Asymptotic Formulas for Extreme Statistics of Escape Times in 1, 2 and 3-Dimensions

  • K. Basnayake
  • Z. Schuss
  • D. HolcmanEmail author


The first of N identical independently distributed (i.i.d.) Brownian trajectories that arrives to a small target sets the time scale of activation, which in general is much faster than the arrival to the target of a single trajectory only. Analytical asymptotic expressions for the minimal time is notoriously difficult to compute in general geometries. We derive here asymptotic laws for the probability density function of the first and second arrival times of a large number N of i.i.d. Brownian trajectories to a small target in 1, 2 and 3-dimensions and study their range of validity by stochastic simulations. The results are applied to activation of biochemical pathways in cellular biology.


Short time asymptotics Diffusion Narrow escape Extreme statistics Transient Calcium dynamics Helmoltz Dendritic spine 

Mathematics Subject Classification

35K08 35J08 35J05 60G70 92C05 92C37 



We thank C. Guerrier for her help in designing the two-dimensional simulations. This research was supported by the Foundation pour la Recherche Médicale—Équipes FRM 2016 grant DEQ20160334882.


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Authors and Affiliations

  1. 1.Applied Mathematics and Computational BiologyEcole Normale SupérieureParisFrance
  2. 2.Department of Applied MathematicsTel-Aviv UniversityTel AvivIsrael

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