Stochastic methods for the neutron transport equation I: Linear semigroup asymptotics

Emma Horton 1 Andreas E. Kyprianou 1 Denis Villemonais 2, 3
2 TOSCA - TO Simulate and CAlibrate stochastic models
CRISAM - Inria Sophia Antipolis - Méditerranée , IECL - Institut Élie Cartan de Lorraine : UMR7502
Abstract : The Neutron Transport Equation (NTE) describes the flux of neutrons through inhomogeneous fissile medium. Whilst well known in the post-Manhattan Project nuclear physics literature, cf. [11, 25, 27, 26, 23, 1, 36], the NTE has had a some- what scattered treatment in mathematical literature with a va- riety of different approaches. The rigorous treatment of Robert Dautray and collaborators, cf. [10, 9], firmly places the analysis of the NTE in the setting of classical c0-semigroup analysis of linear operators. Added to this, Mokhtar-Kharroubi [33] sum- marises the use of perturbation theory therein in order to un- derstand leading spectral properties of the NTE. Within a prob- abilistic framework, the NTE has received somewhat sporadic attention. Although the connection to probabilistic representa- tion has been understood since the early analysis of the NTE, see e.g. [34, 29, 37, 35, 9, 19], our aim is to reintroduce the NTE in the more modern context of Markov branching processes which describes the physical process of nuclear scattering and fission. Our presentation is thus, for the lesser part, a review of foun- dational material and, for the greater part, presentation of new research results. This paper is the first of a two-part instalment. In this first instalment we consider the Perron-Frobenius type decomposition of the linear semigroup associated to the NTE and the role it plays in the classical linear analysis of the un- derlying stochastic process. In the next instalment, we look at stochastic growth properties of the underlying physical process.
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Submitted on : Monday, October 15, 2018 - 1:04:11 PM
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Emma Horton, Andreas E. Kyprianou, Denis Villemonais. Stochastic methods for the neutron transport equation I: Linear semigroup asymptotics. 2018. ⟨hal-01895619⟩

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