| Co-Investigator(Kenkyū-buntansha) |
KIM Dae hong KUMAMOTO UNIV., ENG., LECT., 工学部, 講師 (50336202)
HITSUDA Masuyuki KUMAMOTO UNIV., SCI., PROF., 理学部, 教授 (50024237)
NAITO Koichirou KUMAMOTO UNIV., ENG., PROF., 工学部, 教授 (10164104)
TAKEDA Masayoshi TOHOKU UNIV., SCI., PROF., 理学研究科, 教授 (30179650)
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| Research Abstract |
The main purpose of this research is to construct and give the properties of the diffusion processes on the domains which move depending on time. We assume that not only the domain but also the generator allow depending on time. The typical processes are absorbing or reflecting diffusion processes. In our case, difficulty arises because the boundary changes depending on time. In this research, after reorganizing the basic theory of time dependent Dirichlet forms, we constructed the processes on moving domains and get some properties of them. The fundamental settings are as follows. The domains are related by a time dependent maps of a fixed domain. The generator may depend upon the time. We first construct the diffusion processes on a fixed domain. For the construction, we use the general theory of time dependent Dirichlet forms. After that, we map the process to get the desired processes. In this argument, since the surface elements depend on time, to show that the mapped processes sa
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tisfy the desired property, we need the stochastic calculus including the Girsanov type transformations. Such stochastic calculus is given separately but not systematically in the time dependent case. Since we use such calculus in many places, the fundamental calculus will be published. As an important case, we consider the reflecting diffusion processes. Since the boundary changes depending on time, it is not easy to specify the local time and the normal derivative of the boundary. In this research, by using the map among the domains, we give the notions and define the local time on the boundary. Using the local time, we give the Skorohod representation of the diffusion processes. It is possible to apply our processes. One of the applications is an optimal stopping problem of time inhomogeneous diffusion processes. In the time inhomogeneous case, since there exist semipolar sets, it is not easy to characterize the optimal stopping time. For this problem, we can give the characterization of the time. We are considering that further related applications are possible. Less
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