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

2018/07/13 by Sean Ledger, Ledger, Sean, Andreas Søjmark +1 · 4 citations
Mathematics · Physics and Astronomy · #34B16 #60G57 #60H15 #60H30 #60K35 #82C22 #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Probability (math.PR) #Statistical Mechanics and Entropy #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1807.05126

openalex publication_date 2018/07/13 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28

Abstract

We extend a model of positive feedback and contagion in large mean-field\nsystems, by introducing a common source of noise driven by Brownian motion.\nAlthough the driving dynamics are continuous, the positive feedback effect can\nlead to `blow-up' phenomena whereby solutions develop jump-discontinuities. Our\nmain results are twofold and concern the conditional McKean--Vlasov formulation\nof the model. First and foremost, we show that there are global solutions to\nthis McKean--Vlasov problem, which can be realised as limit points of a\nmotivating particle system with common noise. Furthermore, we derive results on\nthe occurrence of blow-ups, thereby showing how these events can be triggered\nor prevented by the pathwise realisations of the common noise.\n

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