Integrated covariances as excess observables weighted by currents and activities

Near equilibrium, the symmetric part of the time-integrated steady-state covariance, i.e., the time integral of correlation functions, is governed by the fluctuation-dissipation theorem, while the antisymmetric part vanishes due to Onsager reciprocity. Far from equilibrium, where these principles no longer apply, we develop a unified formalism for both symmetric and antisymmetric components of integrated covariances. We derive exact, computationally tractable expressions for these quantities, valid in arbitrary nonequilibrium steady states of Markov jump processes and Fokker-Planck equation. Both components are expressed in terms of excess observables, a notion central to both statistical physics and reinforcement learning. Furthermore, we establish thermodynamic upper bounds for antisymmetric covariances in terms of (pseudo-)entropy production and cycle affinities. Finally, we show that the speed-up of self-averaging induced by nonequilibrium drivings which preserve kinetics (activity) is bounded by the cycle affinities (thermodynamic forces).