| Project/Area Number |
11640171
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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 |
Basic analysis
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| Research Institution | Nanzan University (2000)
Kyushu University (1999) |
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Principal Investigator |
KUNITA Hiroshi Nanzan Univ.Math.Sci., Professor, 数理情報学部, 教授 (30022552)
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| Co-Investigator(Kenkyū-buntansha) |
YASUDA Kumi Kyushu Univ.Math., Assistant, 大学院・数理学研究科, 助手 (40284484)
SUGITA Hiroshi Kyushu Univ.Math., Associate Professor, 大学院・数理学研究科, 助教授 (50192125)
TANIGUCHI Setuo Kyushu Univ.Math., Associate Professor, 大学院・数理学研究科, 助教授 (70155208)
FUKAI Yasunari Kyushu Univ.Math., Assistant, 大学院・数理学研究科, 助手 (00311837)
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| Project Period (FY) |
1999 – 2000
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| Project Status |
Completed (Fiscal Year 2000)
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| Budget Amount *help |
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2000: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1999: ¥2,400,000 (Direct Cost: ¥2,400,000)
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| Keywords | Stochastic differential equation / Levy process / Malliavin calculus / Hormander's hypoellipticity condition / 加法過程 / 準だ円性 / ストカスティック・フロー / p-進体 / ランダム・ウオーク |
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| Research Abstract |
There are extensive works on SDE (stochastic differential equation) based on Brownian motions. Conditions for the existence of the smooth density for the law of the solution have been clearified by using the Malliavin calculus. In this research, we restricted our attention to SDE with jumps based on Levy process and investigated the condition such that the law of the solution has a(smooth) density. As to the equation, we studied the canonical SDE generated by a finite number of vector fields and the same dimensional Levy processes. First, we showed that the law has a smooth density if both the vector fields and Levy processes are nondegenerate. Then we proved that the law has a density function in the case where the vector fields may be degenearate but satisfy Hormanders condition. These results are extensions of the works by Malliavin and Kusuoka-Stroock, who studied the existence of the smooth density in the case of a SDE driven by a Brownian motion.
For the proof, we need the Malliavin calculus on the product of the Wiener space and the Poisson space. We unified the Picard's approach on the Poisson space and the Malliavin's approach on the Wiener space and further we obtained a criterion that the law of the random variable on the product space has a smooth density. The criterion includes Malliavin's on the Wiener space and Picard's on the Poisson space as special cases. We applied the criterion to the solution of SDE with jumps and proved the existence of the density for the law of the solution.
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