McKean-Vlasov SPDEs with coefficients exhibiting locally weak monotonicity: existence, uniqueness, ergodicity, exponential mixing and limit theorems

TL;DR

This paper studies McKean-Vlasov SPDEs with locally weak monotonicity, establishing existence, uniqueness, ergodicity, exponential mixing, and limit theorems, thereby advancing understanding of their long-term behavior and statistical properties.

Contribution

It introduces new methods to prove existence and uniqueness of solutions under weak monotonicity, and derives ergodic and limit theorems for these complex stochastic equations.

Findings

Proved existence and uniqueness of weak solutions.

Established exponential ergodicity and mixing properties.

Derived strong law of large numbers and central limit theorem.

Abstract

This paper investigates the existence and uniqueness of solutions, as well as the ergodicity and exponential mixing to invariant measures, and limit theorems for a class of McKean-Vlasov SPDEs with locally weak monotonicity. In particular, for a class of weak monotonicity conditions, including H$\ddot{\text{o}}$lder continuity, we rigorously establish the existence and uniqueness of weak solutions to McKean-Vlasov SPDEs by employing the Galerkin projection technique and the generalized coupling approach. Additionally, we explore the properties of the solutions, including time homogeneity, the Markov and the Feller property. Building upon these properties, we examine the exponential ergodicity and mixing of invariant measures under Lyapunov conditions. Finally, within the framework of coefficients meeting the criteria of locally weak monotonicity and Lyapunov conditions, alongside the uniform mixing property of invariant measures, we establish the strong law of large numbers and the central limit theorem for the solution and obtain estimates of corresponding convergence rates.