1988 Fiscal Year Final Research Report Summary
Mathematical Study on Stochastic Models
| Project/Area Number |
62460007
|
|---|
| Research Category |
Grant-in-Aid for General Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
|
|---|
| Research Institution | Kyoto University |
|---|
Principal Investigator |
WATANABE Shinzo Faculty of Science, Kyoto University, 理学部, 教授 (90025297)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
TANIGUCHI Masahiko Faculty of Science, Kyoto University, 理学部, 助教授 (50108974)
NISHIDA Takaaki Faculty of Science, Kyoto University, 理学部, 教授 (70026110)
HIRAI Takeshi Faculty of Science, Kyoto University, 理学部, 教授 (70025310)
IKEBE Teruo Faculty of Science, Kyoto University, 理学部, 教授 (00025280)
KUSUNOKI Yukio Faculty of Science, Kyoto University, 理学部, 教授 (90025221)
|
|---|
| Project Period (FY) |
1987 – 1988
|
| Keywords | Stochastic analysis / Malliavin calculus / Generalized Wiener functionals / Asymptotic problems of heat kernels / Lie superalgebra / Periodic solutions of Duffing equations / Fractals / 双曲的変換の離散部分群 |
|---|
| Research Abstract |
An analysis on the Wiener space (Malliavin calculus) has been studied as an analogy of Schwartz distribution theory. In this framework, the pull-back of Schwartz distributions under a non-degenerate Wiener map can be defined as generalized Wiener functionals and by using this, the regularity and the asymptotics with respect to parameters of the law of Winer functionals can be discussed. This method has been applied to obtain asymptotic results for heat kernels (the fundamental solutions of heat equations). Furthermore, by taking a finite measure space as the parameter space, we can discuss the case of geat kernels with boundary conditions. As an application, a probabilistic proof was obtained for the Gauss-Bonnet-Chern therorem in the case of manifolds with boundaries. Also, this method has been applied to asymptotic problems of degenerate heat kernels.
The method of Poisson point processes with values in function spaces has been applied effectively to the study of diffusion processes with boundary conditions. By using this method in the construction problem of diffusions with Wentzell's boundary conditions, we could show the existence and uniqueness of diffusions in certain cases which could not be obtained before by other methods.
A new general method has been established for the construction of irreducible unitary representations of Lie superalgebras.
New results have been obtained for the existence of periodic solutions of Duffing equations. Also, the theory of interval dynamical systems and self-similar sets has been applied to obtain interesting results for the classical nowhere differentiable functions of Weierstrass and Besicovitch. This kind of research on fractals will be an important contact point of probability theory and analysis in future.
Interesting examples have been obtained for discrete groups of hyperbolic motions.
|
