Hypocoercivity of linear kinetic equations via harris's theorem

We study convergence to equilibrium of the linear relaxation Boltz-mann (also known as linear BGK) and the linear Boltzmann equations either on the torus (x, v) ε T_d x R_d or on the whole space (x, v) ε R_d x R_d with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively L¹ or weighted L¹ norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem.