On the convergence to equilibrium for degenerate transport problems

We give a counterexample which shows that the asymptotic rate of convergence to the equilibrium state for the transport equation, with a degenerate cross section and in the periodic setting, cannot be better than $t^{-1/2}$ in the general case. We suggest moreover that the geometrical properties of the cross section are the key feature of the problem and impose, through the distribution of the forward exit time, the speed of convergence to the stationary state.