## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 29, Number 4 (2019), 2338-2373.

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

Ben Hambly, Sean Ledger, and Andreas Søjmark

#### Abstract

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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 29, Number 4 (2019), 2338-2373.

**Dates**

Received: February 2018

Revised: August 2018

First available in Project Euclid: 23 July 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1563869045

**Digital Object Identifier**

doi:10.1214/18-AAP1455

**Mathematical Reviews number (MathSciNet)**

MR3983340

**Zentralblatt MATH identifier**

07120711

**Subjects**

Primary: 35K61: Nonlinear initial-boundary value problems for nonlinear parabolic equations 60H30: Applications of stochastic analysis (to PDE, etc.) 82C22: Interacting particle systems [See also 60K35] 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

**Keywords**

McKean–Vlasov equation nonlinear PDE initial-boundary value problem blow-ups contagion large financial systems supercooled Stefan problem

#### Citation

Hambly, Ben; Ledger, Sean; Søjmark, Andreas. A McKean–Vlasov equation with positive feedback and blow-ups. Ann. Appl. Probab. 29 (2019), no. 4, 2338--2373. doi:10.1214/18-AAP1455. https://projecteuclid.org/euclid.aoap/1563869045