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

We consider the stochastic heat equation $\partial_tZ= \partial_x^2 Z - Z \dot W$ontherealline,where$\dot W$isspace-timewhitenoise\mathrm{.}$h(t,x)=-\log Z(t,x)$isinterpretedasasolutionoftheKPZequation,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 $0<c_1\le c_2<\mathrm{\infty }$ 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 $c_1{t}^{1\mathrm{/}3}\le D(t)\le c_2{t}^{1\mathrm{/}3}\mathrm{.}$ The proof uses approximation by weakly asymmetric simple exclusion processes, for which we obtain the microscopic analogies of the results by coupling.