A flock of starlings, a voting population, and a tray of cells mutating all obey the same mathematical rule: many individual agents interacting to produce a collective pattern. This project tackles a blind spot in the equations used to model such systems. Current multiscale methods work well when the fast-moving part of a system settles into a single stable state—for example, molecules in a droplet vibrating toward equilibrium. But many real systems, from financial markets to bird flocks, can settle into multiple stable patterns. The mathematics for handling that situation does not yet exist. The researcher will derive new equations that compress the fast, messy behaviour of these systems into a simpler description without losing the essential physics. This is fundamental science. It will not directly build a better traffic light or drug trial tomorrow. But the same kind of mathematical work that once turned chaotic weather data into reliable five-day forecasts now underpins every climate model. A rigorous framework for multiscale systems with multiple equilibria could eventually sharpen predictions in epidemiology, collective robotics, and economic modelling—anywhere that many interacting agents produce unexpected, stable patterns.
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Systems with multiple space or time scales are ubiquitous in the applied sciences and an integral part of our everyday experience: matter is made of molecules and molecules are made of atoms, seasons are divided in months and months in days, etc. In many cases of interest it is difficult or even prohibitive to produce mathematical models that fully describe all the involved scales and that are still tractable (a droplet of water contains more than a sextillion atoms!). The purpose of the ensemble of techniques which commonly go under the name of multiscale methods is to find a compromise between microscopic models, which are accurate but too complicated to tackle (analytically intractable or computationally expensive), and macroscopic models, which are less accurate but simpler. This project is primarily concerned with the study of stochastic systems which possess multiple time-scales and that are modelled by stochastic differential equations (SDEs). In their simplest form, the systems we consider are made of two components, commonly referred to as the fast and slow scale. In this case, assuming the fast scale evolves towards a (unique) equilibrium, so called equilibrium measure (EM), established methodologies (which in this scenario go under the name of stochastic averaging) allow one to obtain a reduced (coarse grained) description of the dynamics by substantially replacing the fast process with its behaviour in equilibrium: intuitively, due to the large time-scale separation, the slow part of the system will not have significantly changed its state in the time it takes the fast process to reach equilibrium. The framework of multiscale methods for SDEs and the related analytical tools offered by stochastic averaging and homogenization, are well developed for systems where the fast process (more precisely, the so-called 'frozen process') has a unique EM. Nonetheless many systems in the applied sciences do exhibit multiple equilibria. In particular, when the fast process has multiple EMs, the procedure we have informally described above can no longer be used as is, and even producing an ansatz for the reduced description of the dynamics becomes nontrivial. With this premise, this research project is aimed at the rigorous derivation and analysis of mathematical models which provide a coarse grained description of systems that are multiscale in time, when the fast scale exhibits multiple EMs. An important class of processes exhibiting multiple EMs emerge in the study of Interacting Particle systems (IPSs), in particular in the mean field limit (i.e. in the many particles limit) of such systems, which is described by so-called McKean-Vlasov diffusions. These processes play a pivotal role in consensus formation (voting system, bird flocking), economics (agents interacting through trades), cell biology (quantification of cells' mutation rate), collective navigation etc and in all these fields they have proved tremendously successful modelling tools. Summarizing, the proposed research is in the field of stochastic analysis and its applications and it is concerned with addressing problems in multiscale methods for SDEs, in particular in the context of stochastic averaging and homogenization. The systems we consider include a large class of (degenerate) SDEs, McKean-Vlasov diffusions and associated IPSs. In turn, since McKean Vlasov evolutions are obtained in the mean field limit of IPSs, we will be concerned with understanding the interplay between these two coarse graining techniques, namely multiscale approximations and mean field limits.
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