At the Mercy of the Common Noise: Blow-ups in a Conditional McKean--Vlasov Problem

TL;DR

This paper investigates how common noise influences blow-up phenomena in a conditional McKean--Vlasov model, demonstrating the existence of global solutions and how noise can trigger or prevent solution discontinuities.

Contribution

It extends a mean-field model by incorporating common noise, proving the existence of global solutions and analyzing conditions for blow-ups.

Findings

Global solutions exist as limits of particle systems with common noise.

Blow-ups can be triggered or prevented by the realization of the common noise.

The model captures positive feedback leading to discontinuities despite continuous dynamics.

Abstract

We extend a model of positive feedback and contagion in large mean-field systems, by introducing a common source of noise driven by Brownian motion. Although the driving dynamics are continuous, the positive feedback effect can lead to `blow-up' phenomena whereby solutions develop jump-discontinuities. Our main results are twofold and concern the conditional McKean--Vlasov formulation of the model. First and foremost, we show that there are global solutions to this McKean--Vlasov problem, which can be realised as limit points of a motivating particle system with common noise. Furthermore, we derive results on the occurrence of blow-ups, thereby showing how these events can be triggered or prevented by the pathwise realisations of the common noise.