  Excursion set theory, where density perturbations evolve stochastically with
the smoothing scale, provides a method for computing the mass function of
cosmological structures like dark matter halos, sheets and filaments. The
computation of these mass functions is mapped into the so-called first-passage
time problem in the presence of a moving barrier. In this paper we use the path
integral formulation of the excursion set theory developed recently to
analytically solve the first-passage time problem in the presence of a generic
moving barrier, in particular the barrier corresponding to ellipsoidal
collapse. We perform the computation for both Gaussian and non-Gaussian initial
conditions. The expression of the halo mass function for the ellipsoidal
collapse barrier and with non-Gaussianity is therefore obtained in a fully
consistent way and it does not require the introduction of any form factor
artificially derived from the Press-Schechter formalism based on the spherical
collapse and usually adopted in the literature.
