LOCAL BROWNIAN PROPERTY OF THE NARROW WEDGE SOLUTION OF THE KPZ EQUATION

作者：Quastel Jeremy^*; Remenik Daniel

来源：Electronic Communications in Probability, 2011, 16: 712-719.

• Kardar-Parisi-Zhang equation
• stochastic heat equation
• Brownian motion
• finite variation
• stochastic Burgers equation
• random growth
• asymmetric exclusion process
• directed polymers

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.

• 出版日期2011-11-20