On the first crossing distributions in fractional Brownian motion and
  the mass function of dark matter haloes
  We construct an integral equation for the first crossing distributions for
fractional Brownian motion in the case of a constant barrier and we present an
exact analytical solution. Additionally we present first crossing distributions
derived by simulating paths from fractional Brownian motion. We compare the
results of the analytical solutions with both those of simulations and those of
some approximated solutions which have been used in the literature. Finally, we
present multiplicity functions for dark matter structures resulting from our
analytical approach and we compare with those resulting from N-body
simulations.
  We show that the results of analytical solutions are in good agreement with
those of path simulations but differ significantly from those derived from
approximated solutions. Additionally, multiplicity functions derived from
fractional Brownian motion are poor fits of the those which result from N-body
simulations. We also present comparisons with other models which exist in the
literature and we discuss different ways of improving the agreement between
analytical results and N-body simulations.
