Scaling exponent for the Hopf-Cole solution of KPZ/Stochastic Burgers

TL;DR

This paper establishes that the variance of the logarithm of the solution to the stochastic heat equation with multiplicative noise scales as t^{2/3}, confirming the KPZ universality class behavior.

Contribution

It proves the t^{2/3} variance scaling for the KPZ equation's solution with stationary initial data, using approximation by weakly asymmetric exclusion processes.

Findings

Variance of log Z(t,x) scales as t^{2/3}.

Excess diffusivity D(t) scales as t^{1/3}.

Results confirmed via microscopic coupling with exclusion processes.

Abstract

We consider the stochastic heat equation $\partial_tZ= \partial_x^2 Z - Z \dot W$ on the real line, where $\dot W$ is space-time white noise. $h(t,x)=-\log Z(t,x)$ is interpreted as a solution of the KPZ equation, and $u(t,x)=\partial_x h(t,x)$ as a solution of the stochastic Burgers equation. We take $Z(0,x)=\exp\{B(x)\}$ where $B(x)$ is a two-sided Brownian motion, corresponding to the stationary solution of the stochastic Burgers equation. We show that there exist $0< c_1\le c_2 <\infty$ such that $c_1t^{2/3}\le \Var (\log Z(t,x))\le c_2 t^{2/3}.$ Analogous results are obtained for some moments of the correlation functions of $u(t,x)$. In particular, it is shown that the excess diffusivity satisfies $c_1t^{1/3}\le D(t) \le c_2 t^{1/3}.$ The proof uses approximation by weakly asymmetric simple exclusion processes, for which we obtain the microscopic analogies of the results by coupling.