McKean Vlasov Stochastic Partial Differential Equations

It is a part of everyday experience that physical systems tend to move towards an equilibrium (or steady state): objects fall
on the floor, the pendulum comes to a stop, milk and coffee mix together to make cappuccino. The concept of equilibrium is
somewhat intuitive to us: a given state is an equilibrium for a system if i) when starting in equilibrium, the system doesn't
change its state (if the pen is on the floor, then it stays there); ii) if we let the system evolve, then it will gradually come to a
``stop" or, in other words, it will evolve towards such an equilibrium (if we drop it, the pen will fall on the floor). The concept
of equilibrium can be more dynamic than the one described above: your cappuccino is not going to spontaneously split
back into coffee and milk, i.e. it will stay cappuccino, although each single molecule in it will still be moving. This is where
probability comes into play and it turns out that equilibria, i.e. steady states (SS), are often better described by probability
measures - so called equilibrium measures (EMs) - rather than by points.
Studying the behaviour of dynamics with one EM is the central concern of ergodic theory, which is devoted to establishing
criteria under which the dynamics admits a unique EM and to the analysis of convergence to such an EM. This research
project is aimed at furthering our understanding of the theory of (infinite dimensional) ergodic processes modelled by
(Stochastic) Partial Differential Equations (S)PDEs and of their relation to interacting particle systems (IPS); furthermore,
we will investigate the feasibility of a possible strategy to move some (further) steps towards the ambitious goal of
establishing a theory of non-ergodic processes, i.e. of processes with multiple SS. In particular we will leverage on the
ergodic theory results that we will obtain, in order to produce understanding in the theory of non-ergodic processes.

The class of processes we will consider are so-called McKean-Vlasov (S)PDEs, which underpin the study of interacting
multi-agent system (IMAS). Such processes are of interest within the framework of this proposal because they can be
modelled by (S)PDEs or IPS and they can exhibit multiple SS. More broadly the growing interest of the mathematical
community in these systems stems from their flexibility to describe a vast range of scenarios, and indeed the theory of
IMAS has had a huge impact in a variety of application fields: in economics agents are traders or companies, each of
them characterized by an initial wealth, which is updated after interactions (trades); in biology applications range from fish
schooling and bird flocking, to tumor growth; in the social sciences IMAS have modelled opinion formation and rating
systems. Notably, analogous principles have been used also in control engineering and robotics, e.g. to gain better
control on the motion of large groups of individuals (robots, particles or cells) to create or avoid the formation of
desirable/undesirable patterns. This can have an impact on the way public spaces are designed (e.g. in case of large
gatherings or events), on organization of infrastructures, and on many connected aspects of public interest. IMAS and
processes with multiple SS also arise in robotics, when the aim is to organise the coordinated motion of a group of
relatively simple robots rather than employing fewer non-interacting but more sophisticated machines.