  We establish the existence of generalized Busemann functions and
Gibbs-Dobrushin-Landford-Ruelle measures for a general class of lattice random
walks in random potential with finitely many admissible steps. This class
encompasses directed polymers in random environments, first- and last-passage
percolation, and elliptic random walks in both static and dynamic random
environments in all dimensions and with minimal assumptions on the random
potential.
