  The Neutron Transport Equation (NTE) describes the flux of neutrons through
an inhomogeneous fissile medium. In this paper, we reconnect the NTE to the
physical model of the spatial Markov branching process which describes the
process of nuclear fission, transport, scattering, and absorption. By
reformulating the NTE in its mild form and identifying its solution as an
expectation semigroup, we use modern techniques to develop a Perron-Frobenius
(PF) type decomposition, showing that growth is dominated by a leading
eigenfunction and its associated left and right eigenfunctions. In the spirit
of results for spatial branching and fragmentation processes, we use our PF
decomposition to show the existence of an intrinsic martingale and associated
spine decomposition. Moreover, we show how criticality in the PF decomposition
dictates the convergence of the intrinsic martingale. The mathematical
difficulties in this context come about through unusual piecewise linear motion
of particles coupled with an infinite type-space which is taken as neutron
velocity. The fundamental nature of our PF decomposition also plays out in
accompanying work [20, 9].
