Branching processes and its applications

Debleena. Thacker

The classical Galton-Watson process is a stochastic process, which finds widespread application in biology, computer networks statistical physics and several other fields. Imagine a colony of organisms reproducing; and at every instance an individual gives birth to a random number of organisms. This is the simplest possible model of a branching process,

and a very simple mechanism of obtaining random trees.

The images below show branching process for different branching parameters, and are available here

This project will focus on basic Galton-Watson processes and its applications. Due to its rich

mathematical properties, several mathematical models of independent interest can be coupled

with branching processes. Hence, we can obtain several interesting results for these otherwise complex processes.

The project can potentially lead to many possible directions. It can focus on the basic properties of the single and multi-type Galton-Watson branching process. Other possible directions include using the celebrated Athreya-Karlin embedding to study urn models and related random trees. There are several other directions that this project can lead to, including applications in biology. It is entirely upto the candidate to decide in which direction they would like to drive the project.

Pre-requisite

The pre-requisites for this project are Probability I and II. Knowledge of Markov Chains II is appreciated. Any other technique, such as the martingale theory that may be needed can contribute towards the final evaluation of the project.

References

Interested candidates may look at these references to start with, and then the references therein.

For further details, please write to me at debleena.thacker@durham.ac.uk.

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