Rates of convergence to equilibrium for collisionless kinetic equations in slab geometry

This work deals with free transport equations with partly diffuse stochastic boundary operators in slab geometry. Such equations are governed by stochastic semigroups in L-1 spaces. We prove convergence to equilibrium at the rate O(t(-k/2(k+1)+1)) (t -> +infinity) for L-1 initial data g in a suitable subspace of the domain of the generator T where k is an element of N depends on the properties of the boundary operators near the tangential velocities to the slab. This result is derived from a quantified version of Ingham's tauberian theorem by showing that F-g(s) := lim(epsilon -> 0+ )(is + epsilon - T)(-1) exists as a C-k function on R\textbackslash{0} such that parallel to d(j)/ds(j) F-g(s)parallel to <= C/vertical bar s vertical bar(2(j+1)) near s = 0 and bounded as vertical bar s vertical bar ->infinity(0 <= j <= k). Various preliminary results of independent interest are given and some related open problems are pointed out. (C) 2018 Elsevier Inc. All rights reserved.