Neurons in the brain, particles in a fluid, and traders in an energy market all follow the same kind of mathematics—equations that describe how large groups of interacting components behave over time, but that are notoriously difficult to solve. This project tackles a fundamental gap in mathematics: the equations that model these systems are nonlinear, meaning small changes can produce wildly different outcomes. They also suffer from low regularity (jagged, unpredictable behaviour), noise that doesn’t spread evenly, and high dimensionality (hundreds or thousands of variables interacting at once). Existing mathematical tools cannot handle all these features together. The team will develop new theory—covering stochastic differential equations, rough paths, and gradient flows—to understand when solutions exist, how regular they are, and how to compute them reliably. This is fundamental mathematics. There is no immediate practical application. But the equations the team studies already underpin models of spiking neural networks (used in brain-computer interfaces), particle systems (used in materials science), and financial markets managing renewable energy risk. A deeper mathematical understanding of these systems could, in the long term, make those models more trustworthy and efficient—without which, simulations in neuroscience, hydrodynamics, and energy economics remain uncertain.
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This SE aims at addressing a number of challenging mathematical problems related to stochastic interacting systems, with particular emphasis on the regularity properties of the solutions, their limiting behaviour and numerical computation. The equations we analyse arise from the modelisation of real-world phenomena in several fields of application, including spiking neural systems, hydrodynamics and financial/energy markets, and share nonlinearity as a common underlying trait. Critically, nonlinearity intertwines with other relevant features that include: low regularity of the coefficients, noise degeneracy, jump-diffusion dynamics, and high- dimensionality. The study of stochastic interacting systems is highly multidisciplinary from a two-fold perspective. On one hand, they have become a widespread modelling tool in a variety of applications. For example, they are used to model human neuron interfaces, particle systems, but also interacting agents in economics and finance, in relation to managing risk and decentralised production of renewable energy. On the other hand, the set of mathematical and computational tools needed to reach a holistic understanding of stochastic systems is very vast: ranging from stochastic (partial) differential equations, random measures, rough paths, gradient flows in metric measure spaces, numerical probability and computer simulation. We provide a team of experts that analyse stochastic systems integrating several approaches and techniques. The complementary expertise across the network, together with the consolidated experience and excellence of the key participants in their research areas, places our network in the privileged position to make relevant contributions across interconnected research fields, and to contribute to the training of the early career researchers involved in the project in an exciting field of pure and applied mathematics, with the possibility of boosting their careers in both academic and non-academic sectors
Goncalo Dos Reis (Principal Investigator)Jiawei Li (Co-Investigator)Leonardo Tolomeo (Co-Investigator)Luca Taschini (Co-Investigator)Ofer Busani (Co-Investigator)
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