  We investigate the existence of steady states and exponential decay for
hypocoercive Fokker--Planck equations on the whole space with drift terms that
are linear in the position variable. For this class of equations, we first
establish that hypoellipticity of its generator and confinement of the system
is equivalent to the existence of a unique normalised steady state. These two
conditions also imply hypocoercivity, i.e. exponential convergence of the
solution to equilibrium.
  Since the standard entropy method does not apply to degenerate parabolic
equations, we develop a new modified entropy method (based on a modified,
non-degenerate entropy dissipation-like functional) to prove this exponential
decay in relative entropy (logarithmic till quadratic) - with a sharp rate.
Furthermore, we compute the spectrum and eigenspaces of the generator as well
as flow-invariant manifolds of Gaussian functions.
  Next, we extend our method to kinetic Fokker-Planck equations with a class of
non-quadratic potentials. And, finally, we apply this new method to
non-symmetric, uniformly parabolic Fokker-Planck equations with linear drift.
At least in 2D this always yields the sharp exponential envelopes for the
entropy function. In this case, we obtain even a sharp multiplicative constant
in the decay estimate for the non-symmetric semigroup.
