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