symmetric spaces of rank one have been shown by using the results on the exponential Wiener functionals.
Other investigators gave many suggestions and comments on the works mentioned above and also worked on their own fields. Ihara studied information theory by applying the theory of large deviations. Uemura studied on the local time of multi-dimensional Brownian motions, which should be considered as generalized Wiener functionals. Ueki studied on the spectral properties of Schrodinger operators with random electro-magnetic fields.
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