Stochastic analysis of Markov processes by Dirichlet forms and its applications

2017 Fiscal Year Final Research Report

• Project/Area Number: 26247008
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tohoku University
• Principal Investigator: Takeda Masayoshi 東北大学, 理学研究科, 教授 (30179650)
• Co-Investigator(Kenkyū-buntansha): 上村 稔大 関西大学, システム理工学部, 教授 (30285332) ; 桑田 和正 東北大学, 理学研究科, 教授 (30432032) ; 日野 正訓 京都大学, 情報学研究科, 准教授 (40303888) ; 桑江 一洋 福岡大学, 理学部, 教授 (80243814) ; 会田 茂樹 東京大学, 大学院数理科学研究科, 教授 (90222455)
• Co-Investigator(Renkei-kenkyūsha): KAWABI KOUJI 岡山大学, 大学院自然科学研究科, 教授 (80432904) ; SHIOZAWA YUUICHI 大阪大学, 大学院理学研究科, 准教授 (60454518) ; KUSUOKA SEIICHIRO 岡山大学, 大学院自然科学研究科, 准教授 (20646814) ; KAJINO NAOTAKA 神戸大学, 大学院理学研究科, 准教授 (90700352)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: 対称マルコフ過程 / ディリクレ形式 / 測度空間上の確率解析 / 準定常分布

Outline of Final Research Achievements

For one-dimensional diffusion processes the spectral properties of their semigroups can be studied in terms of Feller's boundary classification. It is important to introduce a class of symmetric Markov processes which possess properties similar to one-dimensional diffusion processes. In this study, we proposed such a class of symmetric Markov processes and obtain some spectral properties of them. If a symmetric Markov process in this class is conservative, it has very strong ergodicity called uniform hyper exponential recurrence. If it is not conservative, it explode very fast. By using these properties we can show that its semigroup becomes a compact operator and every eigenfunction has a bounded continuous version. Moreover, the principal eigenfunction is integrable. As its important application, the existence and uniqueness of the quasi-stationary distributions are derived for non-conservative symmetric Markov processes in this class.

Free Research Field

確率過程論