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Abstract

We present a spectral-theoretic approach to time-average statistical mechanics for general, nonequilibrium 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 we discuss the results from a physical perspective. Feynman-Kac formulas are rederived using Itô 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 timescales. 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.