STOCHASTIC PROCESSES WITH COMPETING REINFORCEMENTS

• 成果类型: Article
• 署名作者: Erhard, Dirk; Reis, Guilherme
• 署名单位: Universidade Federal da Bahia; Instituto Nacional de Matematica Pura e Aplicada (IMPA)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2073
• 发表日期: 2024
• 页码: 4513-4553
• 关键词: random-walk; attracting edge

摘要

We introduce a simple but powerful strategy to study processes driven by two or more reinforcement mechanisms in competition. We apply our method to two types of models: to nonconservative zero range processes on finite graphs, and to multi-particle random walks with positive and negative reinforcement on the edges. The results hold for a broad class of reinforcement functions, including those with superlinear growth. Our strategy consists in a comparison of the original processes with suitable reference models. To implement the comparison we estimate an object reminiscent to the Radon- Nikodym derivative on a carefully chosen set of trajectories. Our results describe the almost sure long time behaviour of the processes. We also prove a phase transition depending on the strength of the reinforcement functions.