2007 Fiscal Year Final Research Report Summary
Research on diffusion processes moving singularly on space-time domains
| Project/Area Number |
18540131
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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 | Kumamoto University |
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Principal Investigator |
OSHIMA Yoichi Kumamoto University, Graduate School of Natural Sciences, Professor (20040404)
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| Co-Investigator(Kenkyū-buntansha) |
NAITO Koichiro Kumamoto University, Graduate School of Natural Sciences, Professor (10164104)
DAEHONG Kim Kumamoto University, Graduate School of Natural Sciences, Professor (50336202)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Dirichlet forms / Time inhomogeneous diffusion processes / Space-time domain / Hitting probability / Recurrence / Simulated annealing |
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| Research Abstract |
The purpose of this paper is to find some conditions on space-time domains and time inhomogeneous diffusion processes under which the processes hit the space-time domains and to study the behavior after the hit. For the purpose, as a first step, we have shown an inequality between a weighted square integral and Dirichlet forms as well as a Poincare type inequality which analytically characterize the transience and the recurrence of time homogeneous Markov processes. By using these inequalities, we considered the theory of simulated annealing. Generally the theory of simulated annealing is considered for the Brownian motion on compact manifolds which allow us to use the estimates on spectral gap. We have shown that our result can be appreciable for the case of general state space and processes.
After that, we considered the possibility of exit of time inhomogeneous processes from the domains varying depending on time, in particular compact domains increasing depending on time. This depend on the speed of the change of the generators or Dirichlet forms and the speed of increase of the domains. Concerning to this problem, for the increasing balls depending on time, we obtained a result on the diffusion coefficients and the order of increase of the radius of the balls for the possibility of exit. In the case of Brownian motion, the result says that, if the radius of the ball increase slowly than the square root of the time, then the Brownian motion exits the balls almost surely. As a converse problem, we considered the possibility of hit near the neighbourhood of the origin before the hit of the increasing balls. These results are obtained by showing general criterion of the hitting probability of space-time domains and applying it. These general results concerning the hitting probability of space-time domains are new.
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