McKean-Vlasov Equations with Positive Feedback through Elastic Stopping Times

TL;DR

This paper studies McKean-Vlasov equations with positive feedback via elastic stopping times, proving existence, uniqueness, and convergence results, and relating them to absorbing stopping times and reflecting Brownian motion.

Contribution

It introduces a novel analysis of McKean-Vlasov equations with elastic stopping times, establishing existence, uniqueness, and convergence properties, and connecting to related stochastic processes.

Findings

Proved existence and uniqueness of solutions.

Established convergence of particle systems.

Linked elastic stopping times to absorbing and reflecting processes.

Abstract

We prove existence and uniqueness of physical and minimal solutions to McKean-Vlasov equations with positive feedback through elastic stopping times. We do this by establishing a relationship between this problem and a problem with absorbing stopping times. We show convergence of a particle system to the McKean-Vlasov equation. Moreover, we establish convergence of the elastic McKean-Vlasov problem to the problem with absorbing stopping times and to a reflecting Brownian motion as the elastic parameter goes to infinity or zero respectively.