Kramers-Fokker-Planck operators with homogeneous potentials

TL;DR

This paper establishes a global subelliptic estimate for Kramers-Fokker-Planck operators with homogeneous potentials, extending previous work to a broader class of potentials and completing the analysis of degenerate ellipticity at infinity.

Contribution

It introduces a new approach to analyze Kramers-Fokker-Planck operators with homogeneous potentials, generalizing prior results and covering all examples of degenerate ellipticity in the Witten Laplacian framework.

Findings

Derived optimal subelliptic lower bounds with logarithmic corrections.

Extended analysis to all homogeneous polynomial potentials with degenerate ellipticity.

Provided a different method from previous studies for a broader class of potentials.

Abstract

In this article we establish a global subelliptic estimate for Kramers-Fokker-Planck operators with homogeneous potentials $V(q)$ under some conditions, involving in particular the control of the eigenvalues of the Hessian matrix of the potential. Namely, this work presents a different approach from the one in [Ben], in which the case $V (q_1, q _2) =-q ^2_1 (q^ 2_1+q^2_2) ^n$ was already treated only for $n=1.$ With this article, after the former one dealing with non homogeneous polynomial potentials, we conclude the analysis of all the examples of degenerate ellipticity at infinty presented in the framework of Witten Laplacian by Helffer and Nier in [HeNi]. Like in [Ben], our subelliptic lower bounds are the optimal ones up to some logarithmic correction.