2D Anisotropic KPZ at stationarity: scaling, tightness and non triviality

TL;DR

This paper investigates the two-dimensional anisotropic KPZ equation at stationarity, demonstrating the necessity of renormalization for limits and showing that the nonlinearity vanishes in the critical regime, leading to a linear limiting behavior.

Contribution

It introduces a regularized version of aKPZ that preserves its invariant measure and analyzes the limiting behavior, revealing the non-triviality and the conditions under which the nonlinearity disappears.

Findings

Renormalization of parameters is necessary for subsequential limits.

In the critical regime, the nonlinearity vanishes, resulting in a linear limit.

The invariant measure is preserved under the regularized dynamics.

Abstract

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((\partial_1 h)^2 - (\partial_2 h)^2) + \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 show that in order to have subsequential limits once the regularisation is removed, it is necessary to suitably renormalise $\lambda$ and $\nu$. Moreover, we prove that, in the regime suggested by the (non-rigorous) renormalisation group computations of [D.E. Wolf, "Kinetic roughening of vicinal surfaces'', Phys. Rev. Lett., 1991], i.e. $\nu$ constant and the coupling constant $\lambda$ converging 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.