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### Open Access Journals

PROC

The linear transport operator associated with abstract bounded
boundary conditions, $T_H$, is considered. It is shown that, in some particular cases, a convergent series similar to Dyson-Phillips series can be defined. Sufficient conditions assuring that the sum of this series is a C$_o$-semigroup generated by $T_H$ itself are given.

DCDS-B

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 $\mathcal{T}_H$ associated to a linear and positive boundary operator $H$ of unit norm. It is known that in this case an extension of $\mathcal{T}_H$ 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.

CPAA

Non-autonomous Honesty
theory in abstract state spaces with applications to linear kinetic equations

We provide a honesty theory of substochastic evolution families in real abstract state space, extending to an non-autonomous setting the result obtained for $C_0$-semigroups in our recent contribution

*[On perturbed substochastic semigroups in abstract state spaces, Z. Anal. Anwend. 30, 457--495, 2011]*. The link with the honesty theory of perturbed substochastic semigroups is established. Application to non-autonomous linear Boltzmann equation is provided.## Year of publication

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