Maximal displacement of branching brownian motion

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.