Schuh, K. J. (2024, July 16). Global contractivity for Langevin dynamics with distribution-depedent forces and uniform in time propagation of chaos [Conference Presentation]. ECM 2024 Sevilla, Sevilla, Spain.
We study the long-time behaviour of the solution to the kinetic Fokker-Planck equation with distribution dependent-forces by analysing the probabilistic counterpart, the second-order Langevin dynamics. Besides the nonlinear Langevin dynamics of McKean-Vlasov type with a distribution-dependent drift we consider its corresponding particle system.
We use a coupling approach to establish global contraction in an L1-Wasserstein distance with an explicit dimension-free rate for pairwise weak interactions. For external forces corresponding to a k-strongly convex potential, a contraction rate of order O(sqrt(k)) is obtained in certain cases. But the contraction result is not restricted to these forces. It rather includes multi-well potentials and non-gradient-type external forces as well as non-gradient-type repulsive and attractive Lipschitz continuous interaction forces. The proof is based on a novel distance function which combines two contraction results for large and small distances and uses a coupling adjusted to the distance. By applying a componentwise adaptation of the coupling we show uniform in time propagation of chaos bounds for the corresponding mean field particle system.