  In molecular dynamics and sampling of high dimensional Gibbs measures
coarse-graining is an important technique to reduce the dimensionality of the
problem. We will study and quantify the coarse-graining error between the
coarse-grained dynamics and an effective dynamics. The effective dynamics is a
Markov process on the coarse-grained state space obtained by a closure
procedure from the coarse-grained coefficients. We obtain error estimates both
in relative entropy and Wasserstein distance, for both Langevin and overdamped
Langevin dynamics. The approach allows for vectorial coarse-graining maps.
Hereby, the quality of the chosen coarse-graining is measured by certain
functional inequalities encoding the scale separation of the Gibbs measure. The
method is based on error estimates between solutions of (kinetic) Fokker-Planck
equations in terms of large-deviation rate functionals.

Analysis of PDEs