Multi-scale projection operator method and coarse-graining of covariant Fokker-Planck theory

A multi-scale projection operator method is developed and applied to study coarse-graining of covariant Fokker-Planck theory and the associated Ito-Langevin dynamics. Explicit expressions for the renormalized kinetic coefficients are obtained. It is also proved that the property of detailed balance is preserved by coarse-graining. These results demonstrate that covariant Fokker-Planck dynamics and covariant Ito-Langevin dynamics emerge naturally as a consequence of coarse-graining of microscopic dynamics. We illustrate our theory using several concrete examples.