Stochastic Analysis and its applications to partial differential operators

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400144
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Aoyama Gakuin University
• Principal Investigator: Matsumoto Hiroyuki 青山学院大学, 理工学部, 教授 (00190538)
• Co-Investigator(Renkei-kenkyūsha): Yuji Hamana 熊本大学, 理工学研究科, 教授 (00243923)
• Research Collaborator: Setsuo Taniguchi 九州大学, 基幹研究員・教授 (70155208)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 確率解析 / 拡散過程 / ベッセル過程 / 到達時刻 / ドリフト付きブラウン運動 / 管状近傍

Outline of Final Research Achievements

Following the former studies on Bessel processes in mind, I carried out studies on the probability distributions of the first hitting times to spheres of Brownian motions with constant drifts. By virtue of the skew-product representation of the Brownian motions and Stroock's representation of the Brownian motions on the spheres as the solutions of some stochastic differential equations, explicit forms of the distribution functions. As applications, we derived the asymptotic behavior of the tail probabilities and the volume of the corresponding Wiener sausage. Moreover, for the Bessel processes, the second term of the asymptotics of the tail probability has been reduced.
As other results, a book on the theory and applications of stochastic analysis was publushed. I studied Kolmogorov's diffusion process from the point of view of classical mechanics and, also investigate some important character of two-dimensional diffusion processes.

Free Research Field

確率解析