We extend a model of  feedback and contagion in large mean-field systems by introducing a common source of noise driven by Brownian motion. Although the dynamics in the model are continuous, the feedback effect can lead to jump discontinuities in the solutions --- that is `blow-ups'. We prove existence of solutions to the corresponding conditional McKean--Vlasov equation, by realising them as suitable limit points of the finite-dimensional  system, and we show that the pathwise realisation of the common noise can both trigger and prevent blow-ups.
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 dynamics in the model are continuous, the feedback effect can lead to jump discontinuities in the solutions , which we refer to as `blow-ups'. We prove existence of solutions to the corresponding conditional McKean--Vlasov equation, by realising them as suitable limit points of the finite-dimensional particle system, and we show that the pathwise realisation of the common noise can both trigger and prevent blow-ups.
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