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We construct an integral equation for the first crossing distributions forfractional Brownian motion in the case of a constant barrier and we present anexact analytical solution. Additionally we present first crossing distributionsderived by simulating paths from fractional Brownian motion. We compare theresults of the analytical solutions with both those of simulations and those ofsome approximated solutions which have been used in the literature. Finally, wepresent multiplicity functions for dark matter structures resulting from ouranalytical approach and we compare with those resulting from N-bodysimulations.  We show that the results of analytical solutions are in good agreement withthose of path simulations but differ significantly from those derived fromapproximated solutions. Additionally, multiplicity functions derived fromfractional Brownian motion are poor fits of the those which result from N-bodysimulations. We also present comparisons with other models which exist in theliterature and we discuss different ways of improving the agreement betweenanalytical results and N-body simulations.

On the first crossing distributions in fractional Brownian motion and the mass function of dark matter haloes