Another look at the Balázs-Quastel-Seppäläinen theorem

Abstract

We study the KPZ equation with a 1 + 1 − 1+1- dimensional spacetime white noise, started at equilibrium, and give a different proof of the main result of Balázs, Quastel, and Seppäläinen [J. Amer. Math. Soc. 24 (2011), pp. 683–708], i.e., the variance of the solution at time t t is of order t 2 / 3 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 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 of Maes and Thiery [J. Stat. Phys. 168 (2017), pp. 937–963].