The mathematical model of age-dependent branching processes is used to describe neutron slowing down and multiplication in an infinite medium. To construct the probability measure of the neutron branching process, it is necessary to determine the probability density for a neutron of age θ(=time elapsed since birth of the fission neutron) to have energy E. This problem, which is equivalent to the time-dependent slowing down problem, is solved for a scattering law of the form v(E)Σs(E → E′)dE′ = aEµh(E′/E) (dE′/E) and an absorption cross section satisfying the relation v(E) Σa(E) = bEµ + c. In this case, it is proved that there always exist particular “invariant” probability densities suffering only contraction during ageing, i.e., having the form . For the time-dependent slowing down problem with a Greuling-Goertzel kernel, the results are compared with those of Koppel. Particular attention is paid to stationary energy spectra.