2022 Fiscal Year Final Research Report

New developments in infinite dimensional stochastic analysis based on constructions of spaces of generalized functionals and applications to quantum information theory

Research Project

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Project/Area Number	19K03592
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Multi-year Fund
Section	一般
Review Section	Basic Section 12020:Mathematical analysis-related
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)
Project Period (FY)	2019-04-01 – 2023-03-31
Keywords	超汎関数空間 / 無限次元確率解析 / 無限次元ラプラシアン / 量子確率過程 / 量子確率論 / ホワイトノイズ理論 / 力学系理論 / 数理ファイナンス
Outline of Final Research Achievements	Through this basic research, especially based on the construction of spaces of white noise distributions, we constructed an infinite dimensional stochastic analysis, and obtained many results in the development of applications to quantum probability theory and quantum information analysis. We have reconsidered the spaces of white noise distributions as a direct Wiener integral, randomized the domain of the higher order Levy Laplacian, and succeeded in representing the infinite dimensional Brownian motion in continuous time. Moreover, we developed a stochastic analysis on delta distributions of an infinite dimensional Brownian motion and constructed a new analysis of white noise distributions.　In addition, we gave the significance of quantum probability theory to the law of Cesaro-type quantum large numbers and a characterization of higher order Levy Laplacian.
Free Research Field	確率解析学
Academic Significance and Societal Importance of the Research Achievements	超関数空間を新しく構成し，無限次元ブラウン運動のべき乗を定義することに成功した結果は，量子場の理論での発散量δ(0)を超関数として定式化することにより繰り込み無しで回避することができるという独自性があり，この超関数空間を基に新しい超汎関数空間を構成し，無限次元確率解析を構築する．本研究は場の理論のYang-Mills 方程式と等価であるLevy Laplace方程式の研究基盤となるものである．本研究における確率解析の理論面の整備，応用展開は他にない独創的な研究であり，無限次元解析に新手法を導入し, 量子確率論、量子情報論，量子力学系理論などへの展開を新しいものとする意義のある研究である．