January 2021 2D anisotropic KPZ at stationarity: Scaling, tightness and nontriviality
Giuseppe Cannizzaro, Dirk Erhard, Philipp Schönbauer
Ann. Probab. 49(1): 122-156 (January 2021). DOI: 10.1214/20-AOP1446


In this work we focus on the two-dimensional anisotropic KPZ (aKPZ) equation, which is formally given by \begin{equation*}\partial _{t}h=\frac{\nu }{2}\Delta h+\lambda \bigl((\partial _{1}h)^{2}-(\partial _{2}h)^{2}\bigr)+\nu ^{\frac{1}{2}}\xi ,\end{equation*} where $\xi $ denotes a noise which is white in both space and time, and $\lambda $ and $\nu $ are positive constants. Due to the wild oscillations of the noise and the quadratic nonlinearity, the previous equation is classically ill posed. It is not possible to linearise it via the Cole–Hopf transformation and the pathwise techniques for singular SPDEs (the theory of regularity structures by M. Hairer or the paracontrolled distributions approach of M. Gubinelli, P. Imkeller, N. Perkowski) are not applicable. In the present work we consider a regularised version of aKPZ which preserves its invariant measure. We prove the existence of subsequential limits once the regularisation is removed, provided $\lambda $ and $\nu $ are suitably renormalised. Moreover, we show that, in the regime in which $\nu $ is constant and the coupling constant $\lambda $ converges to $0$ as the inverse of the square root logarithm, any limit differs from the solution to the linear equation obtained by simply dropping the nonlinearity in aKPZ.


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Giuseppe Cannizzaro. Dirk Erhard. Philipp Schönbauer. "2D anisotropic KPZ at stationarity: Scaling, tightness and nontriviality." Ann. Probab. 49 (1) 122 - 156, January 2021. https://doi.org/10.1214/20-AOP1446


Received: 1 November 2019; Revised: 1 March 2020; Published: January 2021
First available in Project Euclid: 22 January 2021

Digital Object Identifier: 10.1214/20-AOP1446

Primary: 35R60 , 60H15
Secondary: 60F17

Keywords: anisotropic KPZ equation , Criticality , energy solutions , renormalisation

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 1 • January 2021
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