Fluctuation exponent of the KPZ/stochastic Burgers equation

Abstract:

We consider the stochastic heat equation \[ \partial _tZ= \partial _x^2 Z - Z \dot W \] on the real line, where $\dot W$ is space-time white noise. $h(t,x)=-\operatorname {log} Z(t,x)$ 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 \[ c_1t^{2/3}\le \operatorname {Var}(\operatorname {log} Z(t,x) )\le c_2 t^{2/3}. \] Analogous results are obtained for some moments of the correlation functions of $u(t,x)$. In particular, it is shown there that the bulk diffusivity satisfies \[ c_1t^{1/3}\le D_\textrm {bulk}(t) \le c_2 t^{1/3}.\] The proof uses approximation by weakly asymmetric simple exclusion processes, for which we obtain the microscopic analogies of the results by coupling.