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arXiv · 1311.0971

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

Luisa Arlotti·Bertrand Lods
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Abstract

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.

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Luisa Arlotti, Bertrand Lods. 2013-11-05. Transport semigroup associated to positive boundary conditions of unit norm: a Dyson-Phillips approach. https://arxiv.org/abs/1311.0971

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