Convergence to the equilibrium for the kinetic transport equation in the two-dimensional periodic Lorentz Gas

We consider the kinetic transport equation that arises in the Boltzmann-Grad limit of the two-dimensional periodic Lorentz gas. This equation was obtained by extending the phase space of positions and velocities through the introduction of two new variables, representing the time to the next collision and the corresponding impact parameter. Here, we mostly focus on the case of periodic boundary conditions on the position space; we prove that, under suitable conditions, the time evolution of a probability density on the extended phase space converges to the equilibrium state with respect to the $ L^p $ norm ($ ^* $-weakly if $ p = \infty $), if such initial density is $ L^p $. If $ p = 2 $, or if the initial datum does not depend on the position, we also get more precise estimates about the rate of convergence to the equilibrium. Our proof is based on the analysis of the long-time behavior of the Fourier coefficients of the solution.