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We investigate the large time behavior of solutions to the spatiallyhomogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysisis performed in some (weighted) $L^{1}$-spaces. We deal with both the cases ofhard and soft potentials (with angular cut-off). For hard potentials, weprovide a new proof of the fact that, in weighted $L^{1}$-spaces withexponential or algebraic weights, the solutions converge exponentially fasttowards equilibrium. Our approach uses weak-compactness arguments combined withrecent results of the second author on positive semigroups in $L^{1}$-spaces.For soft potentials, in $L^{1}$-spaces, we exploits the convergence to ergodicprojection for perturbed substochastic semigroup to show that, for very generalinitial datum, solutions to the linear Boltzmann equation converges toequilibrium in large time. Moreover, for a large class of initial data, we alsoprove that the convergence rate is at least algebraic. Notice that, for softpotentials, no exponential rate of convergence is expected because of theabsence of spectral gap.

Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: a semigroup approach in $L^1$-spaces