McKean-Vlasov SDE and SPDE with Locally Monotone Coefficients
Wei Hong, Shanshan Hu, Wei Liu
TL;DR
The paper develops a comprehensive theory for McKean–Vlasov SDEs and SPDEs with locally monotone (distribution-dependent) coefficients. It proves strong and weak well-posedness under localized monotonicity via Euler-type approximations and a martingale framework for SDEs, and extends the monotone variational approach to MVSPDEs with compact embeddings. A central contribution is a Freidlin–Wentzell large deviation principle for MVSPDEs under weak, local conditions, using a weak convergence (skeleton) method adapted to distribution-dependent dynamics. The results apply to a broad array of models, including granular media, plasma-type, kinetic, porous media, and 2D Navier–Stokes-type systems, in both finite and infinite dimensions, relaxing global Lipschitz/convexity assumptions and enabling mean-field limit analyses with robust probabilistic control.
Abstract
In this paper we mainly investigate the strong and weak well-posedness of a class of McKean-Vlasov stochastic (partial) differential equations. The main existence and uniqueness results state that we only need to impose some local assumptions on the coefficients, i.e. locally monotone condition both in state variable and distribution variable, which cause some essential difficulty since the coefficients of McKean-Vlasov stochastic equations typically are nonlocal. Furthermore, the large deviation principle is also derived for the McKean-Vlasov stochastic equations under those weak assumptions. The wide applications of main results are illustrated by various concrete examples such as the granular media equations, plasma type models, kinetic equations, McKean-Vlasov type porous media equations and Navier-Stokes equations. In particular, we could remove or relax some typical assumptions previously imposed on those models.