In molecular dynamics and sampling of high dimensional Gibb
s measures coarse-graining is an
important technique to reduce the dimensionality of the pro
blem. We will study and quantify the
coarse-graining error between the coarse-grained dynamic
s and an effective dynamics. The effective
dynamics is a Markov process on the coarse-grained state spa
ce obtained by a closure procedure from
the coarse-grained coefficients. We obtain error estimates b
oth in relative entropy and Wasserstein
distance, for both Langevin and overdamped Langevin dynami
cs. The approach allows for vecto-
rial coarse-graining maps. Hereby, the quality of the chose
n coarse-graining is measured by certain
functional inequalities encoding the scale separation of t
he Gibbs measure. The method is based on
error estimates between solutions of (kinetic) Fokker-Pla
nck equations in terms of large-deviation
rate functionals.