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

Luisa Arlotti (1) , Bertrand Lods (2)

1 Università degli Studi di Udine - University of Udine [Italie]
2 Department of Statistics and Economics

We revisit our study of general transport operator with general force field and general invariant measure by considering, in the $L^1$ setting, the linear transport operator $\T_H$ associated to a linear and positive boundary operator $H$ of unit norm. It is known that in this case an extension of $\T_H$ generates a substochastic (i.e. positive contraction) $C_0$-semigroup $(V_H(t))_{t\geq 0}$. We show here that $(V_H(t))_{t\geq 0}$ is the smallest substochastic $C_0$-semigroup with the above mentioned property and provides a representation of $(V_H(t))_{t \geq 0}$ 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 $J \subseteq [0,\infty)$. New necessary and sufficient conditions for a trajectory to be honest are given in terms of the aforementioned series expansion.

Mots clés: honesty theory; Transport equation; boundary conditions; substochastic semigroups; honesty theory.

Domaines: Equations aux dérivées partielles [math.AP]

Luisa Arlotti, Bertrand Lods. Transport semigroup associated to positive boundary conditions of unit norm: a Dyson-Phillips approach. 2013. ⟨hal-00879798⟩