1998 Fiscal Year Final Research Report Summary
Research on random phenomena by the methods of modern mathematics
| Project/Area Number |
07404005
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
WATANABE Shinzo Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90025297)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHIDA Nobuo Hokkaido Univ., Graduate School of Science, Lecturer, 大学院・理学研究科, 講師 (40240303)
KOKUBU Hiroshi Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50202057)
SHIGEKAWA Ichiro Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (00127234)
TANIGUCHI Masahiko Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (50108974)
NISHIDA Takaaki Kyoto Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70026110)
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| Project Period (FY) |
1995 – 1998
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| Keywords | Sobolev spaces on Wiener space / measure valued branching processes / Brownian snake / innovation problem / bifurcations for equations in fluid dynamics / bifurcation problem in dynamical system / complex dynamics of entirefunctions / lattice spin system |
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| Research Abstract |
We studied on problems of mathematical models in random phenomena ;
stochastic processes, functionals on Wiener path and loop spaces, lattice spin systems, dynamical systems and models in fluid dynamics. We list main results :
1. We defined Sobolev spaces on Wiener space and studied regularity of Wiener path integrals and conditional Wiener path integrals in terms of Sobolev spaces.
We obtained a comparison theorem for Markovian semigroups which can be applied to analysis on Wiener space.
2. We studied structures and operations concerning measure valued branching processes (superprocesses) by using the notion of Brownian snakes due to Le Gall.
We studied the innovation problem for stochastic processes and obtained some example of stochastic processes which generate non-cosy filtrations in the sense of Tsirelson.
3. We gave a computer aided proof in bifurcation problems for equations of fluid dynamics.
4. We studied on a degenerating singularity of vector fields inconnection with bifurcations and chaotic behavior of dynamical systems.
Also we obtained new results on bifurcations of homoclinic orbits.
5. We studied complex dynamical systems by the theory of Teichmuller spaces and constructed a theory of complex dynamics by entire functions.
6. We studied relaxation problem of Glauber dynamics in lattice spin systems.
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