2007 Fiscal Year Final Research Report Summary
Large Deviation and Random Media
| Project/Area Number |
17540133
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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 | Keio University |
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Principal Investigator |
TAMURA Yozo Keio University, Faculty of Science and Tbchnolog, Associate Professor (50171905)
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| Co-Investigator(Kenkyū-buntansha) |
MAEJIMA Makoto Keio Univ., Faculty of Science and Technology, Professor (90051846)
TANAKA Hiroshi Keio Univ., Faculty of Science and Technology, Emeritus Professor (70011468)
SUZUKI Yuki Keio Univ., School of Medicine, Assistant Professor (30286645)
CHIYONOBU Taizo Kansei Gakuin Univ., Fac. Science and Tech., Associate Professor (50197638)
TAKAHASHI Hiroshi the Institute of Physical and Chemical Research, 理化学研究所, Researcher (30413826)
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| Project Period (FY) |
2005 – 2007
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| Keywords | Large Deviation Principle / Current / Random Media |
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| Research Abstract |
We investigated on the stochastic line integrals of differentiable 1-forms on a compact Riemannian manifold. There were only few papers on the large deviation principle for the currents generated by the stochastic line integrals. Then, first we aimed to formulate the large deviation principle for currents generated by stochastic line integrals in an explicit way. For this purpose, we considered the large deviation principle of the pair of the current and the empirical measure generated by a path of the Brownian motion on the compact Riemannian manifold. By considering this pair ; we could have a clear expression of the rate function of the large deviation principle. Furthermore, we succeeded to get the explicit representation of the rate function of the large deviation principle.
We also investigated the topology of the space of currents in which the large deviation principle related to the stochastic line integrals was considered. In former papers, the topology of currents was given by the topology of square integrability. This topology was given in the form depending on the dimension of the Riemannian manifold. By using new estimates, we could get a topology of the space of currents generated of stochastic line integrals, which is independent of the dimension of the manifold.
We also investigated the recurrence and transience of the diffusion processes in multi-dimensional Brownian and reflected Brownian random environments.
We also got the following result related to the multi-dimensional random environments. By using our Wiener-Hopf type path decomposition formula, we got a representation formula of a path decomposition of the Green function of the multi-dimensional general Levy processes in the half space. Applying this formula, we got the explicit formula of the Green functions of the rotation invariant absorbing stable processes in the half space.
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Research Products
(8 results)
