Kramers-Fokker-Planck operators with homogeneous potentials

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 $V (q_1, q _2) =-q ^2_1 (q^ 2_1+q^2_2) ^n$wasalreadytreatedonlyfor$n=1.$ 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.