On transient Bessel processes and planar Brownian motion reflected at their future infima

Let R be a transient Bessel process and let J(t) = infu≥t R(u) be its future infimum process. The main result of this paper is an integral test characterizing the upper functions of R – J, which turn out to be quite different from those of R. The test implies in particular an iterated logarithm law recently obtained by Khoshnevisan et al. (1994), and also solves the problem of characterizing the large gaps between the past supremum and future infimum of R. The corresponding local question for a planar Wiener process is studied.