Research of the potential theory for elliptic type or parabolic type in unbounded domains and its consideration in stochastic analysis

• Project/Area Number: 19540166
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Chiba University
• Principal Investigator: MIYAMOTO Ikuko Chiba University, 大学院・理学研究科, 教授 (00009606)
• Co-Investigator(Kenkyū-buntansha): TANEMURA Hideki 千葉大学, 大学院・理学研究科, 教授 (40217162) ; SHIMOMURA Katunori 茨城大学, 理学部, 准教授 (00201559) ; YOSHIDA Hidenobu 千葉大学, 名誉教授 (60009280)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2008: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 複素解析 / 非有界領域 / ディリクレ問題 / 熱方程式 / ポアソン核 / ガウスワイヤストラス核 / 調和関数 / temperature / minimum princinle / minimum principle / コーン / シリンダー / minimally thin sets / rarefied sets

Research Abstract

Our aim was to research the potential theory for the elliptic type or the parabolic type in unbounded domains. In our study of the behavior of the solutions (harmonic functions) of the elliptic type differential equations i.e.Laplace equations in the neighbouhood of the point at infinity, we could consider qualitative properties and quantitative properties of thin sets on the complimentary sets of which the solutions behave very regularly. But the problems analogous to the solutions (temperatures) of the parabolic type differential equations i.e. heat equations could be almost unsolved except the problem about "the sets of determinations".

