  We study two non-Markovian gene-expression models in which protein production
is a stochastic process with a fat-tailed non-exponential waiting time
distribution (WTD). For both models, we find two distinct scaling regimes
separated by an exponentially long time, proportional to the mean first passage
time (MFPT) to a ground state (with zero proteins) of the dynamics, from which
the system can only exit via a non-exponential reaction. At times shorter than
the MFPT the dynamics are stationary and ergodic, entailing similarity across
different realizations of the same process, with an increased Fano factor of
the protein distribution, even when the WTD has a finite cutoff. Notably, at
times longer than the MFPT the dynamics are nonstationary and nonergodic,
entailing significant variability across different realizations. The MFPT to
the ground state is shown to directly affect the average population sizes and
we postulate that the transition to nonergodicity is universal in such
non-Markovian models.
