| Co-Investigator(Kenkyū-buntansha) |
YOKOI Yoshitaka KUMAMOTO UNIV., ENG., Prof., 工学部, 教授 (50040481)
HITSUDA Masuyuki KUMAMOTO UNIV., SCI., Prof., 理学部, 教授 (50024237)
NAITO Koichiro KUMAMOTO UNIV., ENG., Prof., 工学部, 教授 (10164104)
SAISHO Yasumasa KUMAMOTO UNIV., ENG., A-Prof., 工学部, 助教授 (70195973)
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| Budget Amount *help |
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1999: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1998: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Research Abstract |
The main purpose of this research is to study the ergodic type limit theorem of time inhomogeneous Markov processes. This is considered as an extension of time homogeneous case, but as one can imagine by considering the case of rapidly varying in time, it will be out of the framework and difficult to get the similar results. From this point, we concentrated our attention to the study of getting the degree of the variation under which the similar ergodic theorem holds. In the first year, we started from the study of recurrence which is the basic background of the time homogeneous ergodic theorem. Further we considered if it can be possible to extend the similar argument of time homogeneous case to the present case. Similarly to the time homogeneous case, we obtained a Hopf's type maximal inequality and from which we tried to get the limit theorem. By this method, we could get a classification of the weak sense recurrence of the processes and, under a strong condition, a general result o
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n limit theorem. But, in the general case, to check the condition is not easy. Hence, in the second year, instead of maximal inequality, we tried to use the filling scheme method of Meyer and Fitzsimmons. As a result, under certain condition on hitting probabilities, we could get an fundamental inequality in the time inhomogeneous case. The hypothesis on hitting probabilities can be covered by the hypothesis on transition density. Using this inequality, it is possible to classify the recurrence and transience. Further the existence of F(i, x)=limィイD2n→∞ィエD2GィイD2nィエD2ψ(i,x)/GィイD2nィエD2ψ(i,x) with GィイD2nィエD2ψ(I,x) =ΣィイD3n(/)k=iィエD3EィイD2i,xィエD2 (ψ(κ,XィイD2κィエD2)) was shown. In the time homogeneous case, F(i, x) is equal to the ratio of the integrals of ψ and ψ, with respect to the invariant measure. In the time inhomogeneous case, we cannot expect the existence of time independent invariant measure. But we can give an characterization of F(i,x) by the limit of the ratio of the integrals with respect to some time dependent measures. Less
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