General theories on nonlinear stochastic partial differential equations with renormalizations

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14556
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Osaka University (2021-2022); Kyushu University (2019-2020)
• Principal Investigator: Hoshino Masato 大阪大学, 大学院基礎工学研究科, 准教授 (20823206)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 確率偏微分方程式 / 繰り込み / 正則性構造理論 / パラ制御解析

Outline of Final Research Achievements

We studied stochastic partial differential equations with renormalizations both theoretically and practically. Theoretically, we proved that the two approaches, the theory of regularity structures, which is based on a local analysis, and the theory of paracontrolled calculus, which is based on Fourier analysis, are equivalent in a sense. Practically, we proved renormalizability and local or global in time solvability of several stochastic partial differential equations important in mathematical physics, such as the stochastic quantization associated with the Hoegh-Krohn model, nonlinear damped stochastic wave equations, and quasilinear parabolic equations with general nonlinear terms.

Free Research Field

確率偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

近年，繰り込みという特別な操作が必要な確率偏微分方程式が注目を集めており，場の量子論などの観点から盛んに研究されている．本研究課題の第一の成果は，この分野における正則性構造理論とパラ制御解析という2つの異なるアプローチが本質的に同値であることを厳密に示したことである．また第二の成果は，Hoegh-Krohnモデルなどの重要なモデルの繰り込み可能性を，場合によって一般論を拡張しながら証明したことである．これらの成果により，繰り込みを伴う確率偏微分方程式の一般論としての研究がさらなる発展を遂げることが期待される．