Abstract
The Arcsine laws of Brownian motion are a collection of results describing three different statistical quantities of onedimensional Brownian motion: the time at which the process reaches its maximum position, the total time the process spends in the positive halfspace 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 FeynmanKac formalism and path decomposition techniques we are able to find the exact timedependence 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 halfGaussian form.
Original language  English 

Article number  345002 
Journal  Journal of Physics A: Mathematical and Theoretical 
Volume  56 
Issue number  34 
DOIs  
Publication status  Published  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
 condmat.statmech
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