  We construct space-time stationary solutions of the 1D Burgers equation with
random forcing in the absence of periodicity or any other compactness
assumptions. More precisely, for the forcing given by a homogeneous Poissonian
point field in space-time we prove that there is a unique global solution with
any prescribed average velocity. These global solutions serve as one-point
random attractors for the infinite-dimensional dynamical system associated to
solutions to the Cauchy problem. The probability distribution of the global
solutions defines a stationary distribution for the corresponding Markov
process. We describe a broad class of initial Cauchy data for which the
distribution of the Markov process converges to the above stationary
distribution.
  Our construction of the global solutions is based on a study of the field of
action-minimizing curves. We prove that for an arbitrary value of the average
velocity, with probability 1 there exists a unique field of action-minimizing
curves initiated at all of the Poissonian points. Moreover action-minimizing
curves corresponding to different starting points merge with each other in
finite time.
