Passage of Lévy Processes across Power Law Boundaries at Small Times

We wish to characterise when a L\'{e}vy process $X_t$ crosses boundaries like $t^\kappa$, $\kappa>0$, in a one or two-sided sense, for small times $t$; thus, we enquire when $\limsup_{t\downarrow 0}|X_t|/t^{\kappa}$, $\limsup_{t\downarrow 0}X_t/t^{\kappa}$ and/or $\liminf_{t\downarrow 0}X_t/t^{\kappa}$ are almost surely (a.s.) finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values of $\kappa>0$. Often (for many values of $\kappa$), when the limsups are finite a.s., they are in fact zero, as we show, but the limsups may in some circumstances take finite, nonzero, values, a.s. In general, the process crosses one or two-sided boundaries in quite different ways, but surprisingly this is not so for the case $\kappa=1/2$. An integral test is given to distinguish the possibilities in that case. Some results relating to other norming sequences for $X$, and when $X$ is centered at a nonstochastic function, are also given