Branching interval partition diffusions

This project is situated at the Probability side of the general research theme of "Statistics and Applied Probability" with some interaction with "Mathematical Analysis" and with the Combinatorics side of "Logic and Combinatorics."

Project summary: There is a long history of random branching structures in probability theory. Since early work on discrete models in the 19th century and Feller's work on diffusion limits in the mid-20th century, developments have included age dependency, lifetime characteristics, multi-type and spatial models. They have also been useful as tools to study a priori quite unrelated combinatorial and analytic structures.

Another line of research with some overlap in the study of trees has been the representation theory of exchangeable combinatorial structures. Of particular relevance to this project are exchangeable partitions and compositions of natural numbers. Following work by de Finetti in the 1930s, Kingman and Aldous in the 1970s and 1980s, and Gnedin in 1997 used interval partitions (IPs) to represent exchangeable partitions and composition structures. Of special interest and exemplifying the potential complexity of IPs is the IP formed by the excursion intervals of a Brownian bridge.

The starting point of the present project is a sequence of papers (2018-2021) by the supervisor and co-authors on self-similar IP-valued diffusions that have a branching property. From the perspective of branching structures, these diffusions can be interpreted as Crump-Mode-Jagers-type branching models where individuals have continuously evolving real-valued characteristics in a setting where the characteristics of a countably infinite number of individuals are ordered and represented as interval lengths. The connection to exchangeable combinatorial structures at fixed times has been complemented by scaling limits of composition-valued Markov chains. This recent literature uses representations of IP evolutions by spectrally positive stable Lévy processes whose jumps have been marked by excursions of self-similar real-valued diffusions.

The aim of the present project is to generalise these models by removing the assumption of self-similarity and to develop a theory of general IP-valued diffusions with a branching property. A sufficiently general excursion theory of real-valued diffusions was developed by Pitman and Yor (1982). This does give rise to associated Lévy processes under suitable conditions. Due to accumulations of small jumps, the first main challenge is to find conditions under which the marked Lévy process induces an IP-valued process. While the arguments in the self-similar case remain relevant, there will be further challenges to establish the Markov property and path-continuity as self-similarity has been exploited in the self-similar case.

One of the original motivations for the study of self-similar IP evolutions was that they were instrumental in solving Aldous's conjecture about the existence of a certain continuum-tree-valued diffusion that has the Brownian continuum random tree as its stationary distribution. A somewhat more speculative idea would be to investigate the use of the more general IP evolutions in relation to more general continuum random trees. This could be approached from either end as there are large classes of continuum random trees with explicit spinal IPs.

Probabilistic modelling permeates the Financial services, healthcare, technology and other Service industries crucial to the UK's continuing social and economic prosperity, which are major users of stochastic algorithms for data analysis, simulation, systems design and optimisation. There is a major and growing skills shortage of experts in this area, and the success of the UK in addressing this shortage in cross-disciplinary research and industry expertise in computing, analytics and finance will directly impact the international competitiveness of UK companies and the quality of services delivered by government institutions.
By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to
- sharpening the UK's research lead in this area and
- meeting the needs of industry across the technology, finance, government and healthcare sectors

MATHEMATICS, THEORETICAL PHYSICS and MATHEMATICAL BIOLOGY

The explosion of novel research areas in stochastic analysis requires the training of young researchers capable of facing the new scientific challenges and maintaining the UK's lead in this area. The partners are at the forefront of many recent developments and ideally positioned to successfully train the next generation of UK scientists for tackling these exciting challenges.
The theory of regularity structures, pioneered by Hairer (Imperial), has generated a ground-breaking approach to singular stochastic partial differential equations (SPDEs) and opened the way to solve longstanding problems in physics of random interface growth and quantum field theory, spearheaded by Hairer's group at Imperial. The theory of rough paths, initiated by TJ Lyons (Oxford), is undergoing a renewal spurred by applications in Data Science and systems control, led by the Oxford group in conjunction with Cass (Imperial). Pathwise methods and infinite dimensional methods in stochastic analysis with applications to robust modelling in finance and control have been developed by both groups.
Applications of probabilistic modelling in population genetics, mathematical ecology and precision healthcare, are active areas in which our groups have recognized expertise.

FINANCIAL SERVICES and GOVERNMENT

The large-scale computerisation of financial markets and retail finance and the advent of massive financial data sets are radically changing the landscape of financial services, requiring new profiles of experts with strong analytical and computing skills as well as familiarity with Big Data analysis and data-driven modelling, not matched by current MSc and PhD programs. Financial regulators (Bank of England, FCA, ECB) are investing in analytics and modelling to face this challenge. We will develop a novel training and research agenda adapted to these needs by leveraging the considerable expertise of our teams in quantitative modelling in finance and our extensive experience in partnerships with the financial institutions and regulators.

DATA SCIENCE:

Probabilistic algorithms, such as Stochastic gradient descent and Monte Carlo Tree Search, underlie the impressive achievements of Deep Learning methods. Stochastic control provides the theoretical framework for understanding and designing Reinforcement Learning algorithms. Deeper understanding of these algorithms can pave the way to designing improved algorithms with higher predictability and 'explainable' results, crucial for applications.
We will train experts who can blend a deeper understanding of algorithms with knowledge of the application at hand to go beyond pure data analysis and develop data-driven models and decision aid tools
There is a high demand for such expertise in technology, healthcare and finance sectors and great enthusiasm from our industry partners. Knowledge transfer will be enhanced through internships, co-funded studentships and paths to entrepreneurs