2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/140408Consider the first exit time $T_{a,b}$ from a finite interval $[-a,b]$ for an homogeneous fluctuating functional $X$ of a linear Brownian motion. We show the existence of a finite positive constant $\k$ such that $$\lim_{t\to\infty}t^{-1}\log \p[ T_{ab} > t] = -\k.$$ Following Chung's original approach, we deduce a "liminf" law of the iterated logarithm for the two-sided supremum of $X$. This extends and gives a new point of view on a result of Khoshnevisan and Shi.Revised version, to appear in the Rocky Mountain Journal of MathematicsProbability60F99, 60G17, 60G18, 60J55, 60J65text