Sesión Ecuaciones Diferenciales y Probabilidad

Diciembre 13, 12:00 ~ 12:20

On the Kesten-Stigum theorem in $L^2$ beyond $\lambda$-positivity


We study supercritical branching processes in which all particles evolve according to some general Markovian motion (which may possess absorbing states) and branch independently at a fixed constant rate. Under fairly natural assumptions on the asymptotic distribution of the underlying motion, we first show using only probablistic tools that there is convergence in $L^2$ of the empirical measure (normalized by the mean number of particles) if and only if an associated additive martingale is bounded in $L^2$. This is a significant improvement over previous results, which were mainly restricted to $\lambda$-positive motions. We then investigate under which conditions this limit is strictly positive on the event of non-extinction and show that this occurs whenever, on the latter event, particles do not accumulate on the boundary of the state space. \mbox{In particular,} this property also yields the convergence of the real empirical measure (normalized by the number of particles). Moreover, building on previous results we prove that if the motion is $\lambda$-positive then these limits hold also almost surely whenever the Doob's $h$-transform of this motion admits a suitable Lyapunov functional. Finally, we illustrate our results for a variety of different motions: ergodic motions without absorption, $\lambda$-positive systems either transient or with absorption, and also certain non $\lambda$-positive systems such as the Brownian motion with negative drift killed at $0$.

Autores: JONCKHEERE, Matthieu / Saglietti, Santiago .