Fluctuations in SPDEs and Interacting Particle Systems

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As the name suggests, interacting particle systems are used to model the collective behaviour of a system of particles which interact with one another. Particle systems have a broad applications, for example in economics to describe voters' opinion on a specific topic; in biology to model the spread of an epidemic or in financial markets to model the values of competing firms. We typically assume that there is noise in the particle system, meaning that the particles don't move around in a deterministic way but are subject to random motion. This randomness is typically realised through describing the evolution of individual particles using stochastic differential equations (SDEs). To keep track of the position of the particles it is often useful to consider the empirical measure. The empirical measure characterises the empirical probability that the particles are in a certain region at a given time - it is both a function of time and number of particles. It turns out that under an appropriate joint scaling of both of the above, the empirical measure converges to a limiting measure. Interestingly, the limiting measure satisfies a partial differential equation (PDE), and that is to say that the density of particles evolves in a deterministic way in the limit. To motivate the idea of stochastic partial differential equations (SPDEs) and why they are needed in this context, we need to introduce the notion of fluctuations and large deviation principles. As noted above, we expect that as we increase the number of particles in our system and allow the system to run for a longer time, the empirical measure should converge to a limiting measure. We will be interested in the following question: Given a very large time and large number of particles, what is the probability that the system of particles looks very different to the limiting behaviour we would expect? These fluctuation probabilities can be characterised by SPDEs, and to answer the above question one needs to consider how much "energy" the system of particles needs to deviate from the equilibrium state to the deviated state. Next we briefly outline the first project. Suppose we are looking at a particle system where particles diffuse according to independent Brownian motions on a torus. This means that the particles are indistinguishable and don't interact with one another. The empirical density (not scaled) of this system satisfies a SPDE called the Dean Kawasaki equation. Ferhman and Gess (https://doi.org/10.1007/s00205-019-01357-w) proved the well posedness of a more general class of SPDEs with truncated (low spatial frequency) noise and regularised nonlinearity. Subsequently in arXiv:1910.11860 they also proved a large deviation principle for the SPDE system. Our first goal is to extend the results of these papers by changing the boundary conditions of the particle system from the torus to a bounded domain. We will look at what can be said about the limiting behaviour of the process for different boundary conditions, for example Dirichlet (particles being killed at boundary) or Neumann (particles reflected at boundary) conditions. One motivation for changing the domain is that we may be able to model particle systems that relate more to real life. For example, in a finance application where particles represent value of firms, one may view a Dirichlet boundary condition at spacial point 0 to represent bankruptcy of a firm. We might also consider what happens in the case that the particles evolve on the whole real line, or in the case of more general initial data. Whilst we can't predict what the subsequent projects will look like, they will be of a similar flavour to the topics discussed above. Our project falls within the EPSRC area of 'Mathematical analysis'.