Hypercontractivity for functional stochastic partial differential equations
Hypercontractivity for functional stochastic partial differential equations
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DOI：
10.1214/ejp.v20-4108
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发表时间：
2015-03
影响因子：
1.4
通讯作者：
J. Bao;Feng-Yu Wang;C. Yuan
中科院分区：
文献类型：
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作者：
J. Bao;Feng-Yu Wang;C. Yuan
给出了分别由非退化高斯噪声和退化高斯噪声驱动的两类泛函随机偏微分方程的超收缩性的显式充分条件。因此，这些条件意味着相关的马尔可夫半群是l2 -紧的，并且在熵、方差和总变分范数上指数收敛于平稳分布。由于log-Sobolev不等式在目前的框架下是无效的，我们应用了最近论文[15]中提出的一个准则，该准则利用了平稳分布的Harnack不等式、耦合性质和高斯集中性质。为了验证浓度性质，我们证明了一个无限维高斯过程的Fernique型不等式，这个不等式本身可能很有趣。
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.