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The Neutron Transport Equation (NTE) describes the flux of neutrons over timethrough an inhomogeneous fissile medium. In the recent articles [5, 10], aprobabilistic solution of the NTE is considered in order to demonstrate aPerron-Frobenius type growth of the solution via its projection onto anassociated leading eigenfunction. In [9, 4], further analysis is performed tounderstand the implications of this growth both in the stochastic sense, aswell as from the perspective of Monte-Carlo simulation. Such Monte-Carlosimulations are prevalent in industrial applications, in particular whereregulatory checks are needed in the process of reactor core design. In thatsetting, however, it turns out that a different notion of growth takes centrestage, which is otherwise characterised by another eigenvalue problem. In thatsetting, the eigenvalue, sometimes called k-effective (written$k_\texttt{eff}$), has the physical interpretation as being the ratio ofneutrons produced (during fission events) to the number lost (due to absorptionin the reactor or leakage at the boundary) per typical fission event. In thisarticle, we aim to supplement [5, 10, 9, 4], by developing the stochasticanalysis of the NTE further to the setting where a rigorous probabilisticinterpretation of keff is given, both in terms of a Perron-Frobenius typeanalysis as well as via classical operator analysis. To our knowledge, despitethe fact that an extensive engineering literature and industrial Monte-Carlosoftware is concentrated around the estimation of keff and its associatedeigenfunction, we believe that our work is the first rigorous treatment in theprobabilistic sense (which underpins some of the aforesaid Monte-Carlosimulations).

Stochastic Methods for Neutron Transport Equation III: Generational many-to-one and $k_\texttt{eff}$