2005 Fiscal Year Final Research Report Summary
A study of stochastic analysis - synthesizing and integrating
| Project/Area Number |
14204010
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | KYUSHU UNIVERSITY |
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Principal Investigator |
TANIGUCHI Setsuo Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (70155208)
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| Co-Investigator(Kenkyū-buntansha) |
HAMACHI Toshihiro Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (20037253)
SHIRAI Tomoyuki Kyushu University, Faculty of Mathematics, Associate Professor, 大学院・数理学研究院, 助教授 (70302932)
FUKAI Yasunari Kyushu University, Faculty of Mathematics, Research Assistant, 大学院・数理学研究院, 助手 (00311837)
MATSUMOTO Hiroyuki Nagoya University, Graduate School of Information Science, Professor, 大学院・情報科学研究科, 教授 (00190538)
KUMAGAI Takashi Kyoto University, Research Institute of Mathematical Science, Associate Professor, 数理解析研究所, 助教授 (90234509)
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| Project Period (FY) |
2002 – 2005
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| Keywords | stochastic analysis / path integral / Brownian motion / stochastic oscillatory integral / heat kernel / asymptotic behavior / random walk / Schrodinger operator |
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| Research Abstract |
(1)As for stochastic oscillatory integrals (SOI in short), which are Fourie-Laplace transforms on the path space, studies on exact expression, asymptotic behavior, and application to nonlinear partial differential equation were made. In the case of phase function of quadratic Wiener functional, a Levy-Ito type exponential expression was established with the associated Hilbert-Schmidt operator. Moreover, in the case of phase function of stochastic line integral with polynomial coefficients, a concrete asymptotic behavior was shown. A door to the stationary phase method on the path space was opened by showing the concentration around stationary points of the asymptotic behavior in the case of Gaussian amplitude function. Finally, we found out a bijective relation between the tau function of the KdV hierarchy and SOI' s associated with Ornstein-Uhlenbeck processes, which is a completely new application of stochastic analysis to nonlinear PDE theory. (2)We made studies on concrete functionals on path spaces. We computed the distribution of the exponential functional obtained as time-integral of geometric Brownian motion, which plays a key role in studies of mathematical finance, diffusions in random environments, and Brownian motion on hyperbolic space, and established the recursive formula for its density function. Moreover, the absolute continuity of the distribution of the Wishart process was computed, and a probabilistic proof of Selberg trace formulas for Laplacian on forms on hyperbolic spaces was given. Furthermore, the trace formula on p-adic upper half plane was studied via semi-stable processes. (3)An easily-applicable criterion for heat kernel on general space to have a sub-Gaussian estimate was achieved. Diffusion processes on complexity were constructed, and a detailed estimation of associated heat kernel was established.
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