Interacting particle systems and their mean field limit: phase transitions, control, and inference

In this talk, we will present recent results on the quantitative study of stochastic interacting particle systems and of their mean field limit. We will start by exploring the relationship between the large N limit of the constant in the Logarithmic Sobolev Inequality and the presence or absence of phase transitions for the mean field PDE. We will then present methods for computing both stable and unstable stationary states of the mean field PDE and for steering the dynamics towards a chosen steady state using optimal control methodologies. Finally, we will study the problem of inferring the interaction potential from discrete space-time observations of the McKean-Vlasov PDE.