2000 Fiscal Year Final Research Report Summary
Research on Diffusion Processes in Random Environments
| Project/Area Number |
11640137
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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 | Japan Women's University |
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Principal Investigator |
TANAKA Hiroshi Japan Women's Univ.Science Professor, 理学部, 教授 (70011468)
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| Co-Investigator(Kenkyū-buntansha) |
SUZUKI Yuki Keio Univ. Medicine Lecturer, 医学部, 専任講師 (30286645)
KAWAZU Kiyoshi Yamaguchi Univ. Education Professor, 教育学部, 教授 (70037258)
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| Project Period (FY) |
1999 – 2000
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| Keywords | Brownian motion / Diffusion process / Random environment / Brownian potential / Levy process / Self-similar process / Subordination / Brownian motion with multi-dimensional time |
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| Research Abstract |
1. It was proved that the limit process arising in a suitably scaled reflecting diffusion process, which moves in a Brownian environment of [0, ∞), is a time-inhomogeneous self-similar Levy process. Its Levy measure was computed explicitly.
2. It was shown that the long-term behavior of diffusion process with a one-sided Brownian potential in (-∞, ∞) exhibits a phenomenon quite different from those (other) models studied previously. In fact, under the usual Brownian scaling the limit process is a reflecting Brownian motion on [0, ∞) with "probability 1/2" and vanishes identically with "the other probability 1/2".
3. As a preliminary discussion toward the investigation of a diffusion process in a multi-dimensional Brownian environment. some new results on extrema of a Brownian motion with a multi-dimensional time were obtained.
4. Bochner's subordination of time-homogeneous Markov processes was extended, with modification, to a considerably wider class of time-inhomogeneous Markov processes. In connection with this, some explicit formula for a class of multiplicative functionals of (discontinous) Markov processes was also studied.
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