We develop a categorical framework for the stability of metastable regimes ofMarkov processes under coarse-graining. A category Mark is introduced whoseobjects are generators of Markov semigroups and whose morphisms are Markovkernels satisfying a resolvent-compatibility condition. A subcategory Cgap of admissible morphisms that preserve the spectral gap up to o(1) and are compatiblewith quasi-stationary distributions (QSDs) is dened. We construct a lax functorF : Mark → Spec into the category of spectral data and prove three key lemmas: gap stability, QSD compatibility, and sub-additivity of the composition error.An appendix xes the functional-analytic assumptions (sectorial generators withcompact resolvent, Doob h-transform).
Eldar khubiev (Thu,) studied this question.