Consider a (continuous time) branching Markov process in which:

- One starts with a single particle initially located at some x>0, whose position evolves according to a Brownian motion with negative drift -c which is absorbed upon reaching the origin.
- This particle waits for an independent random exponential time of parameter r>0 and then branches, dying on the spot and giving birth to a random number m\geq0 of new particles at its current position.
- These m new particles now evolve independently, each following the same stochastic behavior of their parent (evolve and then branch, and so on…).

It is well-known that if r(E(m)-1)>c^2/2 then this process is supercritical: with positive probability the process does not die out, in the sense that there are particles strictly above the origin for all times.

In this talk we will show that, whenever the process does not die out, as t\to\infty one has that the number of particles at time t inside any given set B grows like its expectation and, furthermore, that its proportion over their total number behaves like the (minimal) quasi-stationary distribution associated to the Brownian motion with drift -c and absorption at 0. In particular, this proves a result stated by Kesten in [1] for which there was no proof available until now.

Joint work with Oren Louidor.

[1] Kesten, H. (1978). Branching Brownian motion with absorption. *Stochastic Processes and their Applications*, *7*(1), 9-47.

Pontificia Universidad Católica de Chile (PUC-Chile)

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Santiago – Chile