Inhomogeneous spatial branching processes

Thesis

We study several spatial branching processes with inhomogeneities. Branching processes are a fundamental building block in mathematical models related to growth. Often growth happens in space: the spread of an epidemic, a selective sweep in the genetics of a population, the spread of a chemical reaction or the spread of an opinion in a social network. On the level of stochastic processes, branching Brownian motion is used to understand the interplay of effects in a spreading population, and indeed the front of branching Brownian motion exhibits wave-like properties.

We study several variants of branching Brownian motion, in all of them we move away from homogenous Euclidean space. First, we study branching Brownian motion in two-dimensional space where one direction of space improves the reproduction rate of particles. Here we show that this inhomogeneity in the branching rate leads to significant corrections to the speed of the front. Secondly, we study a discrete analogue of branching Brownian in a high dimensional random environment. Here we show that the exponential number of particles produces a law of large numbers effect and that the first order of the speed of the front is linear and non-random. Thirdly, we study branching Brownian motion in two-dimensional hyperbolic space. Here we that the limit of the empirical population measure on the boundary of the space is determined only by typical particles and we use this to compute the Hausdorff dimension of its support. The collection of these results show the emergence of interesting phenomena when inhomogeneities are added to branching Brownian motion.

Authors: Geldbach, D

• DOI: 10.5287/ora-z5r927zpn
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Branching Brownian Motion (BBM)
• Subjects: Stochastic processes; Random walks (Mathematics); Branching processes; Mathematics; Fractals