Hypocoercivity and metastability of degenerate KFP equations at low temperature

TL;DR

This paper establishes hypocoercivity estimates and spectral gap quantification for degenerate Kramers-Fokker-Planck operators at low temperature, advancing understanding of their metastable behavior.

Contribution

It introduces semiclassical hypocoercivity estimates and derives Eyring-Kramers formulas for specific degenerate operators, using adapted WKB methods.

Findings

Proves hypocoercivity estimates for degenerate KFP operators

Derives Eyring-Kramers formulas for spectral analysis

Quantifies spectral gaps in the semiclassical regime

Abstract

We consider Kramers-Fokker-Planck operators with general degenerate coefficients. We prove semiclassical hypocoercivity estimates for a large class of such operators. Then, we manage to prove Eyring-Kramers formulas for the bottom of the spectrum of some particular degenerate operators in the semiclassical regime, and quantify the spectral gap separating these eigenvalues from the rest of the spectrum. The main ingredient is the construction of sharp Gaussian quasimodes through an adaptation of the WKB method.