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This paper deals with collisionless transport equations in bounded open domains Ω ⊂ R d (d ⩾ 2) with C 1 boundary ∂Ω, orthogonally invariant velocity measure m (d v) with support V ⊂ R d and stochastic partly diffuse boundary operators H relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic C 0 -semigroups (U H (t)) t ⩾ 0 on L 1 (Ω × V , d x ⊗ m (d v)). We give a general criterion of irreducibility of (U H (t)) t ⩾ 0 and we show that, under very natural assumptions, if an invariant density exists then (U H (t)) t ⩾ 0 converges strongly (not simply in Cesarò means) to its ergodic projection. We show also that if no invariant density exists then (U H (t)) t ⩾ 0 is sweeping in the sense that, for any density φ , the total mass of U H (t) φ concentrates near suitable sets of zero measure as t → + ∞. We show also a general weak compactness theorem of interest for the existence of invariant densities. This theorem is based on several results on smoothness and transversality of the dynamical flow associated to (U H (t)) t ⩾ 0. [ABSTRACT FROM AUTHOR]

This paper provides a new tauberian approach to the study of quantitative time asymptotics of collisionless transport semigroups with general diffuse boundary operators. We obtain an (almost) optimal algebraic rate of convergence to equilibrium under very general assumptions on the initial datum and the boundary operator. The rate is prescribed by the maximal gain of integrability that the boundary operator is able to induce. The proof relies on a representation of the collisionless transport semigroups by a (kind of) Dyson-Phillips series and on a fine analysis of the trace on the imaginary axis of Laplace transform of remainders (of large order) of this series. Our construction is systematic and is based on various preliminary results of independent interest. [ABSTRACT FROM AUTHOR]

We give a spectral approach to the diffusion approximation for general neutron transport equations on the torus with a particular emphasis on the initial layer problem. By Fourier analysis, we diagonalize the transport operator and deal with diffusive limit for each Fourier mode by a Dunford functional calculus. [ABSTRACT FROM AUTHOR]