Full metastable asymptotic of the Fisher information

TL;DR

This paper provides a detailed asymptotic expansion of the Fisher information for multiwell potentials, revealing a hierarchy of scales linked to metastable states in overdamped Langevin dynamics.

Contribution

It introduces a Gamma-convergence expansion of Fisher information that captures metastable behavior across multiple scales for general multiwell potentials.

Findings

Hierarchical scales depend on concentration on minima, critical points, or general measures.

Full asymptotic description of Fisher information minima over regular probability sets.

Application to diffusion operators and semiclassical analysis on manifolds.

Abstract

We establish an expansion by Gamma-convergence of the Fisher information relative to the reference measure exp(-beta V), where V is a generic multiwell potential and beta goes to infinity. 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 R^d. 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.