  Nonlinear evolution can sometimes be modelled by a deterministic mapping from
initial to final of the local smoothed overdensity. Perturbation theory methods
base on this deterministic and local mapping and ignore the 'cloud-in-cloud'
effect, while the excursion set approach methods take this nonlocality into
account. We compared these methods using the spherical collapse mapping and
showed that, on scales where the rms fluctuation is small, both models give
similar results and they are in good agreement with numerical simulations. If
the deterministic mapping depends on quantities other than overdensity, this
will also manifest as stochasticity if the other quantities are ignored. We
considered the Zeldovich approximation and Ellipsoidal Collapse model, both
include the tidal field in the evolution. Our anaylsis shows that the change in
cell shape effect should be included on scales where the rms is of order of
unity or larger. On scales where the rms is less than 2 methods based on the
spherical collapse model allow a rather accurate reconstruction of the shape of
the initial distribution from the nonlinear field. This can be used as the
basis for constraining the statistical properties of the initial fluctuation
field. (Abridge)
