McKean-Vlasov stochastic differential equations with super-linear measure arguments: well-posedness and propagation of chaos

This paper studies McKean-Vlasov stochastic differential equations (MVSDEs) whose drift coefficients grow super-linearly in both state variables and measure arguments, and whose diffusion coefficients exhibit super-linear growth in the state variables. By constructing an Euler-like sequence, we establish the strong well-posedness of such MVSDEs under a locally monotone condition. Furthermore, the propagation of chaos is studied on both finite and infinite horizons, demonstrating convergence of the interacting particle system to the corresponding non-interacting system. To illustrate the rationality of the theoretical results, we provide examples whose drifts contain the high powers and multiple integrals of distributions, with numerical simulations presented in Section 6.

💡 Research Summary

The paper investigates a class of McKean‑Vlasov stochastic differential equations (MVSDEs) whose drift and diffusion coefficients exhibit super‑linear growth both in the state variable and in the measure (distribution) argument. Classical well‑posedness results for MVSDEs typically require global Lipschitz or at most linear growth conditions on the coefficients; this work relaxes those requirements by imposing only locally monotone (or locally Lipschitz) conditions with respect to the Wasserstein distance.

The authors first motivate the problem with two prototypical examples: (1.2) contains a drift term involving the square of the first moment of the law, and (1.3) involves a double integral over the law. In both cases the dependence on the measure is not globally Lipschitz, yet it can be controlled locally by a function L(R) that grows with the radius R of the state space. By introducing the notion of L‑derivatives for functions on the space of probability measures, they derive estimates (1.5)–(1.11) showing that such measure‑dependent terms satisfy a local Lipschitz bound of the form

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