A McKean--Vlasov equation with positive feedback and blow-ups

Understand

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 c\`{a}dl\`{a}g functions, which confirms the validity of related propagation-of-chaos results in the literature.