  In this paper we study the following Burgers equation
  du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t)
  where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x
and white noise in t. We prove the existence and uniqueness of an invariant
measure by establishing a ``one force, one solution'' principle, namely that
for almost every realization of the force, there is a unique distinguished
solution that exists for the time interval (-infty, +infty) and this solution
attracts all other solutions with the same forcing. This is done by studying
the so-called one-sided minimizers. We also give a detailed description of the
structure and regularity properties for the stationary solutions. In
particular, we prove, under some non-degeneracy conditions on the forcing, that
almost surely there is a unique main shock and a unique global minimizer for
the stationary solutions. Furthermore the global minimizer is a hyperbolic
trajectory of the underlying system of characteristics.
