Research on Diffusion Processes in Random Environments

随机环境中扩散过程的研究

• 批准号: 11640137
• 负责人: TANAKA Hiroshi
• 金额: $ 1.34万
• 依托单位: Japan Women's University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640137/
• 关键词: Brownian motion; Diffusion process; Random environment; Brownian potential; Levy process; Self-similar process; Subordination; Brownian motion with multi-dimensional time; マルコフ過程; 極限分布; レヴィ測度

1. It was proved that the limit process arising in a suitably scaled reflecting diffusion process, which moves in a Brownian environment of [0, ∞), is a time-inhomogeneous self-similar Levy process. Its Levy measure was computed explicitly.2. It was shown that the long-term behavior of diffusion process with a one-sided Brownian potential in (-∞, ∞) exhibits a phenomenon quite different from those (other) models studied previously. In fact, under the usual Brownian scaling the limit process is a reflecting Brownian motion on [0, ∞) with "probability 1/2" and vanishes identically with "the other probability 1/2".3. As a preliminary discussion toward the investigation of a diffusion process in a multi-dimensional Brownian environment. some new results on extrema of a Brownian motion with a multi-dimensional time were obtained.4. Bochner's subordination of time-homogeneous Markov processes was extended, with modification, to a considerably wider class of time-inhomogeneous Markov processes. In connection with this, some explicit formula for a class of multiplicative functionals of (discontinous) Markov processes was also studied.

1. 证明了在[0，∞)布朗环境中运动的适当尺度反射扩散过程的极限过程是一个时间非齐次自相似Levy过程。其征税措施已明确计算。结果表明，在（-∞,∞）中具有单边布朗势的扩散过程的长期行为表现出与以往研究的其他模型完全不同的现象。实际上，在通常的布朗标度下，极限过程是在[0，∞)上以“1/2概率”反映布朗运动，并与“另一个1/2概率”相同地消失。作为对多维布朗环境中扩散过程研究的初步讨论。得到了关于多维时间布朗运动极值的一些新结果。Bochner对时间齐次马尔可夫过程的从属地位经过修改，扩展到更广泛的时间非齐次马尔可夫过程。在此基础上，研究了一类（不连续）马尔可夫过程的乘法泛函的显式公式。

项目成果

Jun Tomiyama: "Structure of ideals and isomorphisms of C-crossed products by single homeomorphisms"Tokyo Journal of Mathematics. Vol.23. 1-13 (2000)

Jun Tomiyama：“单同胚的理想结构和 C 交叉乘积的同构”，东京数学杂志。

Masao Nagasawa and Hiroshi Tanaka: "The principle of variation for relativistic quantum particles"Seminaire de Probabilites XXXV (Lecture Notes in Mathematics). (to appear). (2001)

长泽正雄和田中宏：“相对论量子粒子的变异原理”研讨会 XXXV（数学讲义）。

Jun Tomiyama: "Structure of ideals and isomorphisms of C-crossed products by single homeomorphisms"Tokyo Journal of Mathematics. (to appear).

Jun Tomiyama：“单同胚的理想结构和 C 交叉乘积的同构”，东京数学杂志。

Hiroshi Tanaka: "Some theorems concerning extrema of Brownian motion with d-dimensional time"Osaka Journal of Mathematics. (to appear). (2001)

田中宏：《关于 d 维时间布朗运动极值的一些定理》《大阪数学杂志》。

Masao Nagasawa and Hiroshi Tanaka: "Time dependent subordination and Markov processes with jumps"Seminaire de Probabilites XXXIV (Lecture Notes in Mathematics Vol.1729). 259. 288 (2000)

Masao Nagasawa 和 Hiroshi Tanaka：“时间相关从属关系和带有跳跃的马尔可夫过程”Seminaire de Probabilites XXXIV（数学讲义第 1729 卷）。