×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv