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

Abstract: 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.