Maximal displacement of branching brownian motion

Source abstract

Abstract It is shown that the position of any fixed percentile of the maximal displacement of standard branching Brownian motion in one dimension is 2 1/2 t–3 · 2 −3/2 log t + O (1) at time t , the second‐order term having been previously unknown. This determines (to within O (1)) the position of the travelling wave of the semilinear heat equation, u t =1/2 u xx + f ( u ), in the classic paper by Kolmogorov‐Petrovsky‐Piscounov, “ Étude de l'équations de la diffusion avec croissance de la quantité de la matière et son application à un problème biologique” , 1937.