Abstract:
We consider a (one-dimensional) branching Brownian motion process with a general offspring distribution having at least two finite moments, in which all particles have a drift towards the origin and are immediately killed if they reach it. It is well-known that if and only if the branching rate is sufficiently large, the population survives forever with a positive probability. We show that throughout this super-critical regime, the number of particles inside any given set normalized by the mean population size converges to an explicit limit, almost surely and in L1. As a consequence, we get that, almost surely on the event of survival of the branching process, the empirical distribution of particles converges weakly to the (minimal) quasi-stationary distribution associated with the Markovian motion driving the particles. This proves a result of Kesten from 1978, for which no proof was available until now. Joint work with Oren Louidor.
Biography:
Santiago Saglietti is currently a Postdoctoral Fellow at the Technion, Israel Institute of Technology. He got his Ph.D. in 2014 at the University of Buenos Aires (UBA), under the supervision of Pablo Groisman. His research focuses mainly on mathematical statistical mechanics and stochastic processes. In these fields, he has worked on diverse topics such as perfect simulation of Gibbs measures, random walks in random environments, branching Markov processes, and metastability phenomena. He has also worked in ergodic theory and random dynamical systems.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai