  We present the conceptual and technical background required to describe and
understand the correlations and fluctuations of the empirical density and
current of steady-state diffusion processes on all time scales -- observables
central to statistical mechanics and thermodynamics on the level of individual
trajectories. We focus on the important and non-trivial effect of a spatial
coarse graining. Making use of a generalized time-reversal symmetry we provide
deeper insight about the physical meaning of fluctuations of the coarse-grained
empirical density and current, and explain why a systematic variation of the
coarse-graining scale offers an efficient method to infer bounds on a system's
dissipation. Moreover, we discuss emerging symmetries in the statistics of the
empirical density and current, and the statistics in the central-limit regime.
More broadly our work promotes the application of stochastic calculus as a
powerful direct alternative to Feynman-Kac theory and path-integral methods.
