Deterministic time intervals on which a class of persistent processes are away from their origins

Abstract

There are three results each concerning large but remote deterministic time intervals at which excursions of a process away from the origin must occur. The first result gives a sufficient condition for a persistent random walk with a finite fourth moment. In this instance the aforementioned time intervals include an additional requirement that the walk is far away from the origin. The second result gives a necessary and a sufficient condition for similar excursions in the case of Brownian motion. The third result gives a necessary and a sufficient condition for time intervals to be free of the zeros of a class of persistent natural scale linear diffusions on the line and is equivalent to the determination of recurrent sets at infinity of the inverse local time.