  We present a spectral-theoretic approach to time-average statistical
mechanics for general, non-equilibrium initial conditions. We consider the
statistics of bounded, local additive functionals of reversible as well as
irreversible ergodic stochastic dynamics with continuous or discrete
state-space. We derive exact results for the mean, fluctuations and
correlations of time average observables from the eigenspectrum of the
underlying generator of Fokker-Planck or master equation dynamics, and discuss
the results from a physical perspective. Feynman-Kac formulas are re-derived
using It\^o calculus and combined with non-Hermitian perturbation theory. The
emergence of the universal central limit law in a spectral representation is
shown explicitly on large deviation time-scales. For reversible dynamics with
equilibrated initial conditions we derive a general upper bound to fluctuations
of occupation measures in terms of an integral of the return probability.
Simple, exactly solvable examples are analyzed to demonstrate how to apply the
theory. As a biophysical example we revisit the Berg-Purcell problem on the
precision of concentration measurements by a single receptor. Our results are
directly applicable to a diverse range of phenomena underpinned by time-average
observables and additive functionals in physical, chemical, biological, and
economical systems.
