Passage of Lévy Processes across Power Law Boundaries at Small Times

We wish to characterize when a Lévy process $X_{t}$ crosses boundaries like $t^{\kappa}$, κ > 0, in a one- or two-sided sense, for small times t; thus, we inquire when $\text{lim sup}_{t\downarrow 0}\ |X_{t}|/t^{\kappa}$, $\text{lim sup}_{t\downarrow 0}\ X_{t}/t^{\kappa}$ and/or $\text{lim inf}_{...

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