Hypercontractivity and applications for stochastic Hamiltonian systems

The hypercontractivity is proved for the Markov semigroup associated with a class of stochastic Hamiltonian systems on Hilbert spaces. Consequently, the Markov semigroup converges exponentially to the invariant probability measure in entropy and is compact for large time. These strengthen the hypocoercivity results derived in the literature. Since the log-Sobolev inequality is invalid, we introduce a new argument to prove the hypercontractivity using coupling and dimension-free Harnack inequality. The main results are illustrated by concrete examples of the kinetic Fokker-Planck equation and highly degenerate diffusion processes.