Scaling Exponent for the Hopf-cole Solution of Kpz/stochastic Burgers

We consider the stochastic heat equation ∂tZ = ∂ 2 xZ − ZẆ on the real line, where Ẇ is space-time white noise. h(t, x) = − logZ(t, x) is interpreted as a solution of the KPZ equation, and u(t, x) = ∂xh(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 < c1 ≤ c2 < ∞ such that c1t 2/3 ≤ Var(logZ(t, x)) ≤ c2t. 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 c1t 1/3 ≤ D(t) ≤ c2t. The proof uses approximation by weakly asymmetric simple exclusion processes, for which we obtain the microscopic analogies of the results by coupling.