New theorems of Laplace Kernels with Radon measures in Galerkin Markov-State Models: about time evolution of eigenvalues and about errors

New theorems apt to the methods of the Markov-State Models in the Galerkin representation are derived directly from the Markov-chains theories. The analytical expressions of the time evolution of the eigenvalues and those of the relative error are derived. The new theorems are descended from measuretheory methodologies. The newly-written theorems therefore provide wit the analytical expressions of the time evolution of the eigenvalues and with that of the relative error from the measure-theoretical foundations of the Markov-chains models; the applications to protein folding are originated form the orthogonality of the committor functions in the appropriate description(s). The newly-found theorems are indicated of use for the other necessitated errors calculations. The modellisation used is one apt to recover the items of information about the originating chains. Markov-chains models offer the suitable tools of a wide range of investigation branches: protein-folding, catalysis, polymeric materials, further moleculardynamics processes, kinetic-network models, electron spin resonance, etc.