We discuss some mathematical issues arising in computations where the microscopic Markov process on a lattice is approximated by a hierarchy of coarse-grained processes. We analyse the hierarchy of the coarse models in specific examples of lattice spin dynamics and derive error control of quantities that may also depend on the path. We present mathematical tools for error estimation of the coarse-graining procedure. Numerical evidence guided by analytical results suggests that the coarse-grained Monte Carlo algorithm probes efficiently energy landscape in path-wise agreement to MC simulations at the microscopic level. The presented results are joint work with M. Katsoulakis and A. Sopasakis.
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