Stochastic Analysis on infinite dimensional spaces

• Project/Area Number: 24340023
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tohoku University
• Principal Investigator: Aida Shigeki 東北大学, 理学(系)研究科(研究院), 教授 (90222455)
• Co-Investigator(Renkei-kenkyūsha): Shigekawa Ichiro 京都大学, 大学院理学研究科, 教授 (00127234) ; Hino Masanori 大阪大学, 大学院基礎工学研究科, 教授 (40303888) ; Kawabi Hiroshi 岡山大学, 大学院自然科学研究科, 教授 (80432904) ; Inahama Yuzuru 九州大学, 大学院数理学研究院, 教授 (80431998) ; Hiroshima Fumio 九州大学, 大学院数理学研究院, 教授 (00330358) ; Kuwada Kazumasa 東京工業大学, 大学院理工学研究科, 准教授 (30432032) ; Hariya Yuu 東北大学, 大学院理学研究科, 准教授 (20404030)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥6,890,000 (Direct Cost: ¥5,300,000、Indirect Cost: ¥1,590,000); Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2012: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 確率解析 / 無限次元解析 / ラフパス / 確率微分方程式 / スペクトル / 場の量子論 / 無限次元空間 / 準古典極限 / スペクトルギャップ / 大偏差原理 / 反射境界条件 / スコロホッド方程式

Outline of Final Research Achievements

We studied the spectral property of OU operators on infinite dimensional spaces. This is an infinite dimensional version of the study of semiclassical behaviour
of the spectral gap of the operator. We determined the asymptotic behaviour of the spectral gap of the OU operator on a loop space of the rotationally symmetric Riemannianmanifold by the hessian of the energy function at the geodesic.
Also we studied stochastic differential equations and rough differential equations. We proved that the Wong-Zakai approximation solution of the reflected stochastic differential equation converges to the solution and formulated reflected rough differential equations and proved the existence of the solutions.

Research Products

• [Journal Article] Asymptotics of spectral gaps on loop spaces over a class of Riemannian manifolds (2015)
  • Author(s): Shigeki Aida
  • Journal Title: Journal of Functional Analysis Volume: 269 Issue: 12 Pages: 3714-3764
  • DOI: 10.1016/j.jfa.2015.09.023
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Reflected rough differential equations (2015)
  • Author(s): Shigeki Aida
  • Journal Title: Stochastic Processes and their applications Volume: in press Issue: 9 Pages: 3570-3595
  • DOI: 10.1016/j.spa.2015.03.008
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Wong-Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces II (2014)
  • Author(s): Shigeki Aida
  • Journal Title: Springer Proceedings in Mathematics & Statistics Volume: 100 Pages: 1-23
  • DOI: 10.1007/978-3-319-11292-3_1
  • ISBN: 9783319112916, 9783319112923
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Wong-Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces (2013)
  • Author(s): Shigeki Aida snd Kosuke Sasaki
  • Journal Title: Stochastic Processes and their applications Volume: 123 Issue: 10 Pages: 3800-3827
  • DOI: 10.1016/j.spa.2013.05.004
  • Peer Reviewed
• [Journal Article] Tunneling for spatially cut-off P(φ)_2-Hamiltonians (2012)
  • Author(s): Shigeki Aida
  • Journal Title: Journal of Functional Analysis Volume: 263 Pages: 2689-2753
  • Peer Reviewed
• [Presentation] Asymptotics of spectral gap on loop spaces over a class of Riemannian manifolds (2015)
  • Author(s): Shigeki Aida
  • Organizer: 確率論と幾何学
  • Place of Presentation: 東京工業大学、目黒区大岡山
  • Year and Date: 2015-11-10
• [Presentation] On solutions to reflected rough differential equations (2015)
  • Author(s): Shigeki Aida
  • Organizer: Stochastic Analysis
  • Place of Presentation: 京都大学大学院理学研究科、京都市
  • Year and Date: 2015-09-08
• [Presentation] Asymptotics of the spectral gap on a loop space over a hyperbolic space (2014)
  • Author(s): Shigeki Aida
  • Organizer: Stochastic processes, Analysis and Mathematical Physics
  • Place of Presentation: 関西大学
  • Year and Date: 2014-08-26
  • Invited
• [Presentation] Wong-Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces (2013)
  • Author(s): 会田茂樹
  • Organizer: 確率解析とその周辺
  • Place of Presentation: 京都大学総合人間学部
• [Presentation] Wong-Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces (2013)
  • Author(s): Shigeki Aida
  • Organizer: Stochastic Analysis and Applications
  • Place of Presentation: Oxford University, 英国
• [Presentation] Wong-Zakai approximations for reflecting SDE (2013)
  • Author(s): 会田茂樹
  • Organizer: 確率論シンポジウム
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Tunneling for spatially cut-off P(φ)_2-Hamiltonians (2012)
  • Author(s): 会田茂樹
  • Organizer: 確率論シンポジウム
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2012-12-20
• [Presentation] Tunneling for spatially cut-off P(φ)_2-Hamiltonians (2012)
  • Author(s): 会田茂樹
  • Organizer: 量子場の数理とその周辺
  • Place of Presentation: 京都大学数理解析研究所(招待講演)
  • Year and Date: 2012-11-16
• [Presentation] Tunneling for spatially cut-off P(φ)_2-Hamiltonians (2012)
  • Author(s): 会田茂樹
  • Organizer: 確率解析とその周辺
  • Place of Presentation: 名古屋大学大学院多元数理科学研究科
  • Year and Date: 2012-10-26
• [Presentation] Tunneling for spatially cut-off P(φ) 2-Hamiltonians (2012)
  • Author(s): 会田茂樹
  • Organizer: Stochastic Analysis and applications
  • Place of Presentation: 岡山大学大学院自然科学研究科(招待講演)
  • Year and Date: 2012-09-26
• [Remarks] Tohoku University
  • URL: http://www.math.tohoku.ac.jp/~aida/index-j.html
• [Remarks] 研究紹介
  • URL: http://www.math.tohoku.ac.jp/~aida/paper/paper.html
• [Remarks] Tohoku University
  • URL: http://www.math.tohoku.ac.jp/~aida/index-j.html
• [Remarks]
  • URL: http://www.math.tohoku.ac.jp/~aida/index-j.html
• [Funded Workshop] 確率解析とその周辺 (2015)
  • Place of Presentation: 大阪大学大学院基礎工学研究科
  • Year and Date: 2015-10-20