Chung's Law for Homogeneous Brownian Functionals

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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.