Perturbed and non-perturbed Brownian taboo processes

In this paper we study the Brownian taboo process, which is a version of Brownian motion conditioned to stay within a finite interval, and the α-perturbed Brownian taboo process, which is an analogous version of an α-perturbed Brownian motion.We are particularly interested in the asymptotic behaviour of the supremum of the taboo process, and our main results give integral tests for upper and lower functions of the supremum as t→∞. In the Brownian case these include extensions of recent results in Lambert [4], but are proved in a quite different way.