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We present general results on fluctuations and spatial correlations of thecoarse-grained empirical density and current of Markovian diffusion inequilibrium or non-equilibrium steady states on all time scales. We unravel adeep connection between current fluctuations and generalized time-reversalsymmetry, providing new insight into time-averaged observables. We highlightthe essential role of coarse graining in space from mathematical,thermodynamical, and experimental points of view. Spatial coarse graining isrequired to uncover salient features of currents that break detailed balance,and a thermodynamically "optimal" coarse graining ensures the most preciseinference of dissipation. Defined without coarse graining, the fluctuations ofempirical density and current are proven to diverge on all time scales indimensions higher than one, which has far-reaching consequences for thecentral-limit regime in continuous space. We apply the results to examples ofirreversible diffusion. Our findings provide new intuition about time-averagedobservables and allow for a more efficient analysis of single-moleculeexperiments.

Mathematical, Thermodynamical, and Experimental Necessity for Coarse Graining Empirical Densities and Currents in Continuous Space