  This paper provides a new tauberian approach to the study of quantitative
time asymptotics of collisionless transport semigroups with general diffuse
boundary operators. We obtain an (almost) optimal algebraic rate of convergence
to equilibrium under very general assumptions on the initial datum and the
boundary operator. The rate is prescribed by the maximal gain of integrability
that the boundary operator is able to induce. The proof relies on a
representation of the collisionless transport semigroups by a (kind of)
Dyson-Phillips series and on a fine analysis of the trace on the imaginary axis
of Laplace transform of remainders (of large order) of this series. Our
construction is systematic and is based on various preliminary results of
independent interest.

Analysis of PDEs