Local Brownian property of the narrow wedge solution of the KPZ equation

Jeremy Quastel (University of Toronto)
Daniel Remenik (University of Toronto)

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.

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Pages: 712-719

Publication Date: November 20, 2011

DOI: 10.1214/ECP.v16-1678

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