McKean-Vlasov SDEs under Measure Dependent Lyapunov Conditions

TL;DR

This paper establishes the existence of weak solutions for McKean-Vlasov SDEs with measure-dependent Lyapunov conditions, introducing an integrated Lyapunov criterion and utilizing Lions' measure derivative.

Contribution

It introduces a novel integrated Lyapunov condition for McKean-Vlasov SDEs and proves existence results under less restrictive assumptions than traditional methods.

Findings

Existence of weak solutions under measure-dependent Lyapunov conditions.

Introduction of integrated Lyapunov condition suffices for existence.

Results on uniqueness with weaker assumptions than Lipschitz continuity.

Abstract

We prove the existence of weak solutions to McKean-Vlasov SDEs defined on a domain $D \subseteq \mathbb{R}^d$ with continuous and unbounded coefficients that satisfy Lyapunov type conditions, where the Lyapunov function may depend on measure. We propose a new type of {\em integrated} Lyapunov condition, where the inequality is only required to hold when integrated against the measure on which the Lyapunov function depends , and we show that this is sufficient for the existence of weak solutions to McKean-Vlasov SDEs defined on $D$. The main tool used in the proofs is the concept of a measure derivative due to Lions. We prove results on uniqueness under weaker assumptions than that of global Lipschitz continuity of the coefficients.