Scaling exponent for the Hopf-Cole solution of KPZ/Stochastic Burgers

Abstract

We consider the stochastic heat equation ${\partial }_{t}Z={\partial }_x^{2}Z-Z\dot{W}$ on the real line, where $\dot{W}$ is space-time white noise. $h(t,x)=-\log Z(t,x)$ is interpreted as a solution of the KPZ equation, and $u(t,x)={\partial }_{x}h(t,x)$ as a solution of the stochastic Burgers equation. We take $Z(0,x)=\exp \{B(x)\}$ where $B(x)$ is a two-sided Brownian motion, corresponding to the stationary solution of the stochastic Burgers equation. We show that there exist $0<c_1\le c_2<\infty$ such that $c_1t^{2/3}\le \Var (\log Z(t,x))\le c_2 t^{2/3}.$ Analogous results are obtained for some moments of the correlation functions of $u(t,x)$. In particular, it is shown that the excess diffusivity satisfies $c_1{t}^{1/3}\le D(t)\le c_2{t}^{1/3}.$ The proof uses approximation by weakly asymmetric simple exclusion processes, for which we obtain the microscopic analogies of the results by coupling.

References in corpus (1)

• $t^{1/3}$ Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on $\mathbb{Z}$

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