2011, Roberts, Matthew I.

We give proofs of two results about the position of the extremal particle in a branching Brownian motion, one concerning the median position and another the almost sure behaviour. Our methods are based on a many-to-two lemma which allows us to estimate the effect of the branching structure on the system by considering two dependent Bessel processes.

2011, Döring, Leif, Roberts, Matthew I.

In this article we contribute to the moment analysis of branching processes in catalytic media. The many-to-few lemma based on the spine technique is used to derive a system of (discrete space) partial differential equations for the number of particles in a variation of constants formulation. The long-time behavior is then deduced from renewal theorems and induction.

2011, Harris, Simon S., Roberts, Matthew I.

We develop an extension to the spine theory of branching processes, and use it to give a simple and intuitive identity for calculating additive functionals of such processes, generalizing the well-known many-to-one lemma.