  We introduce the stochastic band structure, a method giving the dispersion
relation for waves propagating in periodic media or along waveguides, and
subject to material loss or radiation damping. Instead of considering an
explicit or implicit functional relation between frequency $\omega$ and
wavenumber $k$, as is usually done, we consider a mapping of the resolvent set
in the dispersion space $(\omega, k)$. Bands appear as as the trace of
Lorentzian responses containing local information on propagation loss both in
time and space domains. For illustration purposes, the method is applied to a
lossy sonic crystal, a radiating surface phononic crystal, and a radiating
optical waveguide. The stochastic band structure can be obtained for any system
described by a time-harmonic wave equation.
