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In molecular dynamics and sampling of high dimensional Gibbs measurescoarse-graining is an important technique to reduce the dimensionality of theproblem. We will study and quantify the coarse-graining error between thecoarse-grained dynamics and an effective dynamics. The effective dynamics is aMarkov process on the coarse-grained state space obtained by a closureprocedure from the coarse-grained coefficients. We obtain error estimates bothin relative entropy and Wasserstein distance, for both Langevin and overdampedLangevin dynamics. The approach allows for vectorial coarse-graining maps.Hereby, the quality of the chosen coarse-graining is measured by certainfunctional inequalities encoding the scale separation of the Gibbs measure. Themethod is based on error estimates between solutions of (kinetic) Fokker-Planckequations in terms of large-deviation rate functionals.

Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics