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

Abstract

qThe (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 ε→0 of Cole–Hopf solutions of the (2 + 1)-dimensional KPZ equation with white noise mollified to spatial scale ε and nonlinearity multiplied by the vanishing factor | log ε|−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.