Constructing a solution of the (2+1) -dimensional KPZ equation

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 ..0 of Cole.Hopf solutions of the (2+1)-dimensional KPZ equation with white noise mollified to spatial scale . and nonlinearity multiplied by the vanishing factor |log.|.12. 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.