1989 Fiscal Year Final Research Report Summary
Study of Stochastic Processes
| Project/Area Number |
62460006
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Nagoya University |
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Principal Investigator |
SATO Ken-iti Nagoya Univ., College of General Education, Professor, 教養部, 教授 (60015500)
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| Co-Investigator(Kenkyū-buntansha) |
ITO Masayuki Nagoya Univ., Coll. of Gen. Educ., Prof., 教養部, 教授 (60022638)
NAGAI Hideo Nagoya Univ., Coll. of Gen. Educ., Ass. Prof., 教養部, 助教授 (70110848)
ICHIHARA Kanji Nagoya Univ., Coll. of Gen. Educ., Ass. Prof., 教養部, 助教授 (00112293)
IHARA Shunsuke Nagoya Univ., Coll. of Gen. Educ., Prof., 教養部, 教授 (00023200)
NOMOTO Hisao Nagoya Univ., Coll. of Gen. Educ., Prof., 教養部, 教授 (40023030)
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| Project Period (FY) |
1987 – 1989
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| Keywords | Birth-and-death Process / Markov Process / Self-similar Process / Gaussian Channel / Brownian Motion / Stochastic Control Problem / Logarithmic Diffusion Kernel / Analytic Capacity |
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| Research Abstract |
1. The distributions of generalized sojourn time for birth-and-death processes are determined. 2. Subordination of a Markov process by increasing Levy processes obtained through exponential transformation with a parameter is studied. Limit theorems for the resulting family of Markov processes are proved, when the parameter approaches infinity. 3. Systematic study of self- similar processes with independent increments is made. Their relation with distributions of class L is discovered. Further it is found that a similar relation exists between operator-self-similar processes with independent increments and operator-self-decomposable distributions. 4. A Gaussian channel with continuous time such that its channel capacity in the case with feedback is twice as large as its capacity in the case without feedback is constructed. This shows that the known inequality that the former is smaller than or equal to twice the latter is the best possible. 5. Estimate from below of the transition proba
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bility for the Brownish motion on a Riemannian manifold is given. It is shown from this estimate that, under some restrictions, the Brownian motion satisfies the global law of iterated logarithm. 6. Ergodic stochastic control problem on the whole Euclidean space is investigated. The problem is connected with the eigenvalue problem of Schrodinger operators; asymptotic behaviors of the solutions of some non-linear partial differential equations are proved. 7. In the axiomatic potential theory, which is closely related with Markov processes, logarithmic diffusion kernels are characterized, boundary behaviors of parabolic potentials are studied, and decomposition of continuous function-kernels into regular and singular parts is investigated. 8. Conditions are given for a perfect set to be the set of singularities such that exceptionally ramified meromorphic functions do not exist. Refinement of the result is made. 9. Estimate of the norms of Cauchy transforms is given. The analytic capacity of sets is thereby estimated and its relation with Buffon needle probability is examined.
10. It has been said that, in repeated measurement of the position of a free mass, there is a limit for sensitivity of measurement called the standard quantum limiti. Controversy has been made on this point, but it is finally settled by mathematically constructing a model of measurement that breaks the standard quantum limit. Less
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