  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.
