Metastable diffusions with degenerate drifts

TL;DR

This paper analyzes the spectrum of the semiclassical Witten Laplacian for confining Morse-Bott functions, revealing exponentially small eigenvalues and deriving an Eyring-Kramers formula using microlocal quasimode constructions.

Contribution

It extends spectral analysis of the Witten Laplacian to Morse-Bott functions and establishes an Eyring-Kramers formula in this setting.

Findings

Existence of exponentially small eigenvalues separated from the spectrum

Derivation of Eyring-Kramers formula for these eigenvalues

Microlocal construction of quasimodes near critical submanifolds

Abstract

We study the spectrum of the semiclassical Witten Laplacian $\Delta_{f}$ associated to a smooth function $f$ on ${\mathbb R}^d$. We assume that $f$ is a confining Morse--Bott function. Under this assumption we show that $\Delta_{f}$ admits exponentially small eigenvalues separated from the rest of the spectrum. Moreover, we establish Eyring-Kramers formula for these eigenvalues. Our approach is based on microlocal constructions of quasimodes near the critical submanifolds.