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

We consider the stochastic heat equation ∂ₜZ= ∂ₓ² Z - Z W on the real line, where 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)=∂ₓ h(t,x) as a solution of the stochastic Burgers equation. We take Z(0,x)=exp(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≤ c_2