Asymptoic theory of the spectral functions of one-dimensional second order differential operators and its applications to diffusion processes

• Project/Area Number: 24540110
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• 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): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 拡散過程 / スペクトル関数 / 正則変動関数 / タウバー型定理 / 推移確率密度 / ２階微分作用素 / hitting time / ベッセル過程

Outline of Final Research Achievements

Stochastic processes are mathematical models which describe various quantities varying randomly. The diffusion process is a basic stochastic process that varies without jumps, and it has applications in various fields. A diffusion corresponds to a second-order differential operator (generator) and its spectral function plays an important role to describe various properties and quantities of the stochastic process.
In our research we clarified the relation between the asymptotic behavior of the spectral function and that of the drift-coefficient of the operator. We also obtained some Tauberian theorems which are tools to extend results on the half line to those of the full line.

Research Products

• [Journal Article] Tauberian theorem for harmonic mean of Stieltjes transforms and its applications to linear diffusions (2016)
  • Author(s): Kasahara Yuji; Kotani Shin'ichi
  • Journal Title: Osaka J. Math. Volume: 53 Pages: 221-249
  • NAID: 120005941427
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Bessel like diffusion の first hitting time について (2016)
  • Author(s): 笠原勇二
  • Journal Title: 統計数理研究所共同研究リポート Volume: 352 Pages: 56-62
  • Acknowledgement Compliant
• [Journal Article] Diffusions with Bessel-like Drifts (2015)
  • Author(s): Kasahara Yuji; Kotani Shin'ichi
  • Journal Title: Kyoto J. Math. Volume: 55 Pages: 773-797
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Stietjes 変換の調和平均に関するタウバー型定理 (2015)
  • Author(s): 笠原勇二
  • Journal Title: 統計数理研究所共同研究レポート Volume: 350 Pages: 28-33
• [Journal Article] Bessel過程に近い拡散過程について (2014)
  • Author(s): 笠原勇二
  • Journal Title: 統計数理研究所共同研究レポート「無限分解可能過程に関連する諸問題(18)」 Volume: 328 Pages: 72-77
• [Journal Article] On the maximum of a one-dimensional diffusion (2013)
  • Author(s): Y. Kasahara, K. Kumanda
  • Journal Title: Kobe J. Math. Volume: 30 Pages: 33-47
  • Peer Reviewed
• [Journal Article] Limiting distribution of the maxmum of a null recurrent diffusion process (2012)
  • Author(s): Yuji Kasahara and Genki Tahara
  • Journal Title: Kodai Math. Jour. Volume: 35 Pages: 629-641
  • Peer Reviewed
• [Journal Article] Spectral function of Krein's and Kotani's string in the class $Gamma$ (2012)
  • Author(s): Kasahara, Yuji
  • Journal Title: Proc. Japan Acad. Ser. A Math. Sci. Volume: 88 Pages: 173-177
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of the transition density of an ergodic linear diffusion. (2012)
  • Author(s): Kasahara, Yuji
  • Journal Title: Publ. Res. Inst. Math. Sci. Volume: 48 Pages: 565-578
  • NAID: 110009487921
  • Peer Reviewed
• [Presentation] Bessel like diffusion の first hitting time について (2015)
  • 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: 2015-12-03
• [Presentation] Bessel-like diffusion の推移密度の漸近挙動 (2015)
  • Author(s): 笠原勇二
  • Organizer: 日本数学会
  • Place of Presentation: 明治大学（東京都千代田区）
  • Year and Date: 2015-03-21
• [Presentation] Stieltjes 変換の調和平均に関するタウバー型定理 (2014)
  • 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（立川市）
  • Year and Date: 2014-11-27
• [Presentation] Bessel like diffusisons (2013)
  • 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（東京都立川市）