Georgia Institute of Technology College of Sciences

Mean-field particle systems are well-understood by now. Typical results involve obtaining a McKean-Vlasov equation for the fluid limit that provides a good approximation for the particle system over compact time intervals. However, when the driving vector field lacks a gradient structure or in the absence of convexity or functional inequalities, the long-time behavior of such systems is far from clear. In this talk, I will discuss two such systems, one arising in the context of flocking and the other in the context of sampling (Stein Variational Gradient Descent), where there is no uniform-in-time control on the discrepancy between the limit and prelimit dynamics. We will explore methods involving Lyapunov functions and weak convergence which shed light on their long-time behavior in the absence of such uniform control.

Based on joint works with Amarjit Budhiraja, Dilshad Imon (UNC, Chapel Hill), Krishnakumar Balasubramanian (UC Davis) and Promit Ghosal (UChicago).