Abstract
Before we embark on our journey to explore the NTE in a stochastic context, we need to lay out some core theory of Markov processes that will appear repeatedly in our calculations. After a brief reminder of some basics around the Markov property, we will focus our attention on what we will call expectation semigroups. These are the tools that we will use to identify neutron density and provide an alternative representation of solutions to the NTE that will be of greater interest to us. As such, in this chapter, we include a discussion concerning the asymptotic behaviour of expectation semigroups in terms of a leading eigentriple.
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Notes
- 1.
Continue à droite, limite à gauche.
- 2.
This means \(f(x)\to 0\) as \(\inf _{y \in \partial E}\lVert x - y \rVert \to 0\), providing the latter limit is possible within E.
- 3.
Recall that a Markov chain is irreducible if, for each pair \((i,j)\) in the state space, there is a non-zero probability that, starting in state i, one will eventually visit state j.
- 4.
Elsewhere in this text we use the notation \(\langle \cdot ,\cdot \rangle \) to denote inner products on other Hilbert spaces.
- 5.
Technically speaking, \(p_t(x,{\mathrm {d}} y)\) is called a kernel rather than a measure because of its dependency on \(x\in E\).
- 6.
In this setting, a connected set simply means that, if \(x,y\in D\), then there is a path joining x to y that remains entirely in D.
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Horton, E., Kyprianou, A.E. (2023). Some Background Markov Process Theory. In: Stochastic Neutron Transport . Probability and Its Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-39546-8_2
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