Branching processes, branching random walks and other applications.
Project supervisor: Debleena. Thacker.
Project research area: Probability theory
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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 description 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
Group project
Group project
The group project will primarily focus on the basics of Galton-Watson branching processes (GWBP) and the knowledge of the mathematical tools and techniques to prove these results for GWBP. Here are some specific goals to be achieved.
Basic definitions for single and multi-type Galton-Watson branching processes
Generating functions for GWBP
Derivation of moments of GWBP using generating functions and conditional expectations.
Extinction probability of a GWBP and its connection to generating functions.
Definition and examples of martingales and the proof of the basic martingale convergence theorem. This is a very crucial step in this project.
The Kesten-Stigum theorem and its proof for a single-type super-critical GWBP using appropriate knowledge from martingales.
Mode of operation and evidence of learning for group project.
Mode of operation and evidence of learning for group project.
The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by reading and filling in the details in the proofs of theorems, lemmas etc, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.
Individual project
Individual project
Due to its rich mathematical properties, GWBP has many applications in theoretical and applied probability and some of these topics can be explored in the individual aspect of the project. Having obtained the requisite knowledge in the group project should enable one to explore more advanced topics related to branching processes. Here are some examples
Multi-type Galton Watson branching process, see Athreya and Ney for more details.
Branching random walks, see Zhan Shi for details.
Urn models and its connections to GWBP, see Mahmoud for more details.
The individual aspect of the project can potentially lead to many possible directions, and it depends on the interest of the individual candidate to decide which direction to pursue . There is also possibility of simulations depending on the direction and topic chosen by an individual.
Mode of operation and evidence of learning for individual project.
Mode of operation and evidence of learning for individual project.
The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by reading and filling in the details in the proofs, exploring examples and theoretical and practical applications of the material, and clearly communicating it in both written and oral formats.
Pre-requisite and Co-requisite
Pre-requisite and Co-requisite
The pre-requisites for this project are Probability I and II, and Analysis I. The knowledge of the material covered in Stochastic Processes III will be useful, but is not mandatory.
References
References
Interested candidates may look at these references to start with, and then the references therein. In addition to these references, you can also have a look at the references given in the section on Individual projects.
Contact details
Contact details
For further details and questions/queries etc, please feel free to write to me at debleena.thacker@durham.ac.uk.
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