Stochastic methods for the neutron transport equation II: Almost sure growth

The neutron transport equation (NTE) describes the flux of neutrons across a planar cross-section in an inhomogeneous fissile medium when the process of nuclear fission is active. Classical work on the NTE emerges from the applied mathematics literature in the 1950s through the work of R. Dautray and collaborators (Méthodes Probabilistes Pour les équations de la Physique (1989) Eyrolles; Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 6: Evolution Problems. II (1993) Springer; Mathematical Topics in Neutron Transport Theory: New Aspects (1997) World Scientific). The NTE also has a probabilistic representation through the semigroup of the underlying physical process when envisaged as a stochastic process (cf. Méthodes Probabilistes pour les équations de la Physique (1989) Eyrolles; Introduction to Monte-Carlo Methods for Transport and Diffusion Equations (2003) Oxford Univ. Press; IMA J. Numer. Anal. 26 (2006) 657.685; Publ. Res. Inst. Math. Sci. 7 (1971/72) 153.179). More recently, Cox et al. (J. Stat. Phys. 176 (2019) 425.455) and Cox et al. (2019) have continued the probabilistic analysis of the NTE, introducing more recent ideas from the theory of spatial branching processes and quasi-stationary distributions. In this paper, we continue in the same vein and look at a fundamental description of stochastic growth in the supercritical regime. Our main result provides a significant improvement on the last known contribution to growth properties of the physical process in (Publ. Res. Inst. Math. Sci. 7 (1971/72) 153.179), bringing neutron transport theory in line with modern branching process theory such as (Ann. Probab. 44 (2016) 235.275; Ann. Probab. 43 (2015) 2545.2610). An important aspect of the proofs focuses on the use of a skeletal path decomposition, which we derive for general branching particle systems in the new context of nonlocal branching generators.