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 approximate to 0 to 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 nonuniversal evolution of the simulated halo mass functions. Compared to standard theoretical estimates, our stochastic theory predicts a halo number density higher by a factor of several toward z greater than or similar to 10, an outcome that 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.
