Uniqueness for contagious McKean--Vlasov systems in the weak feedback regime

TL;DR

This paper proves global and short-time uniqueness for a broad class of McKean-Vlasov systems under weak feedback conditions, using a simple probabilistic comparison approach that does not require regularity of solutions.

Contribution

It introduces a straightforward probabilistic comparison method to establish uniqueness in McKean-Vlasov problems, applicable in both weak feedback and post-blow-up regimes.

Findings

Global uniqueness in weak feedback regime for general random drivers.

Extension of techniques to short-time uniqueness after blow-ups.

Robust comparison argument does not require regularity of solutions.

Abstract

We present a simple uniqueness argument for a collection of McKean-Vlasov problems that have seen recent interest. Our first result shows that, in the weak feedback regime, there is global uniqueness for a very general class of random drivers. By weak feedback we mean the case where the contagion parameters are small enough to prevent blow-ups in solutions. Next, we specialise to a Brownian driver and show how the same techniques can be extended to give short-time uniqueness after blow-ups, regardless of the feedback strength. The heart of our approach is a surprisingly simple probabilistic comparison argument that is robust in the sense that it does not ask for any regularity of the solutions.