Discretization and scaling limit for problems of stochastic calculus of variations

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K03343
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Aoyama Gakuin University
• Principal Investigator: Ichihara Naoyuki 青山学院大学, 理工学部, 教授 (70452563)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 確率的変分問題

Outline of Final Research Achievements

We obtained some mathematical results on the problem of stochastic calculus of variations, which is a stochastic version of the classical problem of calculus of variations originating from geometric optics and brachystochrone. We investigated how the long time behavior of the optimal trajectory changes with a perturbation of the potential term and gave, in terms of the parameters in the model, an explicit condition for a phase transition. We also constructed a discrete model corresponding to the above continuous problem and showed that similar phenomena can be observed in that model.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

連続型確率的変分問題の臨界性理論に関する既知の結果を大幅に改善することができた。特に、粘性ハミルトン・ヤコビ方程式の解の性質や確率的変分問題の最適軌道に関する精密な評価式を得ることができた。また、これまではあまり考察されてこなかった確率的変分問題の離散版に相当するモデルを構築することができた。これらの結果より、確率的変分問題の臨界性理論に関する研究のさらなる学術的な進展が期待される。