Abstract
Explicit sufficient conditions on the hypercontractivity are presented for two classes of functional stochastic partial differential equationsdriven 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 a recent paper 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.
Citation
Jianhai Bao. Feng-Yu Wang. Chenggui Yuan. "Hypercontractivity for functional stochastic partial differential equations." Electron. J. Probab. 20 1 - 15, 2015. https://doi.org/10.1214/EJP.v20-4108
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