Chung's law for homogeneous Brownian functionals

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 $\text{k}$ such that $$\underset{t\to \mathrm{\infty }}{\lim }{t}^{-1}\log \text{p}[T_{ab}>t]=-\text{k}\mathrm{.}$$ 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.