Abstract
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.
AMS subject classification: 60J80.
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Hardy, R., Harris, S.C. (2013). Some Path Large-Deviation Results for a Branching Diffusion. In: Englander, J., Rider, B. (eds) Advances in Superprocesses and Nonlinear PDEs. Springer Proceedings in Mathematics & Statistics, vol 38. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-6240-8_5
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