We study the spatial decorrelation of the solution to the KPZ equation with narrow wedge initial data. For fixed $t>0$, we determine the decay rate of the spatial covariance function, showing that ${ \rm Cov}[h(t,x),h(t,0)]\sim \frac{ t}{ x}$ as $x\to\infty$. In addition, we prove that the finite-dimensional distributions of the properly rescaled spatial average of the height function converge to those of a Brownian motion.
Categories: math.PR