Substochastic semigroups for transport equations with conservative boundary conditions (Q860242)

The authors study the streaming operator associated with conservative boundary conditions. It is well-known that the operator (with classical domain or with strictly contractive boundary conditions) admits an extension \(A\) which generates a \(C_0\)-positive semigroup of contractions in \(L^1\). However, the conservative boundary condition does not have such properties. In the present paper, the authors present a conservative boundary condition under which the streaming operator generates a substochastic semigroup. Moreover, they prove that the substochastic semigroup is stochastic if the conservation boundary operator is positive. The main technique used in this paper is the additive perturbation theory of substochastic semigroups. In particular, the precise domain and necessary and sufficient conditions for a semigroup to be stochastic are given.