Two studies on branching processes

Thesis

We consider two branching processes: a Galton-Watson process and a branching Brownian motion.

In Chapter 2 we study behaviour of a minimax recursion defined on Galton-Watson trees. This recursion corresponds to a two-player combinatorial game. The value associated to the root of a tree truncated at some level $2n$ corresponds to the probability that the first player wins a game played on such a truncated tree. We study convergence of the value at the root as $n \to \infty$, in particular we give distributional limits under suitable rescaling in various cases. We address also a question of endogeny, which can be seen as a property that holds if the game can be played close to optimally for large $n$ by inspecting only the initial section of the tree.

In Chapter 3 we consider a branching Brownian motion in 𝓡^d. We prove that there exists a random subset Θ of 𝕊^d-1 of full measure such that the limit of the derivative martingale exists simultaneously for all directions in Θ almost surely. This allows us to define a random measure on 𝕊^d-1 whose density is given by the derivative martingale. We argue that this measure should correspond to the distribution of the direction of the furthest particle and we prove this claim in dimension one.

In Chapter 4 we present conclusions and further research directions, in particular we discuss a conjecture on the form of the extremal point process of the multidimensional BBM.

Authors: Stasiński, R

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: branching processes; branching Brownian motion; derivative martingale; Galton-Watson trees
• Subjects: Probability