@article{ECP1678,
author = {Jeremy Quastel and Daniel Remenik},
title = {Local Brownian property of the narrow wedge solution of the KPZ equation},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {16},
year = {2011},
keywords = {Kardar-Parisi-Zhang equation; stochastic heat equation; Brownian motion; finite variation; stochastic Burgers equation; random growth; asymmetric exclusion process; directed polymers},
abstract = {Abstract. Let $H(t,x)$ be the Hopf-Cole solution at time t of the Kardar-Parisi-Zhang (KPZ) equation starting with narrow wedge initial condition, i.e. the logarithm of the solution of the multiplicative stochastic heat equation starting from a Dirac delta. Also let $H^{eq}(t,x)$ be the solution at time $t$ of the KPZ equation with the same noise, but with initial condition given by a standard two-sided Brownian motion, so that $H^{eq}(t,x)-H^{eq}(0,x)$ is itself distributed as a standard two-sided Brownian motion. We provide a simple proof of the following fact: for fixed $t$, $H(t,x)-(H^{eq}(t,x)-H^{eq}(t,0))$ is locally of finite variation. Using the same ideas we also show that if the KPZ equation is started with a two-sided Brownian motion plus a Lipschitz function then the solution stays in this class for all time.},
pages = {no. 62, 712-719},
issn = {1083-589X},
doi = {10.1214/ECP.v16-1678},
url = {http://ecp.ejpecp.org/article/view/1678}}