Theoretical analysis demonstrates explicit exponential contraction rates in McKean-Vlasov Langevin dynamics, highlighting uniform propagation of chaos for interacting particle systems.
By the probabilistic coupling approach which combines a new refined basic coupling with the synchronous coupling for L\'evy processes, we obtain explicit exponential contraction rates in terms of the standard L¹-Wasserstein distance for the following Langevin dynamic (Xₜ,Yₜ)t≥0 of McKean-Vlasov type on R²ᵈ: {equation*}\{{array}{l} dX_t=Y_tdt,\\ dY_t=(b(X_t)+∫R^db̃(X_t,z)μ^X_t(dz)-γ Y_t)dt+dL_t, μ^X_t={ Law}(X_t),{array}. {equation*} where γ>0, b:Rᵈᵈ and b̃:R²ᵈᵈ are two globally Lipschitz continuous functions, and (Lₜ)t≥0 is an Rᵈ-valued pure jump L\'evy process. The proof is also based on a novel distance function, which is designed according to the distance of the marginals associated with the constructed coupling process. Furthermore, by applying the coupling technique above with some modifications, we also provide the propagation of chaos uniformly in time for the corresponding mean-field interacting particle systems with L\'evy noises in the standard L¹-Wasserstein distance as well as with explicit bounds.
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Liu et al. (2024) studied this question.
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