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 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.

Expanded introduction, added Theorem 2, and changed title (from "Finite variation of the crossover Airy2 process with respect to Brownian motion"). To appear in Electronic Communications in Probability

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• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• The intermediate disorder regime for directed polymers in dimension $1+1$
• The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1
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Cited by in corpus (6)

• A stochastic Burgers equation from a class of microscopic interactions
• Airy processes and variational problems
• Local behavior and hitting probabilities of the Airy1 process
• Continuum statistics of the Airy2 process
• Brownian absolute continuity of the KPZ fixed point with arbitrary initial condition
• Temporal increments of the KPZ equation with general initial data