1999 Fiscal Year Final Research Report Summary
Symmetric Markov processes and Dirichlet forms
| Project/Area Number |
09640265
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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 | TOHOKU UNIVERSITY (1998-1999)
Osaka University (1997) |
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Principal Investigator |
TAKEDA Masayoshi Mathematics, Tohoku Univ., Professor, 大学院・理学研究科, 教授 (30179650)
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| Co-Investigator(Kenkyū-buntansha) |
TACHIZAWA Kazuya Mathematics, Tohoku Univ., Lecturer, 大学院・理学研究科, 講師 (80227090)
IGARI Satoru Mathematics, Tohoku Univ., Professor, 大学院・理学研究科, 教授 (50004289)
TAKAGI Izumi Mathematics, Tohoku Univ., Professor, 大学院・理学研究科, 教授 (40154744)
NAGAI Hideo Mathematical Science, Osaka Univ., Professor, 大学院・基礎工学研究科, 教授 (70110848)
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| Project Period (FY) |
1997 – 1999
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| Keywords | symmetric Markov process / Dirichlet form / large deviation / additive functional / Feynman-Kac formula |
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| Research Abstract |
The objective of this study is to investigate symmetric Markov processes by using Dirichlet form theory. Symmetric Markov processes are a special class in Donsker-Varadhan type large deviation theory in the sense that the rate functions of large deviation principle are given by the associated Dirichlet forms. In 1984, Fukushima and I showed that symmetric Markov processes can be transformed to ergodic processes by some supermartingale multiplicative functionals even if a symmetric Markov process is explosive or has the killing inside. As a result, Donsker-Varadhan type large deviation principle could be extended to symmetric Markov processes with finite lifetime. In this study, I found a new sufficient condition for the upper estimate holding for not only compact sets but also for closed sets. In fact, I showed that the full large deviation principle holds if the Markov process explodes so fast that the 1-resolvent of the identity function belongs to the space of continuous functions vanishing at infinity. As a corollary of this result, I showed LィイD1pィエD1-independence of the spectral radius of symmetric Markov semigroups. And I applied it to obtain a necessary and sufficient condition for the integrability of Feynman-Kac functionals. This result also gives us an criterion whether a Schrodinger operators is subcritical or not.
We further extended the large deviation principle to Markov processes with Feynman-Kac functional, and consider asymptotic properties of Feynman-Kac semigroups.
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