  We study the existence of densities for distributions of piecewise
deterministic Markov processes. We also obtain relationships between invariant
densities of the continuous time process and that of the process observed at
jump times. In our approach we use functional-analytic methods and the theory
of linear operator semigroups. By imposing general conditions on the
characteristics of a given Markov process, we show the existence of a
substochastic semigroup describing the evolution of densities for the process
and we identify its generator. Our main tool is a new perturbation theorem for
substochastic semigroups, where we perturb both the action of the generator and
of its domain, allowing to treat general transport-type equations with
non-local boundary conditions. A couple of particular examples illustrate our
general results.
