Overviews and constructions of Dirichlet form theory on non-Archimedean space on a basis of hierarchical structure

2017 Fiscal Year Final Research Report

• Project/Area Number: 26400150
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tokyo University of Science
• Principal Investigator: Kaneko Hiroshi 東京理科大学, 理学部第一部数学科, 教授 (90194919)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: マルコフ過程 / 木（ツリー） / 超距離空間 / 関数空間 / 容量 / 統計的推論

Outline of Final Research Achievements

This project began with taking advantage of stochastic counterpart of the Bessel kernels and a framework on Sobolev-Orlicz capacity on ends of a tree has been invented so that capacitary estimates are derived from a spectral analytical classification of eigenfunctions according to design of tree. In second, a modified Van der Corput sequence in the ring of p-adic integers has been introduced so as to be a counterpart of Weyl’s irrational rotation on the unit interval. On the ring, a similar random Weyl sampling to the one by Sugita and Takanobu is also newly built. In third, Ben Amor’s result which had shown an important relationship of Orlicz norm with a capacitary estimate was focused on. As a result, capacitary estimates for fundamental subsets in the ends of a tree have been found in terms of a Radon measure, where a canonical orthonormal basis in the family of square integrable functions can be freed from the orthogonality determined by Dirichlet form in existing formalisms.

Free Research Field

確率過程論