AN INVARIANCE PRINCIPLE FOR THE 1D KPZ EQUATION

• 成果类型: Article
• 署名作者: Adhikari, Arka; Chatterjee, Sourav
• 署名单位: Stanford University; Stanford University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1660
• 发表日期: 2024
• 页码: 2019-2050
• 关键词: end-point distribution; directed polymers; limit

摘要

Consider a discrete one-dimensional random surface whose height at a point grows as a function of the heights at neighboring points, plus an independent random noise. Assuming that this function is equivariant under constant shifts, symmetric in its arguments, and at least six times continuously differentiable in a neighborhood of the origin, we show that, as the variance of the noise goes to zero, any such process converges to the Cole-Hopf solution of the 1D KPZ equation under a suitable scaling of space and time. This proves an invariance principle for the 1D KPZ equation in the spirit of Donsker's invariance principle for Brownian motion.