Research on Markov processes via stochastic analysis

2018 Fiscal Year Final Research Report

• Project/Area Number: 15H03624
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyoto University
• Principal Investigator: Shigekawa ichiro 京都大学, 理学研究科, 教授 (00127234)
• Co-Investigator(Kenkyū-buntansha): 矢野 孝次 京都大学, 理学研究科, 准教授 (80467646)
• Research Collaborator: Kumagai Takashi ; Hino Masanori ; Aida Shigeki
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 確率解析 / マルコフ半群 / エルゴード性 / スペクトル / 対数ソボレフ不等式 / スペクトルギャップ / マルコフ過程 / Dirichlet 形式

Outline of Final Research Achievements

We conducted research on Markov processes using stochastic analysis methods for cases of various state spaces, such as Euclid space, Riemannian manifold, and infinite dimensional space such as Wiener space and path space. In the case of the one-dimensional diffusion process, the spectrum of the Kolmogorov diffusion process was determined in the framework of supersymmetry. Also, in the case of Kummer process, spectra were determined in Zygmundt space or Orlicz space.
Furthermore, we characterized the ultracontractivity using the asymmetric Dirichlet form and applied to the asymptotic behavior of the fundamental solution in the case of compact Riemannian manifolds. We also constructed a non-symmetric diffusion process on the Wiener space as a typical example of an infinite dimensional space.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

拡散過程の研究を行ったが、これらは数理ファイナンス、保険数学に関連するものが含まれる。また拡散過程の不変測度として、統計に表れる分布が出てくるので、統計への応用の道も開かれる。