  Over the years the field of non-Markovian stochastic processes and anomalous
diffusion evolved from a specialized topic to mainstream theory, which
transgressed the realms of physics to chemistry, biology and ecology. Numerous
phenomenological approaches emerged, which can more or less successfully
reproduce or account for experimental observations in condensed matter,
biological and/or single-particle systems. However, as far as their predictions
are concerned these approaches are not unique, often build on conceptually
orthogonal ideas, and are typically employed on an ad hoc basis. It therefore
seems timely and desirable to establish a systematic, mathematically unifying
and clean approach starting from more fine-grained principles. Here we analyze
projection-induced ergodic non-Markovian dynamics, both reversible as well as
irreversible, using spectral theory. We investigate dynamical correlations
between histories of projected and latent observables that give rise to memory
in projected dynamics, and rigorously establish conditions under which
projected dynamics is Markovian or renewal. A systematic metric is proposed for
quantifying the degree of non-Markovianity. As a simple, illustrative but
non-trivial example we study single file diffusion in a tilted box, which, for
the first time, we solve exactly using the coordinate Bethe ansatz. Our results
provide a solid foundation for a deeper and more systematic analysis of
projection-induced non-Markovian dynamics and anomalous diffusion.
