Particles interacting through their hitting times: neuron firing, supercooling and systemic risk
Particles interacting through their hitting times: neuron firing, supercooling and systemic risk
I will discuss a class of particle systems that serve asÌýmodels for supercooling in physics, neuron firing in neuroscience andÌýsystemic risk in finance. The interaction between the particles fallsÌýinto the mean field framework pioneered by McKean and Vlasov in the lateÌý1960s, but many new phenomena arise due to the singularity of the interaction. The most striking of them is the loss of regularity of theÌýparticle density caused by the self-excitation of the system. InÌýparticular, while initially the evolution of the system can be capturedÌýby a suitable Stefan problem, the following irregular behaviorÌýnecessitates a more robust probabilistic approach. Extensions to the setting where the interaction takes place on networks will be also discussed. Based on joint worksÌýwith Sergey Nadtochiy