Transport semigroup associated to positive boundary conditions of unit norm: a Dyson-Phillips approach

We revisit our study of general transport operator with general force field and general invariant measure by considering, in the L1 setting, the linear transport operator TH associated to a linear and positive boundary operator H of unit norm. It is known that in this case an extension of TH generates a substochastic (i.e. positive contraction) C0-semigroup (VH(t))t0. We show here that (VH(t))t0 is the smallest substochastic C0-semigroup with the above mentioned property and provides a representation of (VH(t))t0 as the sum of an expansion series similar to Dyson-Phillips series. We develop an honesty theory for such boundary perturbations that allows to consider the honesty of trajectories on subintervals J ⊆ [0,∞). New necessary and sufficient conditions for a trajectory to be honest are given in terms of the aforementioned series expansion.