Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications

TL;DR

This paper analyzes the tunneling effect in Kramers-Fokker-Planck operators, demonstrating how the system converges to equilibrium with rates linked to small eigenvalues, and explores applications to related supersymmetric operators.

Contribution

It provides new results on the long-time behavior and spectral properties of supersymmetric differential operators with Morse potential functions, including the Kramers-Fokker-Planck operator.

Findings

Convergence to equilibrium with exponentially small eigenvalues

Spectral analysis of operators with Morse functions

Applications to Witten Laplacian and anharmonic oscillators

Abstract

In the first part of this work, we consider second order supersymmetric differential operators in the semiclassical limit, including the Kramers-Fokker-Planck operator, such that the exponent of the associated Maxwellian $\phi$ is a Morse function with two local minima and one saddle point. Under suitable additional assumptions of dynamical nature, we establish the long time convergence to the equilibrium for the associated heat semigroup, with the rate given by the first non-vanishing, exponentially small, eigenvalue. In the second part of the paper, we consider the case when the function $\phi$ has precisely one local minimum and one saddle point. We also discuss further examples of supersymmetric operators, including the Witten Laplacian and the infinitesimal generator for the time evolution of a chain of classical anharmonic oscillators.