Quantitative rates of convergence to equilibrium for the linear relaxation Boltzmann Equation with variable jump rates

I will discuss kinetic equations whose jump rate $\sigma(x)$ depends on the position in space. Exponential convergence to equilibrium for such equations was proved to be equivalent to a geometric control condition by Bernard and Salvarani for the zero potential case and then by Han-Kwan and Leautaud for more complex jump kernels and with non zero potential. These works do not give quantitative rates. I will explain how Doeblin's theorem can be used to give quantitative rates of convergence to equilibrium.