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 [4]. 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 setup of Hardy and Harris [5, 7, 6]. Our proof combines simple martingale ideas with applications of Varadhan’s lemma, and is successful mainly because a ‘spine decomposition’ effectively reduces otherwise extremely difficult calculations on the whole collection of branching-diffusion particles down to just a single diffusing particle (the spine) whose large-deviations behaviour is well known. A similar approach was first used for branching Brownian motion in Hardy and Harris [6]. Importantly, our techniques should be applicable in a much wider class of branching diffusion large-deviation problems. AMS subject classification: 60J80.