Ergodicity for SPDEs driven by divergence-free transport noise

TL;DR

This paper investigates how divergence-free transport noise influences the long-term behavior of McKean-Vlasov equations, demonstrating that sufficiently strong and mixing noise can ensure unique invariant measures despite multiple steady states in the deterministic case.

Contribution

It shows that divergence-free transport noise can induce ergodicity and uniqueness of invariant measures in McKean-Vlasov equations in higher dimensions.

Findings

Strong mixing noise enforces uniqueness of invariant measures

Noise can override multiple steady states in deterministic systems

Results apply to dimensions d ≥ 2

Abstract

We study the ergodic behaviour of the McKean-Vlasov equations driven by common, divergence-free transport noise. In particular, we show that in dimension $d\geq 2$, if the noise is mixing and sufficiently strong it can enforce the uniqueness of invariant probability measures, even if the deterministic part of equation has multiple steady states.