Fluctuation exponent of the KPZ/stochastic Burgers equation

Abstract

We consider the stochastic heat equation \[ ∂ t Z = ∂ x 2 Z − Z W ˙ \partial _tZ= \partial _x^2 Z - Z \dot W \] on the real line, where W ˙ \dot W is space-time white noise. h ( t , x ) = − log ⁡ Z ( t , x ) h(t,x)=-\operatorname {log} Z(t,x) is interpreted as a solution of the KPZ equation, and u ( t , x ) = ∂ x h ( t , x ) u(t,x)=\partial _x h(t,x) as a solution of the stochastic Burgers equation. We take Z ( 0 , x ) = exp ⁡ { B ( x ) } Z(0,x)=\exp \{B(x)\} , where B ( x ) 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 > ∞ 0> c_1\le c_2 >\infty such that \[ c 1 t 2 / 3 ≤ Var ⁡ ( log ⁡ Z ( t , x ) ) ≤ c 2 t 2 / 3 . c_1t^{2/3}\le \operatorname {Var}(\operatorname {log} Z(t,x) )\le c_2 t^{2/3}. \] Analogous results are obtained for some moments of the correlation functions of u ( t , x ) u(t,x) . In particular, it is shown there that the bulk diffusivity satisfies \[ c 1 t 1 / 3 ≤ D bulk ( t ) ≤ c 2 t 1 / 3 . c_1t^{1/3}\le D_\textrm {bulk}(t) \le c_2 t^{1/3}. \] The proof uses approximation by weakly asymmetric simple exclusion processes, for which we obtain the microscopic analogies of the results by coupling.