  We present the quantitative method of the recent work arXiv:2209.09340 in a
simple setting, together with a compactness argument that was not included in
arXiv:2209.09340 and has interest per se. We are concerned with the exponential
stabilization (spectral gap) for linear kinetic equations with degenerate
thermalization, i.e. when the collision operator vanishes on parts of the
spatial domain. The method in arXiv:2209.09340 covers both scattering and
Fokker-Planck type operators, and deals with external potential and boundary
conditions, but in these notes we present only its core argument and restrict
ourselves to the kinetic Fokker-Planck in the periodic torus with unit
velocities and a thermalization degeneracy. This equation is not covered by the
previous results of Bernard and Salvarani (2013), Han-Kwan and L\'eautaud
(2015), Evans and Moyano (arXiv:1907.12836).
