Hypocoercivity and metastability of degenerate KFP equations at low temperature
Loïs Delande
TL;DR
The paper addresses hypocoercivity and metastability for degenerate Kramers–Fokker–Planck operators at low temperature by developing semiclassical tools for a broad class of coefficients and potentials.It combines a semiclassical hypocoercivity framework with a WKB-type Gaussian quasimode construction to obtain sharp information on the bottom of the spectrum, including Eyring–Kramers-type formulas for eigenvalues λ(m,h) associated with minima, and explicit spectral gaps.A global construction then glues local quasimodes into a low-dimensional spectral subspace, whose dynamics yields precise metastable behavior and timescales t_k ∼ h^{μ_k} e^{2S_k/h}, while corollaries give explicit return-to-equilibrium statements in total variation distance.Overall, the work extends hypocoercivity methods to degenerate KFP operators, provides a robust semiclassical analysis framework, and delivers quantitative, metastability-driven descriptions of long-time stochastic dynamics in low-temperature regimes.
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.