Hypercontractivity for Stochastic Hamiltonian Systems
  The hypercontractivity is proved for the Markov semigroup associated to a class of finite/infinite dimensional stochastic Hamiltonian systems.   Consequently, the Markov semigroup is exponentially convergent to the invariant probability measure in entropy (thus, also in $L^2$), and is compact for large time. Since the log-Sobolev inequality is invalid for the associated Dirichlet form, we introduce a general result on the hypercontractivity using the Harnack inequality with power. The main results are illustrated by concrete examples. 
Feng-Yu Wang
math.PR