Brownian motion in two-dimensional regions, using complex analysis and probability

About this project 

Advance our understanding of Brownian motion 
 
If you are excited about starting a career in mathematics and keen to advance knowledge in complex analysis and probability, the University of South Australia – Australia’s University of Enterprise – is offering a hands-on project-based PhD within UniSA STEM.  

This project seeks to develop deeper understanding of the behaviour of Brownian motion in two-dimensional regions, using theory and techniques from the mathematical fields of complex analysis and probability.  

Specifically, when Brownian particles wander from a given starting point in a given region whose shape is known, where and when do the particles first hit the edge? The locations of the first exit are encoded in the harmonic-measure distribution function or “h-function” of the region and the starting point.  

The theory of these h-functions is now well developed for simply connected regions (no holes), and recent theoretical and computational work has enabled us to find the h-functions of many multiply connected regions.  

This research area in complex analysis is tightly linked with probability in two ways. First, an extensive theory of similar problems has been developed in probability, typically with a focus on the time of first exit of Brownian motion, rather than the location.  

Second, our h-functions provide solutions to instances of a variant of a much-studied probability problem, known as the Conformal Skorokhod Embedding Problem, which asks which probability measures can be embedded in Brownian motion in a sense similar to that described above.  

Our project will yield new fundamental knowledge at the intersection of two major mathematical research areas: complex analysis and probability.  

Results from the proposed project will expand our knowledge of h-functions and their properties, and of the Conformal Skorohkod Embedding Problem. This project is expected to result in the solution of some longstanding research questions, and may lead to applications in other fields, similar to those above.  

You will be based in the Maths and Data Science team within UniSA STEM, drawing on its strengths in theoretical and applied mathematics. You will benefit from guidance and mentorship from renowned researchers in their fields.  

This project also involves collaboration with the School of Mathematics at Monash University, Australia, one of Australia’s strongest mathematics groups, and the School of Mathematics, Statistics & Physics at Wichita State University, US, which is a world leader in research in applied and computational complex analysis.  This international collaboration experience will give you a distinct advantage when launching your career.  

What you’ll do 

In this project-based research degree, you will begin by computing the h-functions of new regions, which could be simply or multiply connected, and flat or non-planar, using theoretical and numerical approaches.  

We will teach you how to apply the complex analysis methods already developed to solve problems arising from the Conformal Skorokhod Embedding Problem in probability, and vice versa. Specific open problems include study of the asymptotics of h-functions, as well as determining sufficient conditions on a function to guarantee that it must arise as an h-function.  
A further research area is to develop the theory of h-functions for discrete structures such as networks or lattices, where Brownian motion is replaced by random walks. 

We will support you to develop and submit articles to high impact, peer-reviewed journals. There will also be ample opportunities to discuss your work at conferences and seminar talks.  
 
Where you’ll be based 

You will be based in UniSA STEM. We link engineering, mathematics, science, defence, cybersecurity, construction management, environmental science, aviation, information management, information technology, and project management, harnessing the connections across disciplines to bring big ideas to fruition.  

Our researchers deliver technical expertise and advice to industry, government and community groups to make commercially viable and sustainable impacts. The Future Industries Institute, Innovation and Collaboration Centre and Australian Research Centre for Interactive and Virtual Environments all provide consultancy and access to advanced technology.  

In an information-rich world, our people are thinking big, combatting problems, testing solutions and making data-driven decisions. Our graduates are entrepreneurs and trailblazers, who design and build the world they want to live in. Financial Support  

This project is funded for reasonable research expenses. Additionally, a living allowance scholarship of $35,200 per annum (2025 rate) is available to eligible applicants. Australian Aboriginal and/or Torres Strait Islander applicants will be eligible to receive an increased stipend rate of $50,291 per annum. A fee-offset or waiver for the standard term of the program is also included. For full terms and benefits of the scholarship please refer to our scholarship information for domestic students or international students.

Eligibility and Selection 

This project is open to application from both Domestic and International applicants.

Applicants must meet the eligibility criteria for entrance into a Master of Research or a PhD. Additionally applicants must meet the project selection criteria: 

• Hold an Honours or Masters degree in Mathematics or a related discipline, or an equivalent
• Interest in doing mathematics research

Applicants who can also demonstrate the following will be highly regarded:

• Experience with a programming language such as Matlab or Python
• Experience with complex analysis and/or probability
• Strong communication skills, oral and written

All applications that meet the eligibility and selection criteria will be considered for this project. A merit selection process will be used to determine the successful candidate.

The successful applicant is expected to study full-time and to be based at our Mawson Lakes Campus in the north of Adelaide. Note that international students on a student visa will need to study full-time.

Essential Dates 

Applicants are expected to start in a timely fashion upon receipt of an offer. Extended deferral periods are not available. Applications close on Thursday 21 November 2024.