Variational formulations of Fokker-Planck equations. Theory and applications to stability

I will discuss some recent developments concerning variational formulations of Fokker-Planck-Kolmogorov (FKG) equations, viewed as a dynamic on the space of probability measures. Around 25 years ago, it has been pointed out that the FKG equations associated to reversible diffusion process, can be thought as gradient flows on the space of probability measures. This approach has successfully been used both in the Metric Geometry (e.g. giving a synthetic notion of curvature) and in Mathematical Physics (e.g. proving convergence of some particle systems). I will first review the classical theory, focusing in particular on the finite-dimensional case. The formalism will then be applied to solve the problem of asymptotic trapping of Brownian diffusions. As time allows, the theory will be extended to mean-field and local interactions, and the connection with the Varadhan-Yau methods will be discussed.