Mean field limits for weakly interacting diffusions: phase transitions, multiscale analysis, metastability and inference
We consider a system of N weakly interacting particles driven by white noise. The mean-field limit of this system is described by the (nonlinear and nonlocal) McKean-Vlasov-Fokker-Planck PDE . We present a detailed analysis of continuous and discontinuous phase transitions for the McKeanVlasov PDE on the torus. We study the combined diffusive/mean-field limit of systems of weakly interacting diffusions with a periodic interaction potential. We show that, in the presence of phase transitions, the two limits do not commute. We then show the equivalence between uniform propagation of chaos, a uniform-in-N Logarithmic Sobolev inequality, the absence of phase transitions for the mean-field limit, and of Gaussian fluctuations around the McKean-Vlasov PDE . We discuss about dynamical metastability for systems that exhibit discontinuous phase transitions. Finally, we develop inference methodologies for estimating parameters in the drift of the McKean SDE using either the stochastic gradient descent algorithm or eigenfunction martingale estimators.