2013 Fiscal Year Final Research Report
Infinite dimensional stochastic analysis and geometry
| Project/Area Number |
21340030
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Kyoto University |
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Principal Investigator |
SHIGEKAWA Ichiro 京都大学, 理学(系)研究科(研究院), 教授 (00127234)
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| Co-Investigator(Kenkyū-buntansha) |
KUMAGAI Takashi 京都大学, 数理解析研究所, 教授 (90234509)
SUGITA Hiroshi 大阪大学, 大学院・理学研究科, 教授 (50192125)
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| Co-Investigator(Renkei-kenkyūsha) |
AIDA Shigeki 東北大学, 大学院・理学研究科, 教授 (90222455)
HINO Masanori 大阪大学, 大学院・基礎工学研究科, 教授 (40303888)
MATSUMOTO Hiroyuki 青山学院大学, 理工学部, 教授 (00190538)
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| Project Period (FY) |
2009-04-01 – 2014-03-31
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| Keywords | マルコフ過程 / エルゴード性 / マルコフ半群 / 生成作用素のスペクトル / 対数 Sobolev 不等式 / 超縮小性 / 確率解析 |
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| Research Abstract |
We studied Markov processes by using the stochastic analysis. We are mainly interested in the behavior of the semigroup associated with a Markov process. First we considered diffusion processes on a Riemannian manifold. We give a generator as a sum of the Laplace-Beltrami operator and a vector field. We discussed the uniqueness of the semigroup associated with the generator. The condition was given in terms of the vector field.
Next we consider the conditions for which the semigroup reserves a convex set. We consider this problem in the framework of general Banach space. We give some necessary and sufficient conditions in terms of generator. In Hilbert space setting, we formulate this problem by using Dirichlet forms.
In addition, we considered the rate of convergence of the semigroup under the condition of logarithmic Sobolev inequality. We also discussed the dual ultracontractive property of the semigroup.
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