A spine approach to branching diffusions with applications to \({\mathcal L}^p\)-convergence of martingales. (English) Zbl 1193.60100

Hardy, Robert; Harris, Simon C.

Donati-Martin, Catherine (ed.) et al., Séminaire de probabilités XLII. Berlin: Springer (ISBN 978-3-642-01762-9/pbk; 978-3-642-01763-6/ebook). Lecture Notes in Mathematics 1979, 281-330 (2009).

One of the central elements of the ‘spine’ approach is to interpret the behaviour of a branching process under a certain change of measure. This approach is crystallized within the last twenty years. In the paper, a modified formalization of the ‘spine’ change of measure approach for branching diffusions in the spirit of those found by Kyprianou and Lyons is presented. The approach leads to a proof of a general form of the Many-to-One theorem that enables expectations of sums over particles in the branching process to be calculated in terms of an expectation of a single particle. The authors also exemplify spine proofs of the \({\mathcal L}^p\)-convergence \((p\geq 1)\) 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.
For the entire collection see [Zbl 1166.60002].

Reviewer: Victor V. Goryainov (Volzhsky)

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60F25	\(L^p\)-limit theorems

Keywords:

branching Brownian motion; branching diffusion; spine decomposition