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The Arcsine laws of Brownian motion are a collection of results describingthree different statistical quantities of one-dimensional Brownian motion: thetime at which the process reaches its maximum position, the total time theprocess spends in the positive half-space and the time at which the processcrosses the origin for the last time. Remarkably the cumulative probabilitiesof these three observables all follows the same distribution, the Arcsinedistribution. But in real systems, space is often heterogeneous, and these lawsare likely to hold no longer. In this paper we explore such a scenario andstudy how the presence of a spatial heterogeneity alters these Arcsine laws.Specifically we consider the case of a thin permeable barrier, which is oftenemployed to represent diffusion impeding heterogeneities in physical andbiological systems such as multilayer electrodes, electrical gap junctions,cell membranes and fragmentation in the landscape for dispersing animals. Usingthe Feynman-Kac formalism and path decomposition techniques we are able to findthe exact time-dependence of the probability distribution of the threestatistical quantities of interest. We show that a permeable barrier has alarge impact on these distributions at short times, but this impact is lessinfluential as time becomes long. In particular, the presence of a barriermeans that the three distributions are no longer identical with symmetry abouttheir means being broken. We also study a closely related statistical quantity,namely, the distribution of the maximum displacement of a Brownian particle andshow that it deviates significantly from the usual half-Gaussian form.

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