1997 Fiscal Year Final Research Report Summary
On the recurrence and its applications of time inhomogeneous Markov processes
| Project/Area Number |
08640298
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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 UNIV., ENG., PROF., 工学部, 教授 (20040404)
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| Co-Investigator(Kenkyū-buntansha) |
SAISHO Yasumasa KUMAMOTO UNIV., ENG., A-PROF., 工学部, 助教授 (70195973)
SAKATA Toshio KUMAMOTO UNIV., ENG., A-PROF., 工学部, 助教授 (20117352)
NAITO Koichiro KUMAMOTO UNIV., ENG., PROF., 工学部, 教授 (10164104)
YOKOI Yoshitaka KUMAMOTO UNIV., ENG., PROF., 工学部, 教授 (50040481)
HITSUDA Masuyuki KUMAMOTO UNIV., SCI., PROF., 理学部, 教授 (50024237)
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| Project Period (FY) |
1996 – 1997
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| Keywords | Dirichlet forms / Recurrence / Space-time process / conservativeness / Ergodic theory |
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| Research Abstract |
Under the project "On the recurrence and its applications of time inhomogeneous Markov processes" in 1996-1997, we started a new investigation of a general method to analize the notion of the recurence of time imhomogeneous Markov processes. For a time homogeneous Markov process M=(X_t, P_x), the recurrence of M is defined by the following three equivalent conditions : (s) (a.s.infinity of the occupation time) *^*_ I_C(X_t)dt=0 or * a.s.P_x, (w)(a.s.reachability) P_x(sigmac<*)=0 or 1 and (m) (infinity of the mean occupation time) E_x(*^*_ I_C(X_t)dt)=0 or *. In the time inhomogeneous case, since the motion can change after any time in general, the above conditions are not equivalent any more. We can only show that condition (s) implies (w) and (m). These conditions are invariant under the time change by the additive functionals dependeng only on the states but (s) and (m) are not invariant if the additive functionals depend on the time variable. In this research, we have given the gene
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ral criteria for the recurrence in the sense of (s) and (w). The criteria shows that the recurrences in the sense of (s) and (w) are closely related with the order of the decay of the transition functions of the process killed by an additive functional and at the exit time from a set, respectively. Applying the criterion for the time-changed process X_t=B(psi(t) ) of a Brownian motion B(t) by a function psi(t), we gave a growth condition onpsi(t) for the recurrence of X_t. Further, we gave a condition on rho for the (s)-recurrence of a diffusion processes obtained from a time homogeneous diffusion peocesses via a drift transformation by *rho(t, x). Such processes are important in connection with stochastic mechanics. The recurrence in the sense of (s) will play an important role when we consider limit theorems such as ergodic theorem and ratio limit theorems of additive functionals. In this connection, we provided with a new problems concerning the limit theorems of time inhomogeneous Markov processes. We began the study to solve this interesting problem but we only have a partial result so far. Less
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