Hypercontractivity for Functional Stochastic Partial Differential Equations

TL;DR

This paper establishes explicit conditions for hypercontractivity in certain functional stochastic PDEs driven by Gaussian noise, leading to results on the convergence to stationary distributions using advanced inequalities.

Contribution

It provides new explicit hypercontractivity criteria for functional SPDEs with Gaussian noise, utilizing a Fernique type inequality and Harnack inequalities.

Findings

Markov semigroup is $L^2$-compact and exponentially converges

Conditions imply convergence in entropy, variance, and total variation

Introduces a Fernique type inequality for infinite-dimensional Gaussian processes

Abstract

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.