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 $O\left( t^{-\frac{k}{2(k+1)+1}}\right) \ (t\rightarrow +\infty )$ for $L^{1}$ initial data $g$ in a suitable subspace of the domain of the generator $T$ where $k\in \mathbb{N}$ 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 $F_{g}(s):=\lim_{\varepsilon \rightarrow 0_{+}}\left( is+\varepsilon -T\right) ^{-1}g$ exists as a $C^{k}$ function on $\mathbb{R}\backslash \left\{ 0\right\} $ such that $\ \left\| \frac{d^{j}}{ds^{j}}F_{g}(s)\right\| \leq \frac{C}{| s| ^{2(j+1)}}$ near $s=0$ and bounded as $|s| \rightarrow \infty \ \ \left(0\leq j\leq k\right).$ Various preliminary results of independent interest are given and some related open problems are pointed out.