1996 Fiscal Year Final Research Report Summary
Stochastic processes with applications to statistical mechanics
| Project/Area Number |
06452015
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Nagoya University |
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Principal Investigator |
ICHIHARA Kenji Nagoya University, Graduate School of Polymathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (00112293)
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| Co-Investigator(Kenkyū-buntansha) |
SUGIURA Makoto Nagoya Universty, Graduate School of Polymathematics, Research Assosiate, 大学院・多元数理科学研究科, 助手 (70252228)
KUMAGAI Takasi Nagoya Universty, Graduate School of Polymathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (90234509)
OBATA Nobuaki Nagoya Universty, Graduate School of Polymathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (10169360)
AOMOTO Kazuhiko Nagoya Universty, Graduate School of Polymathematics, Professor, 大学院・多元数理科学研究科, 教授 (00011495)
FUNAKI Tadahisa Tokyo Universty, Graduate School of Mathematical Science, Professor, 大学院数理科学研究科, 教授 (60112174)
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| Project Period (FY) |
1994 – 1996
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| Keywords | hydrodynamical limit / Stochastic partial diflerential equation / Stochastic Burgers equation / Harnack's inequality / random environment / birth and dath process / quauntum stochascic process / fractal |
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| Research Abstract |
T.Funaki firstly studied some basic problems (derivation of nonlinear equations, etc) in hydrodynamical limit for Hamiltonian systems and lattice gas. Then he took up singular limit problem for reaction-diffusion equations with noise as reaction term tends to infinity. It has been shown for the problem that the solutions of the equations approach +1 or -1 pointwise in space and time. Accordingly the generation of random interfaces between two phases +1, -1 occurs. As the third topic, Burgers equations were treated. He proved that non-Gaussian distributions appear as a scaling limit of solutions for Burgers equations with random initial value. Furthermore various properties of solutions for a class of generalized Burgers-type equations with a fractionl power of the Laplacian were studied.
K.Ichihara established a global Harnack inequlity for a class of symmetrizable difference operators on finitely generated groups of polynomial growth order. It was then applied to show a Liouville-property such operators. He secondly introduced birth and death processes in a class of time-dependent random environments. The recurrence and transience problem for the above processes was discussed by means of a renormalization technique. In multi-dimensicnal lattices some examples of recurrent and transient processes have been constructed respectively.
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