Hypercontractivity for functional stochastic differential equations
Hypercontractivity for functional stochastic differential equations
复制标题
函数随机微分方程的超收缩性
DOI：
10.1016/j.spa.2015.04.001
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发表时间：
2015
影响因子：
1.4
通讯作者：
Yuan Chenggui
中科院分区：
文献类型：
--
作者：
Bao Jianhai;Wang Feng-Yu;Yuan Chenggui
给出了一类泛函随机微分方程的马氏半群超压缩性的一个显式充分条件。因此，半群Pt在L2（μ）和全变分范数下都指数收敛到它的唯一不变概率测度μ，并且当t> 0时，它在L2（μ）上是紧的.这提供了一类自然的非对称马尔可夫半群，它们在大时间内是紧的，但在小时间内是非紧的。一个半线性模型，可能不满足这个充分条件进行了研究。由于相关的狄利克雷形式不满足log-Sobolev不等式，使用函数不等式的标准论证不起作用。
An explicit sufficient condition on the hypercontractivity is derived for the Markov semigroup associated with a class of functional stochastic differential equations. Consequently, the semigroup P t converges exponentially to its unique invariant probability measure μ in both L 2 (μ) and the totally variational norm‖⋅‖ var, and it is compact in L 2 (μ) for sufficiently large t> 0. This provides a natural class of non-symmetric Markov semigroups which are compact for large time but non-compact for small time. A semi-linear model which may not satisfy this sufficient condition is also investigated. As the associated Dirichlet form does not satisfy the log-Sobolev inequality, the standard argument using functional inequalities does not work.