Extreme Value Statistics and Arcsine Laws of Brownian Motion in the Presence of a Permeable Barrier

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Abstract

The Arcsine laws of Brownian motion are a collection of results describing three different statistical quantities of one-dimensional Brownian motion: the time at which the process reaches its maximum position, the total time the process spends in the positive half-space and the time at which the process crosses the origin for the last time. Remarkably the cumulative probabilities of these three observables all follows the same distribution, the Arcsine distribution. But in real systems, space is often heterogeneous, and these laws are likely to hold no longer. In this paper we explore such a scenario and study how the presence of a spatial heterogeneity alters these Arcsine laws. Specifically we consider the case of a thin permeable barrier, which is often employed to represent diffusion impeding heterogeneities in physical and biological systems such as multilayer electrodes, electrical gap junctions, cell membranes and fragmentation in the landscape for dispersing animals. Using the Feynman-Kac formalism and path decomposition techniques we are able to find the exact time-dependence of the probability distribution of the three statistical quantities of interest. We show that a permeable barrier has a large impact on these distributions at short times, but this impact is less influential as time becomes long. In particular, the presence of a barrier means that the three distributions are no longer identical with symmetry about their means being broken. We also study a closely related statistical quantity, namely, the distribution of the maximum displacement of a Brownian particle and show that it deviates significantly from the usual half-Gaussian form.
Original languageEnglish
Article number345002
JournalJournal of Physics A: Mathematical and Theoretical
Volume56
Issue number34
DOIs
Publication statusPublished - 2 Aug 2023

Bibliographical note

Funding Information:
T K and L G acknowledge funding from, respectively, an Engineering and Physical Sciences Research Council (EPSRC) DTP student grant and the Biotechnology and Biological Sciences Research Council (BBSRC) Grant No. BB/T012196/1 and NERC Grant No. NE/W00545X/1. This work was carried out using the computational facilities of the Advanced Computing Research Centre, University of Bristol— www.bristol.ac.uk/acrc/ .

Funding Information:
T K and L G acknowledge funding from, respectively, an Engineering and Physical Sciences Research Council (EPSRC) DTP student grant and the Biotechnology and Biological Sciences Research Council (BBSRC) Grant No. BB/T012196/1 and NERC Grant No. NE/W00545X/1. This work was carried out using the computational facilities of the Advanced Computing Research Centre, University of Bristol—www.bristol.ac.uk/acrc/.

Publisher Copyright:
© 2023 The Author(s). Published by IOP Publishing Ltd.

Structured keywords

  • Engineering Mathematics Research Group

Keywords

  • cond-mat.stat-mech

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