2002 Fiscal Year Final Research Report Summary
Deformation of 2-dimensional diffusion processes which preserves recurrence
| Project/Area Number |
12640127
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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 | HIROSHIMA UNIVERSITY |
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Principal Investigator |
IWATA Koichiro Hiroshima Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20241292)
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| Co-Investigator(Kenkyū-buntansha) |
TAKEDA Masayoshi Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (30179650)
KUBO Izumi Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (70022621)
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| Project Period (FY) |
2000 – 2002
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| Keywords | 2-dimensional diffusion / recurrence and boundary / deformation of complex structures |
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| Research Abstract |
The uniformization theorem foe Riemann surfaces claims that complex structures on non-compact simply connected surfaces are equivalent to either the standard complex structure on the complex plain or the one on the unit disk. In terms of Brownian motions this classification is completely described by the recurrence of the process on the domain. On the complex plain the L^∞-norm of Beltrami-coefficients must be 1 as for those complex structures not equivalent to the standard one, in other words, if the corresponding Brownian motion is transient then the L^∞-norm of Beltrami-coefficient is 1. However this does not mean that every complex structure on the complex plain whose Beltrami-coefficient has unit L^∞-norm corresponds to transient Brownian motions. The aim of the present research project is to study such complex structures. One can construct a one-parameter family of complex structures with critical point and at the critical point the recurrence and the transience switch to the other. The model of the complex structures discussed in the present research is parameterized by the closed unit disk and the modulus of the parameter describes the L^∞-norm of the corresponding Beltrami-coefficient. When the parameter lies in the open unit disk the prescribed complex structure is equivalent to the standard one. On the boundary of the unit disk except for one point the complex structure is equivalent to the standard one on the unit disk and thus the Brownian motion is transient. At the critical point the complex structure is recurrent but the freedom of deformation that preserves recurrence is relatively low. A natural direction for further direction is as follows : Study the action of the modular group on the space of Beltrami-coefficients and the action of the modular group on the boundary.
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