Well-posedness and propagation of chaos for McKean-Vlasov stochastic variational inequalities
Ning Ning, Jing Wu
TL;DR
The paper addresses MVSVIs where both drift and diffusion depend on time, state, and law, and proves strong well-posedness under weak regularity conditions, including super-linear growth and local Hölder continuity. It introduces a two-pronged approach: first solving the SVI via Yosida–Moreau approximation and generalized Itô calculus, then solving the MVSVI using a Picard iteration framework and Wasserstein-space convergence, with uniqueness secured by the Yamada–Watanabe function and Osgood's lemma. A key contribution is the first propagation of chaos result for MVSVIs, showing that the empirical measure of the $N$-particle system converges to the MVSVI limit in the mean-field regime. The methods unify subdifferential analysis, fixed-point arguments, and Wasserstein-distance techniques to establish existence, uniqueness, a priori estimates, and mean-field limits for a broad class of non-Lipschitz, distribution-dependent stochastic systems with reflection. Overall, the work extends the MVSVI theory to weaker coefficient regularity and provides rigorous mean-field convergence results with potential implications for related stochastic optimization and mean-field games models.
Abstract
In this paper, we study a broad class of McKean-Vlasov stochastic variational inequalities (MVSVIs), where both the drift coefficient $b$ and the diffusion coefficient $σ$ depend on time $t$, the state $X_t$ and its distribution $μ_t$. We establish the strong well-posedness, when $b$ is superlinear growth and locally Lipschitz continuous, and $σ$ is locally Hölder continuous, both with respect to $X_t$ and $μ_t$. Additionally, we present the first propagation of chaos result for MVSVIs.