Finer limit theorems for stochastic models on lattices with spatio-temporal interactions

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03514
• 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: Yokohama National University
• Principal Investigator: Takei Masato 横浜国立大学, 大学院工学研究院, 准教授 (60460789)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: パーコレーション / ランダムウォーク / 高木関数

Outline of Final Research Achievements

Percolation process was originally introduced as a model of penetration of fluids into porous media. Nowadays it is one of the most fundamental stochastic models concerning random geometry. From the viewpoint of percolation theory, we studied finer limit theorems for several stochastic models with spatio-temporal interactions. We obtained fine limit theorems for first-passage percolation, random walks with memory effect, and so on. Among others we briefly describe our result on the linearly edge-reinforced random walk on the half-line: Assign weight one to each edge of the half-line. The walker jumps to one of the neighboring vertices with probability proportional to weight of the edge connecting them (with reflection at 0). After crossing an edge, its weight is increased by one. The position of the walker at time n is denoted by S_n. We proved that liminf S_n=0 and limsup S_n/(log_4 n)=1 with probability one.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

空間構造をもった確率モデルは，物理・化学・生物現象の研究においてのみならず，人々の意見が合意に達するか否かといった社会現象の研究等においても重要な役割を果たしており，多様な現象のモデル構築と解析を可能にすることが求められている．本研究では，浸透現象の数学的解析における様々な着想を基盤とし，記憶があり学習しながら歩むランダムウォーク等に関する成果を得て，この方面の研究に一定の寄与をした．また，本研究で得られた知見の副産物として，至るところ微分不可能な連続関数の性質を記述する極限定理が得られており，数学の中での周辺分野にも貢献することができた．