A single particle moving at random traces a path; this project studies what happens when infinitely many such particles move, meet, split, and vanish together. The problem is that mathematicians have only one well-understood model for the large-scale behaviour of such infinite random path systems—the Brownian web. That single model has successfully described phenomena in population genetics, drainage networks, and traffic flow, but it cannot capture the full range of behaviours that real interacting particle systems produce. This project will build a family of new "webs"—limit objects that describe what emerges when particles coalesce, branch, or jump across space, without forcing them to follow a prescribed motion. This is fundamental mathematics, not applied engineering. There is no immediate practical application. But the Brownian web’s existing connections to real-world systems—from how genes spread through a population to how water runs through a drainage basin—show that understanding these abstract limits can eventually give researchers a precise language for describing complex, large-scale behaviour across physics, biology, and engineering. A deeper catalogue of such webs could reveal hidden mathematical structure in systems that currently resist analysis.
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Stochastic processes are used throughout mathematical modelling in a vast array of applications, both in their own right and as the building blocks for more sophisticated systems in which many elements interact. To give an intuition that is easily visualized, a single stochastic process is often described as a small particle that makes random movements within some space. The path traced out is, formally, the stochastic process. The project focuses on systems that are naturally viewed not just as a single path, but as an infinite random set of paths. Typically such models involve many different particles moving around in space, each creating a path, whilst interacting with one another. For example, particles that meet each other might coalesce together, or a particle might branch (i.e. split) into two, or perhaps jump across space or disappear entirely. A key topic of interest in such a model is to determine what behaviour emerges over large spatial and large time scales. Such behaviour can often be characterized by rescaling both space and time (and perhaps other parameters too) and showing that the rescaled system approximates some known ‘limit’ object. The Brownian web was the first example of a model represented as an infinite set of random paths. This representation proved key to establishing that the Brownian web could describe the long-term large-scale behaviour of a wide variety of models with diverse applications – from population genetics, aggregation and interface growth, drainage networks and traffic analysis, as well as models of interest within abstract probability theory. In one sense these connections are a great success: a wide range of models are now known to share a key common feature. In another sense, it carries a visible limitation: both the applications and the theory that has developed have so far focused almost exclusively on just one limit object i.e. on just one possible type of large-scale behaviour of random infinite sets of paths. The project will provide natural examples of webs, in analogy to the Brownian web, but without a prescribed behaviour for the particle motions. It will provide robust conditions for rescaling models to such webs, thereby allowing connections to be made between far wider classes of physical systems.
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