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 .
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 .
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,
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dynamics in the model
driving dynamics
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positive
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jump discontinuities in the solutions, which we refer to as `blow-ups' . We prove existence of solutions to the corresponding conditional
`blow-up' phenomena whereby solutions develop jump-discontinuities. Our main results are twofold and concern the conditional
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equation, by realising them as suitable
formulation of the model. First and foremost, we show that there are global solutions to this McKean--Vlasov problem, which can be realised as
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the finite-dimensional particle system , and we show that the pathwise realisation
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
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can both trigger and prevent blow-ups