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Local Brownian property of the narrow wedge solution of the KPZ equation. (English) Zbl 1243.60054
Summary: 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.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics
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