Summary. We present a modified formalization of the 'spine' change of measure approach for branching diffusions in the spirit of those found in Kyprianou [40] and Lyons et al. [44,43,41]. We use our formulation to interpret certain 'GibbsBoltzmann' weightings of particles and use this to give an intuitive proof of a general 'Many-to-One' result which enables expectations of sums over particles in the branching diffusion to be calculated purely in terms of an expectation of one 'spine' particle. We also exemplify spine proofs of the L p -convergence (p ≥ 1) of some key 'additive' martingales for three distinct models of branching diffusions, including new results for a multi-type branching Brownian motion and discussion of left-most particle speeds.

Let X be the branching particle diffusion corresponding to the operator Lu + β(u 2 − u) on D ⊆ R d (where β ≥ 0 and β ≡ 0). Let λc denote the generalized principal eigenvalue for the operator L + β on D and assume that it is finite. When λc > 0 and L+β−λc satisfies certain spectral theoretical conditions, we prove that the random measure exp{−λct}Xt converges almost surely in the vague topology as t tends to infinity. This result is motivated by a cluster of articles due to Asmussen and Hering dating from the mid-seventies as well as the more recent work concerning analogous results for superdiffusions of [ET, EW]. We extend significantly the results in [AH76, AH77] and include some key examples of the branching process literature. As far as the proofs are concerned, we appeal to modern techniques concerning martingales and 'spine' decompositions or 'immortal particle pictures'.

We develop a simple and intuitive identity for calculating expectations of weighted k-fold sums over particles in branching processes, generalising the well-known many-to-one lemma.

At the heart of this article will be the study of a branching Brownian motion (BBM) with killing, where individual particles move as Brownian motions with drift −ρ, perform dyadic branching at rate β and are killed on hitting the origin.Firstly, by considering properties of the right-most particle and the extinction probability, we will provide a probabilistic proof of the classical result that the 'one-sided' FKPP travelling-wave equation of speed −ρ with solutions f : [0, ∞) → [0, 1] satisfying f (0) = 1 and f (∞) = 0 has a unique solution with a particular asymptotic when ρ < √ 2β, and no solutions otherwise. Our analysis is in the spirit of the standard BBM studies of Harris [18] and Kyprianou [27] and includes an intuitive application of a change of measure inducing a spine decomposition that, as a by product, gives the new result that the asymptotic speed of the right-most particle in the killed BBM is √ 2β − ρ on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, W λ , that appears of fundamental importance as well as facilitating some new results on the almost-sure exponential growth rate of the number of particles of speed λ ∈ (0, √ 2β − ρ). Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed λ > √ 2β − ρ. This result combined with Chauvin and Rouault's [9] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals W λ as the limiting Radon-Nikodým derivative when conditioning the right-most particle to travel at speed λ into the distant future.

We outline a completely probabilistic study of travelling-wave solutions of the FKPP reaction-diffusion equation that are monotone and connect 0 to 1. The necessary asymptotics of such travelling-waves are proved using martingale and Brownian motion techniques. Recalling the connection between the FKPP equation and branching Brownian motion through the work of McKean and Neveu, we show how the necessary asymptotics and results about branching Brownian motion combine to give the existence and uniqueness of travelling waves of all speeds greater than or equal to the critical speed.

The Neutron Transport Equation (NTE) describes the flux of neutrons through inhomogeneous fissile medium. Whilst well treated in the nuclear physics literature (cf. [8, 28]), the NTE has had a somewhat scattered treatment in mathematical literature with a variety of different approaches (cf. [7,26]). Within a probabilistic framework it has somewhat undeservingly received little attention in recent years; nonetheless, probabilistic treatments can be found see for example [18,27,23,30,4,3]. In this article our aim is threefold. First we want to introduce a slightly more general setting for the NTE, which gives a more complete picture of the different species of particle and radioactive fluxes that are involved in fission. Second we consolidate the classical c 0 -semigroup approach to solving the NTE with the method of stochastic representation which involves expectation semigroups. Third we provide the leading asymptotic of our multi-species NTE, which will turn out to be crucial for further stochastic analysis of the NTE in forthcoming work [14,12,5]. The methodology used in this paper harmonises the culture of expectation semigroup analysis from the theory of stochastic processes against c 0 -semigroup theory from functional analysis. In this respect, our presentation is thus part review of existing theory and part presentation of new research results based on generalisation of existing results.1. Introduction. The neutron transport equation (NTE) describes the flux of neutrons across a directional planar cross-section in an inhomogeneous fissile medium (typically measured is number of neutrons per cm 2 per second). As such, flux is described as a function of time, t, Euclidian location, r ∈ R 3 , direction of travel, Ω ∈ S 2 , speed c > 0 (and hence velocity υ = cΩ), and neutron energy, E ∈ R. It is not uncommon in the physics literature, as indeed we shall do here, to assume that energy is a function of velocity (E = m|υ| 2 /2), thereby reducing the number of variables by one. This allows us to describe the dependency of flux more simply in terms of time and, what we call, the configuration variables (r, υ) ∈ D × V where D ⊆ R 3 is a smooth, open, connected and bounded domain of concern such that ∂D has zero Lebesgue measure and V is the velocity space, which can now be taken to be V = {v ∈ R 3 : υ min < |v| < υ max }, where 0 < υ min < υ max < ∞.

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