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In this paper, we consider the long time behaviour of collisionless kineticequation with stochastic diffuse boundary operators for velocities bounded awayfrom zero. We show that under suitable reasonable conditions, the semigroup iseventually compact. In particular, without any irreducibility assumption, thesemigroup converges exponentially to the spectral projection associated to thezero eigenvalue as $t \to \infty.$ This contrasts drastically to the caseallowing arbitrarily slow velocities for which the absence of a spectral gapyields at most algebraic rate of convergence to equilibrium. Some openquestions are also mentioned.

On eventual compactness of collisionless kinetic semigroups with velocities bounded away from zero