Arcsine Laws for Brownian motion in the presence of Permeable Barriers #

Toby Kay

13:50 Tuesday in 3Q16.

Part of the Stochastic processes and random walks session.

Abstract #

The Arcsine laws of Brownian motion are a collection of results describing three different statistical quantities of one-dimensional Brownian motion: the proportion of time the process spends in the positive half-space, the time taken for the process to reach its maximum, 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. In real systems space is often no longer homogeneous, and these laws are likely to no longer hold. In this talk we explore how the presence of a spatial heterogeneity alters these Arcsine laws. Specifically the heterogeneity we study is a permeable barrier. Permeable barriers appear in many biological and physical systems across multiple scales such that understanding their impact on the motion of Brownian particles is paramount to modelling various situations. Using the Feynman-Kac formalism and path decomposition techniques we are able to find the exact form of the probability distribution of the three statistical quantities of interest. We discover these quantities are no longer governed by the same distribution and that the presence of a permeable barrier has a large impact at short times, but this impact is lost as time becomes infinite. Finally, we also study a closely related statistical quantity, the maximum displacement of a Brownian particle and show how the inclusion of a permeable barrier has a large effect on the monotonicity of the distribution.