On the exponential decay to equilibrium of the degenerate linear Boltzmann equation

In this paper we study the decay to the equilibrium state for the solution of the linear Boltzmann equation in the torus by allowing that the non-negative cross section can vanish in a subregion X of the domain with positive measure with respect to the Lebesgue measure. We show that the geometrical characterization of X is the key property to produce exponential decay to equilibrium.