Analysis of statistical mechanical models evolving in random media

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K13423
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Nagoya University
• Principal Investigator: Nakashima Makoto 名古屋大学, 多元数理科学研究科, 准教授 (60635902)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 確率熱方程式 / KPZ方程式 / ランダム媒質中のディレクティドポリマー

Outline of Final Research Achievements

It is known that there exit singular SPDEs. We studied high dimensional stochastic heat equations (SHE) and KPZ equations among them. For singular SPDEs, we usually define their solutions as the limit of modified SPDEs with renormalizations. In the case SHE and KPZ eq, we can find a certain regime of parameters in which the limits are solutions of the standard heat equations. We focused on the solutions' fluctuations and proved that they converged to Gaussian random variables.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

高次元SHEやKPZ方程式は特異確率偏微分方程式に有効な正則構造理論などは現時点では適用できていない. 今回は自明な解に収束するようなパラメータ領域ではあるがそれらに対して摂動を調べることで解析を試みた点で非常に意味がある. またこのような収束が成り立つと予想される全ての領域で示せたことも完全な解決を与えたという意味で重要である.