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 $ε→0$ 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.

Duke Scholars

Author Alexander J Dunlap Mathematics