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August 2019 A McKean–Vlasov equation with positive feedback and blow-ups
Ben Hambly, Sean Ledger, Andreas Søjmark
Ann. Appl. Probab. 29(4): 2338-2373 (August 2019). DOI: 10.1214/18-AAP1455


We study a McKean–Vlasov equation arising from a mean-field model of a particle system with positive feedback. As particles hit a barrier, they cause the other particles to jump in the direction of the barrier and this feedback mechanism leads to the possibility that the system can exhibit contagious blow-ups. Using a fixed-point argument, we construct a differentiable solution up to a first explosion time. Our main contribution is a proof of uniqueness in the class of càdlàg functions, which confirms the validity of related propagation-of-chaos results in the literature. We extend the allowed initial conditions to include densities with any power law decay at the boundary, and connect the exponent of decay with the growth exponent of the solution in small time in a precise way. This takes us asymptotically close to the control on initial conditions required for a global solution theory. A novel minimality result and trapping technique are introduced to prove uniqueness.


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Ben Hambly. Sean Ledger. Andreas Søjmark. "A McKean–Vlasov equation with positive feedback and blow-ups." Ann. Appl. Probab. 29 (4) 2338 - 2373, August 2019.


Received: 1 February 2018; Revised: 1 August 2018; Published: August 2019
First available in Project Euclid: 23 July 2019

zbMATH: 07120711
MathSciNet: MR3983340
Digital Object Identifier: 10.1214/18-AAP1455

Primary: 35K61 , 60H30 , 82C22 , 91G80

Keywords: blow-ups , contagion , Initial-boundary value problem , large financial systems , McKean–Vlasov equation , nonlinear PDE , supercooled Stefan problem

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.29 • No. 4 • August 2019
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