  Consider 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.
