Abstract:

The mean-field approximation provides a derivation of a macroscopic model, which typically corresponds to a
non-local transport PDE, from microscopic dynamics, which correspond to a system of ODEs. The mean-field
limit regime arises naturally in kinetic theory, mathematical biology, collective dynamics and many other large
particle systems models. Recently, various models in collective dynamics considered the scenario where time
evolving weights of influence are associated with each particle of the underlying dynamics. At the microscopic
level, this means that the ODE governing the particles is coupled with an ODE governing the weights. At the
macroscopic level, the inclusion of time dependent weights translates to the appearance of a non-local source
term in the limit PDE. The existence theory, Wasser- stein stability estimates and rigorous justification of the
mean-field limit are considerably more complicated in comparison to the case where the weights are constant
in time. In a series of joint works with Jos´e A. Carrillo, Sondre Galtung, Alexandra Holzinger and Pierre-
Emmanuel Jabin we studied the mean-field limit for flows with time evolving weights using a range of different
techniques, including commutator estimates and re-normalized energy, graph limits and the method of relative
entropy. I will survey some of the results that have been achieved and indicate possible future research
directions.