Solvability of linear kinetic equations with multi-energetic inelastic scattering

In this paper we shall analyse the linear Boltzmann equation describing the motion of test particles through a background of heavy field particles that can appear in several energy states. The inelastic scattering process consists in the exchange of quanta of energy between the field and test particles. The well-posedness of the problem is investigated by means of the substochastic semigroup theory and the conditions on the scattering collision frequencies are given for the evolution to keep the collision rate finite and to preserve the total number of particles.