  Explicit sufficient conditions on the hypercontractivity are presented for
two classes of functional stochastic partial differential equations driven by,
respectively, non-degenerate and degenerate Gaussian noises. Consequently,
these conditions imply that the associated Markov semigroup is $L^2$-compact
and exponentially convergent to the stationary distribution in entropy,
variance and total variational norm.
  As the log-Sobolev inequality is invalid under the framework, we apply a
criterion presented in the recent paper \cite{Wang14} using Harnack inequality,
coupling property and Gaussian concentration property of the stationary
distribution. To verify the concentration property, we prove a Fernique type
inequality for infinite-dimensional Gaussian processes which might be
interesting by itself.
