We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $ (x,v) \in \mathbb{T}^d \times \mathbb{R}^d $ or on the whole space $ (x,v) \in \mathbb{R}^d \times \mathbb{R}^d $ with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively $ L^1 $ or weighted $ L^1 $ 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.
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