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.
|Title of host publication||Advances in Superprocesses and Nonlinear PDEs|
|Editors||Janos Englander, Brian Rider|
|Place of Publication||New York|
|Publication status||Published - 2013|
|Name||Springer Proceedings in Mathematics & Statistics|
Hardy, R., & Harris, S. C. (2013). Some path large-deviation results for a branching diffusion. In J. Englander, & B. Rider (Eds.), Advances in Superprocesses and Nonlinear PDEs (pp. 61-91). (Springer Proceedings in Mathematics & Statistics; Vol. 38). New York: Springer. https://doi.org/10.1007/978-1-4614-6240-8_5