Multiplicative SHE limit of random walks in space-time random environments

• 成果类型: Article; Early Access
• 署名作者: Das, Sayan; Drillick, Hindy; Parekh, Shalin
• 署名单位: University of Chicago; Columbia University; University System of Maryland; University of Maryland College Park
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01339-4
• 发表日期: 2024
• 关键词: kpz equation; heat-equation; UNIVERSALITY; fluctuations; MODEL

摘要

We show that under a certain moderate deviation scaling, the multiplicative-noise stochastic heat equation (SHE) arises as the fluctuations of the quenched density of a 1D random walk whose transition probabilities are iid [0,1]-valued random variables. In contrast to the case of directed polymers in the intermediate disorder regime, the variance of our weights is fixed rather than vanishing under the diffusive rescaling of space-time. Consequently, taking a naive limit of the chaos expansion fails for this model, and a nontrivial noise coefficient is observed in the limit. Rather than using chaos techniques, our proof instead uses the fact that in this regime the quenched density solves a discrete SPDE which resembles the SHE. As a byproduct of our techniques, it is shown that independent noise is generated in the limit, in the sense that the prelimiting noise field does not converge to the driving noise of the limiting SPDE.