  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.
