McKean-Vlasov SDE and SPDE with Locally Monotone Coefficients

TL;DR

This paper establishes strong and weak well-posedness for McKean-Vlasov stochastic (partial) differential equations under local monotonicity conditions, and derives a large deviation principle, broadening applicability to various models.

Contribution

It introduces new existence and uniqueness results for McKean-Vlasov equations with locally monotone coefficients, relaxing previous global assumptions.

Findings

Proved well-posedness under local monotonicity conditions.

Derived large deviation principles for the equations.

Applied results to models like Navier-Stokes and plasma equations.

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.