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March 2020 Constructing a solution of the $(2+1)$-dimensional KPZ equation
Sourav Chatterjee, Alexander Dunlap
Ann. Probab. 48(2): 1014-1055 (March 2020). DOI: 10.1214/19-AOP1382


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 $d\ge 3$ 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 $d\ge 3$ 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 $\varepsilon \to 0$ of Cole–Hopf solutions of the $(2+1)$-dimensional KPZ equation with white noise mollified to spatial scale $\varepsilon $ and nonlinearity multiplied by the vanishing factor $|\log \varepsilon |^{-\frac{1}{2}}$. 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.


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Sourav Chatterjee. Alexander Dunlap. "Constructing a solution of the $(2+1)$-dimensional KPZ equation." Ann. Probab. 48 (2) 1014 - 1055, March 2020.


Received: 1 November 2018; Revised: 1 May 2019; Published: March 2020
First available in Project Euclid: 22 April 2020

zbMATH: 07199868
MathSciNet: MR4089501
Digital Object Identifier: 10.1214/19-AOP1382

Primary: 35R60, 60H15, 81T15

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 2 • March 2020
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