Stochastic PDEs, interacting particle systems and large deviations

This project falls within the EPSRC research areas Mathematical Analysis, Statistics and Applied Probability, Mathematical Physics.

Partial differential equations are one of the key tools to describe our reality. However, any model of the real world should take into account uncertainty or random fluctuations, so that most often the correct description of a physical phenomenon is in fact given by a stochastic PDE. Indeed, stochastic PDEs have crucial relevance for mathematical physics (quantum field theory, statistical mechanics, fluid dynamics), mathematical biology, mathematical finance, statistics and data science.

Diffusion equations are a particular type of deterministic or stochastic PDEs used to describe the macroscopic behavior of many micro-particles interacting with each other as time passes. The study of these interacting particle systems is on its own of great interest for all of the above-mentioned subjects and many others such as computational neuroscience or population dynamics. These systems consider one or more equations for each single particle, and one should regard them as the discrete analogous of SPDEs. Indeed, a general feature of particle systems is that, in the limit as the number of particles increases up to infinity, the common behavior of the particles can be described by a single deterministic equation, which is often a nonlinear diffusion PDE. The associated stochastic PDE arises then to describe the fluctuations of the particle process about this deterministic limit.

The convergence of the particle system towards the limiting model prescribed by a single equation is of interest both for a better understanding of the PDE, its stochastic version and the system itself, and for computational reasons, allowing for a quick, compact description of the phenomenon under consideration. In this setting, understanding and quantifying the occurrence of rare events that differ from the limiting behavior suggested by this convergence result is of course relevant for theoretical reasons, and even more crucial for the concrete application of these models in the real world. This is the object of study of large deviation theory.

From a technical point of view, the main difficulty in the study of stochastic PDEs is their lack of regularity. Generally speaking, solutions to linear SPDEs are not function valued and need to be made sense of in the space of distributions. In the nonlinear setting, most often it is not even clear how to make sense of these equations and what to call a 'solution'.

In this framework, the aim of the project is the understanding of these stochastic diffusion equations, both from the point of view of abstract PDE theory, answering questions such as well-posedness, regularity and continuity with respect to the initial data; and in terms of their connection with particle systems, addressing problems about the modelling of natural phenomena such as the convergence towards a limiting behavior or the occurrence of rare events not predicted by the limiting model.

The mathematics used to tackle these problems lies at the interface of probability and analysis. At the continuum level, the combination of deep PDE theory and recent tools from probability is needed to handle the roughness of these SPDEs; the need for sharp probabilistic arguments is even more evident at the discrete level. Finally, both aspects crucially rely on a good understanding of the physical phenomena one is trying to describe.