  There are multiple ways in which a stochastic system can be out of
statistical equilibrium. It might be subject to time-varying forcing; or be in
a transient phase on its way towards equilibrium; it might even be in
equilibrium without us noticing it, due to insufficient observations; and it
even might be a system failing to admit an equilibrium distribution at all. We
review some of the approaches that model the effective statistical behavior of
equilibrium and non-equilibrium dynamical systems, and show that both cases can
be considered under the unified framework of optimal low-rank approximation of
so-called transfer operators. Particular attention is given to the connection
between these methods, Markov state models, and the concept of metastability,
further to the estimation of such reduced order models from finite simulation
data. We illustrate our considerations by numerical examples.
