New development of theory of generalized functionals, an infinite dimensional stochastic analysis and applications to theory of quantum dynamical system

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K04940
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Meijo University
• Principal Investigator: Saito Kimiaki 名城大学, 理工学部, 教授 (90195983)
• Co-Investigator(Kenkyū-buntansha): 三町 祐子 名城大学, 理工学部, 准教授 (00218629)
• Co-Investigator(Renkei-kenkyūsha): ICHIHARA Kanji 名城大学, 理工学部, 教授 (00112293) ; OBATA Nobuaki 東北大学, 大学院・情報科学研究科, 教授 (10169360) ; HIBINO Yuji 佐賀大学, 工学系研究科, 准教授 (50253589)
• Research Collaborator: ACCARDI Luigi ローマ第Ⅱ大学, 経済学部, 教授 ; KUO Hui-Hsiung ルイジアナ州立大学, 理学部, 教授
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 無限次元確率過程 / 超関数空間 / 超汎関数空間 / 無限次元ラプラシアン / ホワイトノイズ理論 / 力学系理論 / 量子確率過程 / 数理ファイナンス

Outline of Final Research Achievements

We researched new developments of theory of generalized functionals, an infinite dimensional stochastic analysis based on the theory,and its applications.We obtained remarkable results:①We constructed spaces of generalized functionals from new spaces consisting of distributions that are closed in the product of distributions,and also obtained an integral expressions of the higher order Levy Laplacians.Moreover,we obtained a SDE including random directional differentiations induced from the delta distribution centered at powers of an infinite dimensional Brownian motion.②We initiated the new method of stochastic integral for instantly independent stochastic processes,and proved the Doob-Meyer type decomposition theorem. ③We proved that the nuclearity of the space of test functionals for infinite dimensional distributions is a necessary and sufficient condition for a characterization of the S-transform.The result is the breakthrough as an answer for an unsolved problem.

Free Research Field

確率解析学