82 research outputs found
Maximal displacement in the -dimensional branching Brownian motion
We consider a branching Brownian motion evolving in . We prove
that the asymptotic behaviour of the maximal displacement is given by a first
ballistic order, plus a logarithmic correction that increases with the
dimension . The proof is based on simple geometrical evidence. It leads to
the interesting following side result: with high probability, for any , individuals on the frontier of the process are close parents if and only if
they are geographically close.Comment: 12 pages, 2 figure
Branching random walk with selection at critical rate
We consider a branching-selection particle system on the real line. In this
model the total size of the population at time is limited by . At each step , every individual dies while reproducing
independently, making children around their current position according to
i.i.d. point processes. Only the rightmost
children survive to form the generation. This process can
be seen as a generalisation of the branching random walk with selection of the
rightmost individuals, introduced by Brunet and Derrida. We obtain the
asymptotic behaviour of position of the extremal particles alive at time by
coupling this process with a branching random walk with a killing boundary.Comment: Updated versio
Maximal displacement of a branching random walk in time-inhomogeneous environment
Consider a branching random walk evolving in a macroscopic time-inhomogeneous
environment, that scales with the length of the process under study. We
compute the first two terms of the asymptotic of the maximal displacement at
time . The coefficient of the first (ballistic) order is obtained as the
solution of an optimization problem, while the second term, of order ,
comes from time-inhomogeneous random walk estimates, that may be of independent
interest. This result partially answers a conjecture of Fang and Zeitouni. Same
techniques are used to obtain the asymptotic of other quantities, such as the
consistent maximal displacement.Comment: 51 pages, to appear in SP
Infinitely ramified point measures and branching L\'evy processes
We call a random point measure infinitely ramified if for every , it has the same distribution as the -th generation of some branching
random walk. On the other hand, branching L\'evy processes model the evolution
of a population in continuous time, such that individuals move in space
independently, according to some L\'evy process, and further beget progenies
according to some Poissonian dynamics, possibly on an everywhere dense set of
times. Our main result connects these two classes of processes much in the same
way as in the case of infinitely divisible distributions and L\'evy processes:
the value at time of a branching L\'evy process is an infinitely ramified
point measure, and conversely, any infinitely ramified point measure can be
obtained as the value at time of some branching L\'evy process.Comment: To appear in Annals of Probabilit
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