  Inelastic scattering techniques provide a powerful approach to studying
electron and nuclear dynamics, via reconstruction of a propagator that
quantifies the time evolution of a system. There is now growing interest in
applying such methods to very low energy excitations, such as lattice
vibrations, but in this limit the cross section is no longer proportional to a
propagator. Significant deviations occur due to the finite temperature Bose
statistics of the excitations. Here we consider this issue in the context of
high-resolution electron energy loss experiments on the copper-oxide
superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$. We find that simple division of a
Bose factor yields an accurate propagator on energy scales greater than the
resolution width. However, at low energy scales, the effects of resolution and
finite temperature conspire to create anomalies in the dynamics at long times.
We compare two practical ways for dealing with such anomalies, and discuss the
range of validity of the technique in light of this comparison.
