Various properties relative to intermittency of space-time fractional SPDEs

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05197
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Shinshu University
• Principal Investigator: XIE Bin 信州大学, 学術研究院理学系, 准教授 (50510038)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 確率偏微分方程式 / 無限次元Harnack不等式 / Intermittency / 確率カーン＝ヒリアード方程式 / Hypercontractivity / 特異性

Outline of Final Research Achievements

In this research project, we mainly devoted to various important behaviors relative to the intermittency of singular stochastic partial differential equations. In particular, we proved the intermittent property of the solution of stochastic partial differential equations driven by inhomogeneous space-time white noises. We also studied the ergodicity of some singular stochastic partial differential equations, which include the stochastic Cahn-Hilliard equations with logarithmic potentials as an important example.

Free Research Field

確率解析

Academic Significance and Societal Importance of the Research Achievements

非適切(ill-posed)な確率偏微分方程式の解の性質を考察すべき問題が多く残されている．本研究ではある種の特異性を持つ確率偏微分方程式の解の長時間にわたる振る舞い，特にintermittencyや半群の超圧縮性の研究を行ったので，非適切な確率偏微分方程式についてのエルゴード理論の確立に寄与する意義がある．また，得られた研究成果は多くの応用分野での応用が期待できる．さらに，共著の書籍「確率偏微分方程式」を出版することにより，確率偏微分方程式論の研究の普及や発展に貢献する．