A Stochastic Theory of the Hierarchical Clustering II. Halo progenitor
  mass function and large-scale bias
  We generalize the stochastic theory of hierarchical clustering presented in
paper I by Lapi & Danese (2020) to derive the (conditional) halo progenitor
mass function and the related large-scale bias. Specifically, we present a
stochastic differential equation that describes fluctuations in the mass growth
of progenitor halos of given descendant mass and redshift, as driven by a
multiplicative Gaussian white noise involving the power spectrum and the
spherical collapse threshold of density perturbations. We demonstrate that, as
cosmic time passes, the noise yields an average drift of the progenitors toward
larger masses, that quantitatively renders the expectation from the standard
extended Press & Schechter (EPS) theory. We solve the Fokker-Planck equation
associated to the stochastic dynamics, and obtain as an exact, stationary
solution the EPS progenitor mass function. Then we introduce a modification of
the stochastic equation in terms of a mass-dependent collapse threshold
modulating the noise, and solve analytically the associated Fokker-Planck
equation for the progenitor mass function. The latter is found to be in
excellent agreement with the outcomes of $N-$body simulations; even more
remarkably, this is achieved with the same shape of the collapse threshold used
in paper I to reproduce the halo mass function. Finally, we exploit the above
results to compute the large-scale halo bias, and find it in pleasing agreement
with the $N-$body outcomes. All in all, the present paper illustrates that the
stochastic theory of hierarchical clustering introduced in paper I can describe
effectively not only halos' abundance, but also their progenitor distribution
and their correlation with the large-scale environment across cosmic times.
