Asymptotic Formulas for Extreme Statistics of Escape Times in 1, 2 and 3-Dimensions
- 3 Downloads
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.
KeywordsShort time asymptotics Diffusion Narrow escape Extreme statistics Transient Calcium dynamics Helmoltz Dendritic spine
Mathematics Subject Classification35K08 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.
- Guerrier, C., Korkotian, E., Holcman, D.: Calcium dynamics in neuronal microdomains: Modeling, stochastic simulations, and data analysis. In: Jaeger, D., Jung, R. (eds.) Encyclopedia of Computational Neuroscience, pp. 486–516. Springer, New York, NY (2015)Google Scholar
- Hille, B.: Ion Channels of Excitable Membranes, vol. 507. Sinauer, Sunderland (2001)Google Scholar
- Katz, B., Voolstra, O., Tzadok, H., Yasin, B., Rhodes-Modrov, E., Bartels, J.-P., Strauch, L., Huber, A., Minke, B.: The latency of the light response is modulated by the phosphorylation state of drosophila trp at a specific site. Channels 37, 1–8 (2017)Google Scholar
- Majumdar, S.N., Pal, A.: Extreme value statistics of correlated random variables. (2014). arXiv preprint arXiv:1406.6768
- Schuss, Z.: Diffusion and Stochastic Processes: An Analytical Approach. Springer Series on Applied Mathematical Sciences, vol. 170. Springer, New York (2010)Google Scholar
- Schuss, Z.: Brownian Dynamics at Boundaries and Interfaces. Springer, New York (2015)Google Scholar