2015/10/09 by Michel Pain, Pain, Michel
Economics, Econometrics and Finance · Mathematics · Medicine · Social Sciences · #60J70 #60J80 #60K35 #FOS: Mathematics #FOS: Physical sciences #Insurance, Mortality, Demography, Risk Management #Mathematical Physics (math-ph) #Mathematical and Theoretical Epidemiology and Ecology Models #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1510.02683
openalex publication_date 2015/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a branching-selection system of particles on the real line that evolves according to the following rules: each particle moves according to a Brownian motion during an exponential lifetime and then splits into two new particles and, when a particle is at a distance L of the highest particle, it dies without splitting. This model has been introduced by Brunet, Derrida, Mueller and Munier in the physics literature and is called the L-branching Brownian motion. We show that the position of the system grows linearly at a velocity vL almost surely and we compute the asymptotic behavior of vL as L tends to infinity: vL = √(2) - π2 / 2 √(2) L2 + o(1/L2), as conjectured by Brunet, Derrida, Mueller and Munier. The proof makes use of results by Berestycki, Berestycki and Schweinsberg concerning branching Brownian motion in a strip.