Another look at the Bal\'azs-Quastel-Sepp\"al\"ainen theorem

We study the KPZ equation with a $1+1-$dimensional spacetime white noise, started at equilibrium, and give a different proof of the main result of \cite{bqs}, i.e., the variance of the solution at time $t$ is of order $t^{2/3}$. Instead of using a discrete approximation through the exclusion process and the second class particle, we utilize the connection to directed polymers in a random environment. Along the way, we show the annealed density of the stationary continuum directed polymer equals to the two-point covariance function of the stationary stochastic Burgers equation, confirming the physics prediction in \cite{MT}.