## 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.

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## Related Subject Headings

- Statistics & Probability
- 4905 Statistics
- 4904 Pure mathematics
- 0104 Statistics
- 0101 Pure Mathematics

### Citation

*Annals of Probability*,

*48*(2), 1014–1055. https://doi.org/10.1214/19-AOP1382

*Annals of Probability*48, no. 2 (March 1, 2020): 1014–55. https://doi.org/10.1214/19-AOP1382.

*Annals of Probability*, vol. 48, no. 2, Mar. 2020, pp. 1014–55.

*Manual*, doi:10.1214/19-AOP1382.

## Published In

## DOI

## EISSN

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- Statistics & Probability
- 4905 Statistics
- 4904 Pure mathematics
- 0104 Statistics
- 0101 Pure Mathematics