| Project/Area Number |
59460006
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kyushu University |
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Principal Investigator |
KUNITA Hiroshi Professor at Faculty of Engineering, Kyushu University, 工学部, 教授 (30022552)
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| Co-Investigator(Kenkyū-buntansha) |
TANIGUCHI Setsuo Lecturer at Faculty of Engineering, Kyushu University, 工学部, 講師 (70155208)
WATANABE Hisao Professor at Faculty of Engineering, Kyushu University, 工学部, 教授 (40037677)
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| Project Period (FY) |
1984 – 1986
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| Project Status |
Completed (Fiscal Year 1986)
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| Budget Amount *help |
¥7,500,000 (Direct Cost: ¥7,500,000)
Fiscal Year 1986: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 1985: ¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 1984: ¥3,000,000 (Direct Cost: ¥3,000,000)
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| Keywords | Stochastic differential equations / Stochastic flows / Central limit theorems / The Malliavin calculus / 偏微分方程式の境界値問題 |
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| Research Abstract |
1. The relationship between a stochastic differential equation and the stochastic flow of diffeomorphisms defined by it has been studied intensively since 1975. In particular, in case where the stochastic differential equation is based on Brownian motions, the problem was solved at the beginning of 1980. In this research, we studied the case where the stochastic differetntial equation is defined based on point processes or Levy processes with jumps, and showed that the solution defines a stochastic flow of the semigroup of smooth maps. Further we gave a sufficienct condition that it becomes a stochastic flow of diffeomorphisms.
2. Asymptotic behaviors of solutions of differential equations with random coefficients are important objects of the research in the systems theory in engineering and the theory of population genetics in mathematical biology. Further they are also interesting as a mathematical theory, since these can be regarded as an application of the law of the large numbers a
… More
nd the central limit theorems to stochastic differential equations and stochastic partial differential equations. In this research, we studied the problems in the following cases. (1) The limits of stochastic differential equations or difference equations are represented by a Brownian flow. (2) Limit theorems for solutions of partial differential equations with random coefficients. (3) Fluctuation theorems and central limit theorems for stochastic ordinary differential equations and partial differential equations.
3. Malliavin calculus was proposed by a French mathematician Malliavin in order to get the smoothness of the solutions of hypoelliptic partial differential equations and has been developed very rapidly. In this research, we showed that the Malliavin calculus cn be applied to a time-dependent system of hypoelliptic differential operator and that the calculus is also applicable to the study of the spectrum of a certian differential operator.
4. We studied the boundary value problems of a degenerate elliptic operator and the problems of the Silov boundary from the point of the views of probability theory. Less
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