An almost sure functional limit theorem at zero for a class of Lévy processes normed by the square root function, and applications

A recent result of Bertoin, Doney and Maller (Ann. Prob., 2007) gives an integral condition to characterize the class of Lévy processes X(t) for which lim sup_t↓0 |X (t)|/√t ∈ (0, ∞) occurs almost surely (a.s.). For such processes we have a kind of almost sure "iterated logarithm" result, but without the logs. In the present paper we prove a functional version of this result, which then opens the way to various interesting applications obtained via a continuous mapping theorem. We set these out in a rigorous framework, including a characterisation of the existence of an a.s. cluster set for the interpolated process, appropriate to the continuous time situation. The applications relate to functional laws for the supremum, reflected and a variety of other processes, including a class of stochastic differential equations, where we aim to give as informative a description as we can of the functional limit sets.