2024/12/19 by Zhou, Datong
#05C22 #05C60 #05C90 #35Q70 #35Q83 #35R02 #Analysis of PDEs (math.AP) #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2412.14512
We study a non-exchangeable multi-agent system and rigorously derive a strong form of the mean-field limit. The convergence of the connection weights and the initial data implies convergence of large-scale dynamics toward a deterministic limit given by the corresponding extended Vlasov PDE, at any later time and any realization of randomness. This is established on what we call a bi-coupling distance defined through a convex optimization problem, which is an interpolation of the optimal transport between measures and the fractional overlay between graphs. The proof relies on a quantitative stability estimate of the so-called observables, which are tensorizations of agent laws and graph homomorphism densities. This reveals a profound relationship between mean-field theory and graph limiting theory, intersecting in the study of non-exchangeable systems.