On L1L1 exponential trend to equilibrium for conservative linear kinetic equations on the torus

We compute the growth bound of weighted advection semigroups on the n  -dimensional flat torus with general (not necessarily bounded) measurable nonnegative collision frequencies and give a compactness theorem in L1L1 spaces of interest for the stability of essential type of kinetic semigroups under perturbation by scattering operators. This provides us with a characterisation of the existence of a spectral gap for general conservative neutron transport like equations. Various related questions are also dealt with.