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

• 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
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000); Fiscal Year 2017: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000); Fiscal Year 2016: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• 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 / 擬似乱数 / 固有関数 / 木構造 / 階層構造 / 理想境界 / 非アルキメデス的距離 / ディリクレ空間

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.

Research Products

• [Journal Article] A pairwise independent random sampling method in the ring of p-adic integers (2016)
  • Author(s): Hiroshi Kaneko,Hisaaki Matsumoto
  • Journal Title: Osaka Journal of Mathematics Volume: 53 Pages: 775-798
  • NAID: 120005986278
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] CAPACITARY ESTIMATE ON THE SPACE OF ENDS OF TREE BASED ON ORLICZ NORM (2016)
  • Author(s): Ryutaro, Iijima, Hiroshi Kaneko
  • Journal Title: Facta Universitatis, Series Physics, Chemistry and Technology Volume: 14 Pages: 120-127
  • Peer Reviewed
• [Journal Article] Orlicz norm and sobolev-Orlicz capacity on ends of tree based on probabilisitic Bessel kernels (2015)
  • Author(s): Chaiki Hara, Ryutaro Iijima, Hiroshi Kaneko, Hisaaki Matsumoto
  • Journal Title: p-adic numbers ultrametric analysis, and applicationｓ Volume: 7 Issue: 1 Pages: 24-38
  • DOI: 10.1134/s2070046615010033
  • Peer Reviewed
• [Journal Article] A Dirichlett space on ends of tree and Dirichlet forms with a nodewise orthogonal property (2014)
  • Author(s): Hiroshi Kaneko
  • Journal Title: Potential Analysis Volume: 41 Issue: 1 Pages: 245-268
  • DOI: 10.1007/s11118-013-9372-7
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] 木に付随する末端空間における非線形容量 (2017)
  • Author(s): 金子 宏
  • Organizer: ディリクレ形式と対称マルコフ過程
• [Presentation] An Orlicz space on ends of tree and superposition of nodewise given Dirichlet forms with tier-linkage (2015)
  • Author(s): Hiroshi Kaneko
  • Organizer: International conference on p-adic mathematical physics and its applications
  • Place of Presentation: Mathematical Institute of the Serbian Academy of Science and Arts
  • Year and Date: 2015-09-10
  • Int'l Joint Research / Invited
• [Presentation] Dirichlet space and Orlicz space on ends of tree associated with nodewise given Dirichlet forms with tier-linkage (2014)
  • Author(s): Hiroshi Kaneko
  • Organizer: Stochoastic Processes, Analysis and Mathematical Pｈysics
  • Place of Presentation: Kansai Universtiy
  • Year and Date: 2014-08-25
  • Invited
• [Book] Festschrift Masatoshi Fukushima (2015)
  • Author(s): Sergio Albeverio, Zhi-Ming Ma, Michael Roeckner, Martin Barlow, Nicolas Bouleau, Guillaume Poly, Mu-Fa Chen, Xin Chen, Feng-Yu Wang Jian Wang, Shizan Fang Andrey Pilipenko, Patrick J Fitzsimmons, ,Irina Penner, Wolfhard Hansen, Hiroshi Kaneko,他
  • Total Pages: 620
  • Publisher: World Scinetific
• [Book] Stochastic Analysis and Applications 2014 (2014)
  • Author(s): Domique Bakry, Erich Bauer, Jean Bertoin, Rene Carmona, Fransois Delarue, Ana Bella Curzerio, Remi Lasselle, Alexander Davie, Joscha Diehl, Peter K. Friz, Harald Oberhauser, Yidong Dong, Ronnie Sircar, David Elworthy, Hans Follmer, Claudia Kluppelberg, Masatoshi Fukushima, Hiroshi Kaneko,他
  • Total Pages: 503
  • Publisher: Springer Verlag