STUDY OF SELF-SIMILAR PROCESSES

自相似过程的研究

• 批准号: 08454038
• 负责人: KASAHARA Yuji
• 金额: $ 2.94万
• 依托单位: OCHANOMIZU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454038/
• 关键词: self-similar; fractional Brownian motion; extremal process; local time; Gaussian process; occupation times; limit theorem; Gaussian process; 局所時間; モーメント; 指数分布

・We studied the limiting processes for occupation times of fractional Brownian motion, which is a typical self-similar stochastic process. We see that the limiting process degenerates with the usual linear normalization but if consider the processes with the log-scale, we do have a meaningful process, which is the inverse of the so-called extremal process.・We generalized the above result to a certain class of Gaussian processes, where the occupation time increases slowly.・We studied the asymptotic behavior of the local times of fractional Brownian motion. As the index times the dimension approaches 1, we see that the one-dimensional marginal distributions of the local time at the origin converge to the exponentioal distribution, and furthermore, the processes, with a suitable time scale, converge to the inverse extremal process.・The invariant set under a family of functions has self-similarity. We studied its topological aspects.

・我们研究了分数布朗运动占据时间的极限过程，这是一种典型的自相似随机过程，我们发现极限过程随着通常的线性归一化而退化，但如果考虑对数尺度的过程，我们确实有。一个有意义的过程，它是所谓极值过程的逆过程。・我们将上述结果推广到某一类高斯过程，其中占据时间缓慢增加。・我们研究了局部时间的渐近行为当指数乘以维数接近 1 时，我们看到原点处的局部时间的一维边缘分布收敛到指数分布，而且，具有适当时间尺度的过程收敛到逆极值过程。・函数族下的不变量集具有自相似性。

项目成果

A.Kaneko: "On continuation of Gevrey class solutions of linear differential equations" Proc.7th Int.Colloq.on Diff.Eq.197-204 (1997)

A.Kaneko：“关于线性微分方程的 Gevrey 类解的延续”Proc.7th Int.Colloq.on Diff.Eq.197-204 (1997)

F.Takeo: "The topology of fractal sets induced by a certain type of multivalued functions" Nonlinear Analysis,Theory,Methods & Applications. 30. 3071-3080 (1997)

F.Takeo：“由某种类型的多值函数引起的分形集的拓扑”非线性分析、理论、方法

Y.Kasahara & N.Kosugi: "A Limit theorem for occupation times of fractional Brownian motion" Stoch.Proc.Appl.67. 161-175 (1997)

笠原勇

Y.Kasahara and N.Kosugi: "A limit theorem for occupation times of Gaussian processes" Trends in Probability and Related Analysis. 197-208 (1997)

Y.Kasahara 和 N.Kosugi：“高斯过程占据时间的极限定理”概率趋势及相关分析。

F.Takeo: "Topological property of an invariant set with respect to a family of functions" Sci.Bull.Josai Univ.special Issue. 1. 45-61 (1997)

F.Takeo：“关于函数族的不变集的拓扑性质”Sci.Bull.Josai Univ.special Issue。