Annals of Probability

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

Sourav Chatterjee and Alexander Dunlap

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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.

Article information

Ann. Probab., Volume 48, Number 2 (2020), 1014-1055.

Received: November 2018
Revised: May 2019
First available in Project Euclid: 22 April 2020

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Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 81T15: Perturbative methods of renormalization 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

KPZ equation Cole–Hopf transformation renormalization


Chatterjee, Sourav; Dunlap, Alexander. Constructing a solution of the $(2+1)$-dimensional KPZ equation. Ann. Probab. 48 (2020), no. 2, 1014--1055. doi:10.1214/19-AOP1382.

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