Stochastic Methods for the Neutron Transport Equation I: Linear Semigroup asymptotics

TL;DR

This paper develops a mathematical framework connecting the neutron transport equation to spatial branching processes, using Perron-Frobenius decomposition to analyze growth, criticality, and martingale convergence.

Contribution

It introduces a Perron-Frobenius decomposition for the neutron transport equation, revealing eigenfunction dominance and martingale properties in a complex stochastic setting.

Findings

Existence of a leading eigenfunction governing growth

Development of a spine decomposition for the process

Criticality determines martingale convergence

Abstract

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].