Extremes in Branching Random Walk and Branching Brownian Motion

2013 ◽  
Vol 10 (2) ◽  
pp. 1205-1251
Author(s):  
Louigi Addario-Berry ◽  
Nathanaël Berestycki ◽  
Nina Gantert
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Branching Random Walk ◽  
Branching Brownian Motion

scholarly journals On the maximal displacement of near-critical branching random walks

2021 ◽  
Author(s):  
Eyal Neuman ◽  
Xinghua Zheng
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Gumbel Distribution ◽  
Convergence Result ◽  
Support Point ◽  
Branching Random Walk ◽  
Branching Random Walks ◽  
Tail Distribution ◽  
Branching Law ◽  
Super Brownian Motion

AbstractWe consider a branching random walk on $$\mathbb {Z}$$ Z started by n particles at the origin, where each particle disperses according to a mean-zero random walk with bounded support and reproduces with mean number of offspring $$1+\theta /n$$ 1 + θ / n . For $$t\ge 0$$ t ≥ 0 , we study $$M_{nt}$$ M nt , the rightmost position reached by the branching random walk up to generation [nt]. Under certain moment assumptions on the branching law, we prove that $$M_{nt}/\sqrt{n}$$ M nt / n converges weakly to the rightmost support point of the local time of the limiting super-Brownian motion. The convergence result establishes a sharp exponential decay of the tail distribution of $$M_{nt}$$ M nt . We also confirm that when $$\theta >0$$ θ > 0 , the support of the branching random walk grows in a linear speed that is identical to that of the limiting super-Brownian motion which was studied by Pinsky (Ann Probab 23(4):1748–1754, 1995). The rightmost position over all generations, $$M:=\sup _t M_{nt}$$ M : = sup t M nt , is also shown to converge weakly to that of the limiting super-Brownian motion, whose tail is found to decay like a Gumbel distribution when $$\theta <0$$ θ < 0 .

Self-avoiding random walk: A Brownian motion model with local time drift

1987 ◽  
Vol 74 (2) ◽  
pp. 271-287 ◽  
Author(s):  
J. R. Norris ◽  
L. C. G. Rogers ◽  
David Williams
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Local Time ◽  
Motion Model ◽  
Time Drift ◽  
Brownian Motion Model

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

scholarly journals Critical branching Brownian motion with absorption: Particle configurations

2015 ◽  
Vol 51 (4) ◽  
pp. 1215-1250 ◽  
Author(s):  
Julien Berestycki ◽  
Nathanaël Berestycki ◽  
Jason Schweinsberg
Keyword(s):  
Brownian Motion ◽  
Branching Brownian Motion

Tightness for the minimal displacement of branching random walk

2007 ◽  
Vol 2007 (07) ◽  
pp. P07010-P07010 ◽  
Author(s):  
Maury Bramson ◽  
Ofer Zeitouni
Keyword(s):  
Random Walk ◽  
Branching Random Walk

scholarly journals An Application of the Backbone Decomposition to Supercritical Super-Brownian Motion with a Barrier

2012 ◽  
Vol 49 (03) ◽  
pp. 671-684
Author(s):  
A. E. Kyprianou ◽  
A. Murillo-Salas ◽  
J. L. Pérez
Keyword(s):  
Brownian Motion ◽  
Wave Equation ◽  
Relative Ease ◽  
Branching Brownian Motion ◽  
Analytical Properties ◽  
Super Brownian Motion ◽  
Exit Measure ◽  
The Right

We analyse the behaviour of supercritical super-Brownian motion with a barrier through the pathwise backbone embedding of Berestycki, Kyprianou and Murillo-Salas (2011). In particular, by considering existing results for branching Brownian motion due to Harris and Kyprianou (2006) and Maillard (2011), we obtain, with relative ease, conclusions regarding the growth in the right-most point in the support, analytical properties of the associated one-sided Fisher-Kolmogorov-Petrovskii-Piscounov wave equation, as well as the distribution of mass on the exit measure associated with the barrier.

scholarly journals Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk

2001 ◽  
Vol 186 (2) ◽  
pp. 239-270 ◽  
Author(s):  
Amir Dembo ◽  
Yuval Peres ◽  
Jay Rosen ◽  
Ofer Zeitouni
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Planar Brownian Motion ◽  
Thick Points
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