Existence of invariant probability measures for functional McKean-Vlasov SDEs

TL;DR

This paper proves the existence of invariant probability measures for certain functional McKean-Vlasov stochastic differential equations without requiring monotonicity conditions, using Kakutani's fixed point theorem.

Contribution

It introduces weaker conditions than previous studies and applies a fixed point theorem to establish invariant measures for a broad class of functional McKean-Vlasov SDEs.

Findings

Existence of invariant probability measures established under weaker assumptions.

Application of Kakutani's fixed point theorem to functional McKean-Vlasov SDEs.

No need for monotonicity conditions in the proof.

Abstract

We show existence of an invariant probability measure for a class of functional McKean-Vlasov SDEs by applying Kakutani's fixed point theorem to a suitable class of probability measures on a space of continuous functions. Unlike some previous works, we do not assume a monotonicity condition to hold. Further, our conditions are even weaker than some results in the literature on invariant probability measures for functional SDEs without dependence on the law of the solution.