New approach to spectral theory of generalized second-order differential operators and its applications to probability theory

• Project/Area Number: 21540109
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: University of Tsukuba
• Principal Investigator: KASAHARA Yuji 筑波大学, 数理物質系, 教授 (60108975)
• Co-Investigator(Kenkyū-buntansha): LIANG Song 筑波大学, 数理物質系, 准教授 (60324399)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 確率論 / Kreinの対応 / 1次元拡散過程 / 極限円 / 推移確率密度 / 推移確率 / スペクトル関数 / スペクトル関数の漸近挙動

Research Abstract

It is well known that a linear diffusion is described by a generalized second-order differential operator. Therefore, the study of various quantities of linear diffusions is reduced to problems on the spectral functions. In the present study we obtained a necessary and sufficient condition on the asymptotic behavior of the spectral function. Furthermore, we also obtained some related results on the maximum process of the diffusion.

Research Products

• [Journal Article] 拡散過程の最大値に関する極限定理 (2012)
  • Author(s): 笠原勇二, 田原元気
  • Journal Title: 統計数理研究所共同研究リポート Volume: 275 Pages: 115-119
• [Journal Article] On the maximum of a one-dimensional diffusion (2012)
  • Author(s): Kasahara, Yuji ; Kumanda, Kosuke
  • Journal Title: Kobe J.Math. Volume: (掲載決定)
  • Peer Reviewed
• [Journal Article] A classical mechanical model of Brownian motion with one particle coupled to a random wave field (2012)
  • Author(s): Kusuoka, Shigeo ; Liang, Song
  • Journal Title: Stochastic Analysis and Applications Volume: (掲載決定)
  • Peer Reviewed
• [Journal Article] 1次元拡散過程の最大値について (2011)
  • Author(s): 笠原勇二
  • Journal Title: 統計数理研究所共同研究リポート Volume: 262 Pages: 103-107
• [Journal Article] Asymptotic behavior of spectral measures of Krein's and Kotani's strings (2010)
  • Author(s): Kasahara, Yuji and Watanabe, Shinzo
  • Journal Title: Kyoto J. Math. 50 Volume: no.3 Pages: 623-644
  • Peer Reviewed
• [Journal Article] A classical mechanical model of Brownian motion with plural particles (2010)
  • Author(s): Kusuoka, Shigeo and Liang, Song
  • Journal Title: Rev. Math. Phys. 22 Volume: no.7 Pages: 733-838
  • Peer Reviewed
• [Journal Article] Brownian functionals attracted to stable Levy processes (2010)
  • Author(s): 笠原勇二
  • Journal Title: 統計数理研究所共同研究リポート Volume: 247 Pages: 22-26
• [Journal Article] Asymptotic behavior of spectral measures of Krein's and Kotani's strings (2010)
  • Author(s): Kasahara, Y, Watanabe, S
  • Journal Title: Kyoto J.Math. Volume: 50 Pages: 623-644
  • Peer Reviewed
• [Journal Article] Remarks on Krein-Kotani's correspondence between strings and Herglotz functions (2009)
  • Author(s): Kasahara, Yuji and Watanabe, Shinzo
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci. 85 Volume: no.3 Pages: 22-26
  • NAID: 120007131205
  • Peer Reviewed
• [Journal Article] Occupation time theorems for one-dimensional random walks anddiffusion processes in random environments (2009)
  • Author(s): Kasahara, Yuji and Watanabe, Shinzo
  • Journal Title: Stochastic Process. Appl. 119 Volume: no.2 Pages: 347-372
  • Peer Reviewed
• [Journal Article] A formula to compute implied volatility, with error estimate (2009)
  • Author(s): Liang, Song ; Tahara, Yoshihiro
  • Journal Title: Interdiscip. Inform. Sci15 Volume: no.2 Pages: 267-272
  • Peer Reviewed
• [Journal Article] Krein-Kotaniの対応のレヴィ過程への応用 (2009)
  • Author(s): 笠原勇二, 渡辺信三
  • Journal Title: 統計数理研究所共同研究リポート Volume: 225 Pages: 1-6
• [Journal Article] Remarks on Krein-Kotani's correspondence between strings and Herglotz functions. (2009)
  • Author(s): Kasahara, Y.; Watanabe, S.
  • Journal Title: Proc.Japan Acad.Ser. A Math.Sci. 85 Pages: 22-26
  • NAID: 120007131205
  • Peer Reviewed
• [Journal Article] Occupation time theorems for one-dimensional random walks and diffusion processes in random environments. (2009)
  • Author(s): Kasahara, Y.; Watanabe, S.
  • Journal Title: Stochastic Process.Appl. 119 Pages: 347-372
  • NAID: 120007138343
  • Peer Reviewed
• [Journal Article] Limiting distribution of the maximum of a null recurrent diffusion process
  • Author(s): Yuji Kasahara and Genki Tahara
  • Journal Title: Kodai Mathematical Journal, to appear
  • NAID: 130004687977
  • Peer Reviewed
• [Journal Article] On the maximum of a one-dimensional diffusion
  • Author(s): Kasahara, Yuji and Kumanda, Kosuke
  • Journal Title: Kobe J. Math., to appear
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of spectral measures of Krein-Kotani's strings
  • Author(s): Kasahara, Y.; Watanabe, S.
  • Journal Title: J.Math.Kyoto Univ. (To appear)(印刷中)
  • Peer Reviewed
• [Presentation] 拡散過程の最大値に関する極限定理 (2011)
  • Author(s): 笠原勇二
  • Organizer: 統計数理研究所共同研究集会
  • Place of Presentation: 統計数理研究所
  • Year and Date: 2011-11-12
• [Presentation] 拡散過程の最大値に関する極限定理 (2011)
  • Author(s): 笠原勇二
  • Organizer: 統計数理研究所共同研究集会
  • Place of Presentation: 統計数理研究所(東京都立川市)
  • Year and Date: 2011-11-12
• [Presentation] Brownian functionals attracted to stable L'evy processes (2010)
  • Author(s): 笠原勇二
  • Organizer: 統計数理研究所共同研究集会「無限分解可能過程に関連する諸問題」
  • Place of Presentation: 統計数理研究所
  • Year and Date: 2010-11-19
• [Presentation] 1次元拡散過程の最大値について (2010)
  • Author(s): 笠原勇二
  • Organizer: 共同研究集会「無限分解可能過程に関連する諸問題」
  • Place of Presentation: 統計数理研究所
  • Year and Date: 2010-10-23
• [Presentation] 1次元拡散過程の推移確率の漸近挙動 (2010)
  • Author(s): 笠原勇二, 渡辺信三
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-22
• [Presentation] 1次元拡散過程の推移確率の漸近挙動 (2010)
  • Author(s): 笠原勇二・渡辺信三
  • Organizer: 日本数学会 秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-22
• [Presentation] Brownian functionals attracted to stable L'evy processes (2009)
  • Author(s): 笠原勇二
  • Organizer: 統計数理研究所共同研究集会
  • Place of Presentation: 統計数理研究所
  • Year and Date: 2009-11-19