Full Metastable Asymptotic of the Fisher Information

We establish an expansion by Γ-convergence of the Fisher information relative to the reference measure e−βV dx, where V is a generic multiwell potential and β → ∞. The expansion reveals a hierarchy of multiple scales reflecting the metastable behavior of the underlying overdamped Langevin dynamics: distinct scales emerge and become relevant depending on whether one considers probability measures concentrated on local minima of V , probability measures concentrated on critical points of V , or generic probability measures on Rd. We thus fully describe the asymptotic behavior of minima of the Fisher information over regular sets of probabilities. The analysis mostly relies on spectral properties of diffusion operators and the related semiclassical Witten Laplacian and covers also the case of a compact smooth manifold as underlying space.