[ABE-L] =?ISO-8859-1?Q?_Semin=E1rio_de_Prob?=abilidade - IM/UFRJ - 13/06

[Maria Eulalia Vares]

Caros colegas, 

Segue abaixo a informação sobre o próximo seminário de probabilidade no
IM-UFRJ, na próxima segunda-feira. Todos são bem-vindos.

Saudações,

Eulalia

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SEMINÁRIO DE PROBABILIDADE

Data: 13 de junho de 2016 (segunda-feira)

Hora: 15:30 h

Local: Sala C 119 - Instituto de Matemática - Bloco C - CT - Campus do Fundão

Palestrante: Santiago Juan Saglietti (PUC, Chile)

Título: The Kesten-Stigum theorem in L^2 for BABMD

Resumo:

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 BABMD(c,r).
It is well known that if r > (c^2)/2 then the BABMD(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

--
Maria Eulalia Vares
Instituto de Matemática - UFRJ
http://www.im.ufrj.br/~eulalia
#fica MCTI