Fluctuation exponent of the KPZ/stochastic Burgers equation

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. $h(t,x)=-\operatorname{log} Z(t,x)$ 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

$$c_1t^{2/3}\le \operatorname{Var}(\operatorname{log} Z(t,x) )\le c_2 t^{2/3}.$$

Analogous results are obtained for some moments of the correlation functions of $u(t,x)$. In particular, it is shown there that the bulk diffusivity satisfies

$$c_1t^{1/3}\le D_{\rm bulk}(t) \le c_2 t^{1/3}.$$

The proof uses approximation by weakly asymmetric simple exclusion processes, for which we obtain the microscopic analogies of the results by coupling.