Branching particle systems with interaction, and their scaling limits

The main theme of my research is the study of branching particle systems with interaction, and their scaling limits. We have started working in two directions.

A first project concerns the fluctuations of a model of population with competition. In this model, particles branch, diffuse, and coalesce at a rate c when they meet. As the growth is linear while the competition is quadratic, this model converges to an equilibrium state. When c is sent to 0, the equilibrium population grows like 1/c. In that setting, we would like to understand the fluctuations of the process around its equilibrium.
When a particle branches, it produces a random number of offsprings. If a second moment exists for this reproduction law, then the appropriately rescaled fluctuations process converges to a stochastic heat equation with dissipation and additive Gaussian white noise. When branching is binary, it is expected that the hydrodynamic limit is given by the FKPP equation, and that the fluctuations at the macroscopic scale satisfy the stochastic cable equation. This is of potential interest in areas as diverse as stochastic quantization (Parisi et al. 1981), the quantification of tumor growth during treatment (Swanson et al. 2003), and computational neuroscience (Tuckwell 1989, Walsh 1981, and Fox 1987). This process is also the dual to the stochastic FKPP equation (Doering et al. 2003). On the other hand, if a second moment does not exist, then we expect to see a Levy noise driving the dynamics of the fluctuations. We further aim to understand what happens when the mean is infinite, possibly after conditioning on non-explosion and studying Yaglom type limits.

A second direction of research concerns particle systems with diffusion, branching and selection. In those models, the size of the population is fixed or controlled by a parameter and branching events are offset by the removal of particles. Examples include the Brownian bees model (J. Berestycki et al. 2020, Addario-Berry et al. 2020), the Fleming-Viot particle systems (Fleming et al. 1979), and the N-BBM (N. Berestycki et al. 2013, Groisman et al. 2019, De Masi et al. 2017). We are interested in the macroscopic behavior of the density of particles when the population size goes to infinity (hydrodynamic limit, fluctuations), the long-time behavior of the system, and the role played by the parameters of the model (spatial dimension, offspring distribution, selection rule).
In 2017, De Masi et al. computed the hydrodynamic limit of the N-BBM with killing of the leftmost particle in one spatial dimension. The limit point is described by a free boundary problem (FBP), whose global existence was shown by J. Berestycki et al. in 2018. We wish to work on the conjecture of De Masi et al. that a strong selection principle holds for this N-BBM recentered by the leftmost particle, in the sense that the large N limit of the invariant measure of the recentered particle system is a continuous probability measure with density given by the profile of the minimal travelling wave of the hydrodynamic limit.
We next introduce the Bernoulli Brownian bees, where at a branching event, the particle closest to the origin is removed with probability p, and the particle furthest from the origin is removed with probability 1-p. If p<1/2, then we expect to have stationary solutions of the hydrodynamic limit, and an invariant measure for the process on N bees. Our main goal is to prove that a selection principle holds, with different macroscopic behaviour depending on the spatial dimension and on p. As highlighted in Bramson et al. 1986, these problems may help shed light on universal rules in pattern formation and selection. In addition, they are relevant to the study of FBPs where they provide representations of solutions in terms of particle systems.

This project falls within the following EPSRC research areas: mathematical analysis, and statistics and applied probability.

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