Some path large-deviation results for a branching diffusion

We give an intuitive proof of a path large-deviations result for a typed branching diffusion as found in Git, J.Harris and S.C.Harris (Ann. App. Probab. 17(2):609-653, 2007). Our approach involves an application of a change of measure technique involving a distinguished infinite line of descent, or spine, and we follow the spine set-up of Hardy and Harris (Séminaire de Probabilités XLII:281-330, 2009). Our proof combines simple martingale ideas with applications of Varadhan's lemma and is successful mainly because a "spine decomposition" effectively reduces otherwise difficult calculations on the whole collection of branching diffusion particles down to just a single particle (the spine) whose large-deviations behaviour is well known. A similar approach was used for branching Brownian motion in Hardy and Harris (Stoch. Process. Appl. 116(12):1992-2013, 2006). Importantly, our techniques should be applicable in a much wider class of branching diffusion large-deviations problems.