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

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

2011-05-04

arxiv.org/abs/1105.0952v3

Mathematics

Probability

Expanded introduction, added Theorem 2, and changed title (from "Finite variation of the crossover Airy2 process with respect

Scientific paper

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(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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