Local Brownian property of the narrow wedge solution of the KPZ equation
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 Heq(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 Heq(t,x)-Heq(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(t,x)-Heq(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.