  We consider the Burgers equation on the real line with forcing given by
Poissonian noise with no periodicity assumption. Under a weak concentration
condition on the driving random force, we prove existence and uniqueness of a
global solution in a certain class. We describe its basin of attraction that
can also be viewed as the main ergodic component for the model. We establish
existence and uniqueness of global minimizers associated to the variational
principle underlying the dynamics. We also prove the diffusive behavior of the
global minimizers on the universal cover in the periodic forcing case.