Research Products

• [Journal Article] Caloric morphism for rotation invariant metric (2010)
  • Author(s): M. Nishio, K. Shimomura
  • Journal Title: Hiroshima Math.J. 40(to appear)
  • Peer Reviewed
• [Journal Article] Non-equilibrium dynamics of Dyson's model with an infinite number of particles (2010)
  • Author(s): M. Katori, H. Tanemura
  • Journal Title: 293 Pages: 469-497
  • Peer Reviewed
• [Journal Article] Caloric morphism for roration invariant metric (2010)
  • Author(s): M.Nishio, K.shimomura
  • Journal Title: Hiroshima Math.J. 40(to appear)
  • Peer Reviewed
• [Journal Article] Non-equilibrium dynamics of Dyson's model with an infinite number of particles (2010)
  • Author(s): M.Katori, H.Tanemura
  • Journal Title: Commun.Math.Phys. 293 Pages: 469-497
  • Peer Reviewed
• [Journal Article] On a covering Property of Minimally thin sets at infinity in a cone (2009)
  • Author(s): I. Miyamoto, H. Yoshida
  • Journal Title: Advances Appl.Math.Sci. 1 Pages: 361-380
  • Peer Reviewed
• [Journal Article] Zeros of Airy function and relaxation process (2009)
  • Author(s): M. Katori, H. Tanemura
  • Journal Title: J.Stat.Phys. 136 Pages: 1177-1204
  • Peer Reviewed
• [Journal Article] On a covering Property of Minimally thin sets at infinity in a cone (2009)
  • Author(s): I.Miyamoto, H.Yoshida
  • Journal Title: Advances Appl.Math.Sci. 1 Pages: 361-380
  • Peer Reviewed
• [Journal Article] On a covering Property of Minimally thin sets at infinity in a cylinder (2008)
  • Author(s): I. Miyamoto, H. Yoshida
  • Journal Title: Mathematica Montisnigri 21 Pages: 35-54
  • Peer Reviewed
• [Journal Article] On a covering property of minimally thin sets at infinity in a cylinder (2008)
  • Author(s): I. Miyamoto, H. Yoshida
  • Journal Title: Mathematica Montinigri 21 Pages: 35-54
  • Peer Reviewed
• [Journal Article] On a-minimally thin sets at infinity in a cone (2007)
  • Author(s): I. Miyamoto, H. Yoshida
  • Journal Title: Hiroshima Math.J. 37 Pages: 61-80
  • NAID: 110006197821
  • Peer Reviewed
• [Journal Article] Two criteria of Wiener type for minimally thin sets and rarefied sets in a cylinder (2007)
  • Author(s): I. Miyamoto
  • Journal Title: Hokkaido Math.J. 36 Pages: 507-534
  • Peer Reviewed
• [Journal Article] Noncolliding Brownian motion and determinantal processes (2007)
  • Author(s): M. Katori, H. Tanemura
  • Journal Title: J.Statist.Phys. 129 Pages: 1233-1277
  • Peer Reviewed
• [Journal Article] Infinite Systems of Non-Colliding Generalized Meanders and Riemann-Liouville Differintegrals (2007)
  • Author(s): M. Katori, H. Tanemura
  • Journal Title: Probab.Th.Rel.Fields 138 Pages: 113-156
  • Peer Reviewed
• [Journal Article] On a-minimally thin sets at infinity in a cone (2007)
  • Author(s): I. Miyamoto, H. Yoshida
  • Journal Title: Hiroshima Math. J 37 Pages: 61-80
  • NAID: 110006197821
  • Peer Reviewed
• [Journal Article] Two criteria of Wiener type for minimally thin sets and rarefied sets in a cylinder (2007)
  • Author(s): I. Miyamoto
  • Journal Title: Hokkaido Math. J 36 Pages: 507-534
  • Peer Reviewed
• [Journal Article] Noncolliding Brownian motion and determinantal processes (2007)
  • Author(s): M. Katori, H. Tanemura
  • Journal Title: J. Stat. Phys 129 Pages: 1233-1277
  • Peer Reviewed
• [Journal Article] 非衝突過程・行列値過程・行列式過程
  • Author(s): H. Tanemura
  • Journal Title: 数学 (to appear)
  • NAID: 10026415164
  • Peer Reviewed
• [Presentation] A result of Azar in and rarefied sets at infinity in a cone (2009)
  • Author(s): I.Miyamoto, H.Yoshida
  • Organizer: Analysis Seminar
  • Place of Presentation: バル・イラン大学(イスラエル、テルアビブ)
  • Year and Date: 2009-10-26
• [Presentation] Liouville type theorem associate with the wave equation (2009)
  • Author(s): K.Shimomura
  • Organizer: International Workshop on Potential Theory 2009
  • Place of Presentation: 北海道大学
  • Year and Date: 2009-09-04
• [Presentation] Noncolliding processes and Random Matrices (2009)
  • Author(s): M.Katori, H.Tanemura
  • Organizer: Selfsimilar processes and their applications
  • Place of Presentation: ボン・パスツールセンター(フランス、アンジェ)
  • Year and Date: 2009-07-20
• [Presentation] A result of Azarin and rarefied sets at infinity in a cone (2009)
  • Author(s): 宮本育子, 吉田英信
  • Organizer: Analysis Seminar
  • Place of Presentation: バル・イラン大学
• [Presentation] Sets of determination for harmonic functions in an abstract harmonic space (2009)
  • Author(s): 吉田英信
  • Organizer: Internationak Workshop on Potential theory
  • Place of Presentation: 北大
• [Presentation] Liouville type theorem associate with the wave equation (2009)
  • Author(s): 下村勝孝
  • Organizer: International Workshop on Potential Theory 2009
  • Place of Presentation: 北大
• [Presentation] Noncolliding processes and Random Matrices (2009)
  • Author(s): 香取眞理, 種村秀紀
  • Place of Presentation: ボン・パスツールセンター
• [Presentation] Non-equilibrium dynamics fo determinantal processes with infinite particles (2009)
  • Author(s): 香取眞理, 種村秀紀
  • Place of Presentation: 京都大学理学研究科
• [Presentation] Non-equilibrium dynamics of Dyson's model (2009)
  • Author(s): 香取眞理, 種村秀紀
  • Place of Presentation: 忠北国立大学
• [Presentation] 波動方程式の解を保つ変換について (2008)
  • Author(s): 下村勝孝
  • Place of Presentation: 名城大学ポテンシャル論セミナー
• [Presentation] Infinite Dyson's model (2008)
  • Author(s): 香取眞理, 種村秀紀
  • Organizer: Stochastic Analysis on Large Scale Interacting System
  • Place of Presentation: 東京大学大学院数理科学研究科大講義室
• [Presentation] Dyson's model with infinite particles (2008)
  • Author(s): 香取眞理, 種村秀紀
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Qualitative or quantitative properties of minimally thin sets at infinity on a cone (2008)
  • Author(s): 宮本育子, 吉田英信
  • Organizer: Analysis Seminar
  • Place of Presentation: モスクワ大学
• [Presentation] シリンダー内の無限遠点での minimally thin な集合の定量的な特徴付け (2008)
  • Author(s): 宮本育子, 吉田英信
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東工大
• [Presentation] Caloric morphism : Transformation preserving the solutions of heat equation (2008)
  • Author(s): 下村勝孝
  • Organizer: Analysis Seminar
  • Place of Presentation: Charmars工科大学
• [Presentation] Property of Minimally thin sets at infinity in a cylinder (2008)
  • Author(s): 宮本育子, 吉田英信
  • Organizer: Analysis Seminar
  • Place of Presentation: モンテネグロ大学
• [Presentation] シリンダー内の無限遠点でのminimally thinな集合の定量的な特徴付け (2008)
  • Author(s): 宮本育子
  • Organizer: 日本数学会
  • Place of Presentation: 東京工業大学
• [Presentation] Infinite Dyson's model (2008)
  • Author(s): M. Katori, H. Tanemura
  • Organizer: Stochastic Analysis on Large Scale Interacting System
  • Place of Presentation: 東京大学
• [Presentation] コーン内の無限遠点での minimally thin な集合の定量的な特徴付け (2007)
  • Author(s): 宮本育子, 吉田英信
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東北大
• [Presentation] On a covering property of minimally thin sets at infinity in a cylinder (2007)
  • Author(s): 宮本育子, 吉田英信
  • Organizer: フィンランド・日本合同解析セミナー
• [Presentation] Caloric morphism for rotation invariant metric on Euclidean spaces (2007)
  • Author(s): 下村勝孝
  • Organizer: 2007 Finland-Japan Joint Seminar on Analysis
• [Presentation] R^n (n≧3) の回転不変計量に関するcaloric morphism (2007)
  • Author(s): 下村勝孝
  • Organizer: 日本数学会年会
• [Presentation] コーン内のa-minimal な集合の被覆定理 (2007)
  • Author(s): 宮本育子, 吉田英信
  • Organizer: ポテンシャル論研究集会
  • Place of Presentation: 千葉大
• [Presentation] コーン内の無限遠点でのminimally thin な集合の定量的な特徴付け (2007)
  • Author(s): 宮本 育子
  • Organizer: 日本数学会
  • Place of Presentation: 東北大学
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