Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential

We consider the Fokker-Planck equation with a confining or anti-confining potential which behaves at infinity like a possibly high-degree homogeneous function. Hypoellipticity techniques provide the well-posedness of the weak Cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding Witten Laplacian, with detailed applications.