  We consider various problems related to the persistence probability of
fractional Brownian motion (FBM), which is the probability that the FBM $X$
stays below a certain level until time $T$. Recently, Oshanin et al. study a
physical model where persistence properties of FBM are shown to be related to
scaling properties of a quantity $J_N$, called steady-state current. It turns
out that for this analysis it is important to determine persistence
probabilities of FBM with a moving boundary. We show that one can add a
boundary of logarithmic order to a FBM without changing the polynomial rate of
decay of the corresponding persistence probability which proves a result needed
in Oshanin et al. Moreover, we complement their findings by considering the
continuous-time version of $J_N$. Finally, we use the results for moving
boundaries in order to improve estimates by Molchan concerning the persistence
properties of other quantities of interest, such as the time when a FBM reaches
its maximum on the time interval $(0,1)$ or the last zero in the interval
$(0,1)$.
