Chilean Probability Seminar: “Stochastic particle system approximation of the Keller-Segel PDE and suppression of blow-up”

Abstract: In this talk, we will first present a system of moderately interacting stochastic particle system (singular, attractive interaction potential) whose empirical measure converges to the solution of the Keller-Segel PDE, which is a PDE known for exhibiting finite-time blow-up. In a second part, we then introduce a birth-and-death mechanism for each particle, involving a death rate proportional to the local concentration of particles. We will show that the empirical measure of this new system converges to a Keller-Segel equation with a logistic term (of KPP- type), for which explosion in finite time can be avoided.

Time permitting, we may finally consider the collisions of the particle system with birth-and- death mechanism and non-mollified kernel, and show that as the number of particles goes to infinity, the probability of critical collisions between particles vanishes, leading to convergence
of the empirical measure along subsequences.

Based on joint works with T.Cavallazzi (CentraleSupélec), C.Olivera (Unicamp), S.Popat (Polytechnique), Y.Tardy (Polytechnique) and M.Tomasevic (Polytechnique).
Join the seminar via Zoom: https://reuna.zoom.us/j/84521834914?pwd=OTZ6Y0NWM3pYTGtTbEt3c0luTG96UT09