Consider the following branching dynamics in R_+. At time 0 one particle is positioned at some starting point x > 0 and evolves according to an ABMD(c) process, that is a Brownian motion with negative drift -c which is absorbed upon reaching the origin. This particle waits for an random exponential time of parameter r and then branches, dying and giving birth to two identical particles at his current position. These new particles now evolve independently, following the same stochastic behavior of their ancestor (evolve and then branch, and so on…). We call this dynamics the ABBMD(c,r). It is well known that if r > (c^2)/2 then the ABBMD(c,r) is supercritical, i.e. with positive probability the process lives on forever. It was stated by Kesten in [1] that, in this supercritical case, there exists a random variable W such that for any Borel set B in R_+ the empirical density of the process on B divided by the mean number of particles in B converges almost surely, as t goes to infinity, to W. But a proof of this fact was not included in [1] and no other proof of this convergence has been obtained since.

In this talk we will focus on studying the convergence above in the L^2 sense. We will show that the Kesten’s theorem holds in L^2 if and only if r > c^2, so that the dynamics can be supercritical but the normalized empirical density may still not converge in L^2. If time permits, we will discuss how to extend this result to other types of absorbed Markov processes and also how to apply this result to obtain efficient simulation algorithms for quasi-stationary distributions.

**[1] Kesten, Harry**. Branching Brownian motion with absorption. *Stochastic Processes Appl.* **7 **(1978), no. 1, 9–47

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