Transport equations with singularity

Abstract

Test particle dynamics in the vicinity of singularities of a scattering medium are important for a number of physical applications, one of which is the coupling of kinetic and continuum equations in numerical simulations. Out of several relevant questions arising in this context, the paper concerns boundary conditions for the related hyperbolic system of equations in a slab between two singularities. For a linear model problem, we investigate first exit times from the slab and give a complete characterization of the stochastic particle dynamics, which provides a classification of the hyperbolic systems into Cauchy problems and into those which have to be supplemented with boundary conditions. Properties of the corresponding process like ergodicity, recurrence and asymptotic behaviour are investigated.