Band theory for partially coherent light is introduced by using the formalism
of second-order classical coherence theory under paraxial approximation. It is
demonstrated that the cross-spectral density function, describing correlations
between pairs of points in the field, can have bands and gaps and form a
correlation band structure. The propagation of a partially coherent beam in
non-Hermitian periodic structures is considered to elucidate the interplay
between the degree of coherence and the gain/loss present in the lattice. We
apply the formalism to study partially coherent Bloch oscillations in lattices
having parity-time symmetry and demonstrate that the oscillations can be
sustained in such media but they are strongly dependent upon the spatial
correlations of the beam. A transition between breathing and oscillating modes
is shown to be induced by the degree of spatial coherence