2020/09/04 by Leonid Mytnik, Jean‐Michel Roquejoffre, Mytnik, Leonid +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · Medicine · #Analysis of PDEs (math.AP) #FOS: Mathematics #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.2009.02042
openalex publication_date 2020/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the limiting extremal process mathcal X of the particles of\nthe binary branching Brownian motion. We show that after a shift by the\nlogarithm of the derivative martingale Z, the rescaled "density" of\nparticles, which are at distance n+x from a position close to the tip of\n mathcal X, converges in probability to a multiple of the exponential ex\nas n\→+\∞. We also show that the fluctuations of the density, after\nanother scaling and an additional random but explicit shift, converge to a\n1-stable random variable. Our approach uses analytic techniques and is\nmotivated by the connection between the properties of the branching Brownian\nmotion and the Bramson shift of the solutions to the Fisher-KPP equation with\nsome specific initial conditions initiated in citeBD1,BD2 and further\ndeveloped in the present paper. The proofs of the limit theorems for mathcal\nX rely crucially on the fine asymptotics of the behavior of the Bramson shift\nfor the Fisher-KPP equation starting with initial conditions of "size"\n0<\ε\≪ 1, up to terms of the order [(\log\n\ε-1)]-1-\γ, with some \γ>0.\n