In this note, under weak monotonicity and weak coercivity, we address strong well-posedness of McKean–Vlasov stochastic differential equations (SDEs) driven by Lévy jump processes, where the coefficients are Lipschitz continuous (with respect to the measure variable) under the \(L^\beta \) -Wasserstein distance for \(\beta \in [1,2].\) Moreover, the issues of on the weak propagation of chaos (i.e., convergence in distribution via convergence of the empirical measure) and strong propagation of chaos (i.e., at the level paths by coupling) are explored simultaneously. To treat the strong well-posedness of McKean–Vlasov SDEs we are interested in, we investigate strong well-posedness of classical time-inhomogeneous SDEs with jumps under a local weak monotonicity and a global weak coercivity. Such a result is of independent interest, and, most importantly, can provide an available reference on strong well-posedness of Lévy-driven SDEs under the monotonicity condition, which has been is missing for a long time. Based on the theory derived, along with the interlacing technique and the Banach fixed point theorem, the strong well-posedness of McKean–Vlasov SDEs driven by Lévy jump processes can be established. Additionally, as a potential extension, strong well-posedness and conditional propagation of chaos are treated for Lévy-driven McKean–Vlasov SDEs with common noise under a weak monotonicity.
