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

Understand

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.

• $h(t,x)=-\log Z(t,x)$ 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 $u(t,x)$.