  In the framework of the stochastic theory for hierarchical clustering, we
investigate the time-dependent solutions of the Fokker-Planck equation
describing the statistics of dark matter halos, and discuss the typical
timescales needed for these to converge toward stationary states, far away
enough from initial conditions. Although we show that the stationary solutions
can reproduce the outcomes of state-of-the-art $N-$body simulations at
$z\approx 0$ to a great accuracy, one needs to go beyond to fully account for
the cosmic evolution of the simulated halo mass function toward high-redshift.
Specifically, we demonstrate that the time-dependent solutions of the
Fokker-Planck equation can describe, for reasonable initial conditions, the
non-universal evolution of the simulated halo mass functions. Compared to
standard theoretical estimates, our stochastic theory predicts a halo number
density higher by factor of several toward $z\gtrsim 10$, an outcome which can
be helpful in elucidating early and upcoming data from JWST. Finally, we point
out the relevance of our approach in designing, interpreting and emulating
present and future $N-$body experiments.
