CONSTRUCTING A SOLUTION OF THE (2+1)-DIMENSIONAL KPZ EQUATION

• 成果类型: Article
• 署名作者: Chatterjee, Sourav; Dunlap, Alexander
• 署名单位: Stanford University; Stanford University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1382
• 发表日期: 2020
• 页码: 1014-1055
• 关键词: stochastic heat-equation; UNIVERSALITY; inequalities; wiener; limit; MODEL

摘要

The (d + 1)-dimensional KPZ equation is the canonical model for the growth of rough d-dimensional random surfaces. A deep mathematical understanding of the KPZ equation for d = 1 has been achieved in recent years, and the case d >= 3 has also seen some progress. The most physically relevant case of d = 2, however, is not very well understood mathematically, largely due to the renormalization that is required: in the language of renormalization group analysis, the d = 2 case is neither ultraviolet superrenormalizable like the d = 1 case nor infrared superrenormalizable like the d >= 3 case. Moreover, unlike in d = 1, the Cole-Hopf transform is not directly usable in d = 2 because solutions to the multiplicative stochastic heat equation are distributions rather than functions. In this article, we show the existence of subsequential scaling limits as epsilon -> 0 of Cole-Hopf solutions of the (2 + 1)-dimensional KPZ equation with white noise mollified to spatial scale epsilon and nonlinearity multiplied by the vanishing factor vertical bar log epsilon vertical bar(-1/2). We also show that the scaling limits obtained in this way do not coincide with solutions to the linearized equation, meaning that the nonlinearity has a nonvanishing effect. We thus propose our scaling limit as a notion of KPZ evolution in 2 + 1 dimensions.