Hypercontractivity for functional stochastic partial differential equations

Abstract

Explicitly sufficient conditions on the hypercontractivity are presented for two classes of functional stochastic partial differential equations driven by, respectively, nondegenerate 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 present framework, we apply a criterion presented in the recent paper [15] 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.