  We consider the large time behavior of strong solutions to a kind of
stochastic Burgers equation, where the position x is perturbed by a Brownian
noise. It is well known that both the rarefaction wave and viscous shock wave
are time-asymptotically stable for deterministic Burgers equation since the
pioneer work of A. Ilin and O. Oleinik [21] in 1964. However, the stability of
these wave patterns under stochastic perturbation is not known until now. In
this paper, we give a definite answer to the stability problem of the
rarefaction and viscous shock waves for the 1-d stochastic Burgers equation.
That is, the rarefaction wave is still stable under white noise perturbation
and the viscous shock is not stable yet. Moreover, a time-convergence rate
toward the rarefaction wave is obtained. To get the desired decay rate, an
important inequality (denoted by Area Inequality) is derived. This inequality
plays essential role in the proof, and may have applications in the related
problems for both the stochastic and deterministic PDEs.
