  We revisit functional central limit theorems for additive functionals of
ergodic Markov diffusion processes. Translated in the language of partial
differential equations of evolution, they appear as diffusion limits in the
asymptotic analysis of Fokker-Planck type equations. We focus on the square
integrable framework, and we provide tractable conditions on the infinitesimal
generator, including degenerate or anomalously slow diffusions. We take
advantage on recent developments in the study of the trend to the equilibrium
of ergodic diffusions. We discuss examples and formulate open problems.
