Metastable hierarchies without explicitly known invariant measures

Establish complete metastable hierarchies for diffusions whose invariant measures are not explicitly known, thereby extending the verification of nontrivial metastable hierarchies beyond models with explicitly known stationary measures.

Background

The paper’s full Gamma-expansion theorem is conditional on a complete metastable hierarchy, defined through successive metastable time scales, finite-state effective generators, and resolvent convergence. The transfer theorem shows that once this hierarchy is available, it yields the successive Gamma-limits of the level-two large-deviation rate functionals.

For the generalized overdamped Langevin model, existing results verify the hierarchy under additional assumptions. In contrast, the underdamped and third-order Langevin models considered in the paper only receive unconditional pre-metastable results; their full expansions require the hierarchy to be established. The authors note that currently available verification results for nontrivial hierarchies concern diffusions with explicitly known invariant measures, leaving the corresponding problem for models without such explicit stationary measures unresolved.