arXiv2014
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.