Abstract
In this thesis, we prove fundamental results on branching processes, and apply the theory of branching processes to prove results on urn processes and chase-escape. In particular, we are interested in the long-term behaviour of such processes.Firstly, we consider urn processes which have a large initial composition (large number of initial balls). The continuous time embedding of an urn process is a multi-type branching process. By studying the long-term behaviour of multi-type branching processes, we show the long-term first and second order behaviour of urn processes with large initial composition.
Next, we consider a class of non-local branching processes. This class consists of spatial branching processes in which the offspring can be displaced from the parent particle at birth. One important example of a branching process in this class is the neutron branching process. The neutron branching process models the behaviour of neutrons undergoing fission. Thus, has many applications in nuclear physics. We prove results on the fluctuations of non-local branching processes in the supercritical regime.
Finally, we study the long-term behaviour of an interacting particle system called chaseescape. Chase-escape is a two-type (red and green type) growth model that evolves on graphs. In particular, the growth of red is restricted to sites occupied by green. Therefore, for infinite graphs, understanding when red occupies infinitely many vertices is a non-trivial problem. We use the theory of branching processes to solve this problem when chase-escape evolves on the configuration model.
| Date of Award | 24 Apr 2024 |
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| Original language | English |
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| Supervisor | Cecile Mailler (Supervisor) & Mathew Penrose (Supervisor) |