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Does propagation of chaos hold quantitatively for memory-dependent particle systems on the same timescales as their memoryless counterparts, and if so with what rate?

Related: McKean-Vlasov propagation of chaos theorem, Sznitman coupling argument for mean-field limits, Dobrushin stability and contractivity for interacting diffusions

Consider a large system of interacting particles where each particle's future motion depends not only on its current position and interactions with others, but also on its entire history of past positions. The central open problem is whether one can establish sharp, quantitative rates at which the empirical measure of such a system converges to the solution of the corresponding McKean-Vlasov equation with memory, uniformly over long or even infinite time horizons, and whether the rates degrade gracefully or catastrophically as time grows. This goes beyond showing that the system eventually decorrelates; it asks for precise polynomial or exponential bounds in both the number of particles and the time variable simultaneously.

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