论文标题
高斯随机字段的索博莱夫规律性
Sobolev regularity of Gaussian random fields
论文作者
论文摘要
在本文中，我们充分表征可测量的高斯过程$（u（x））_ {x \ in \ Mathcal {d}} $，其示例路径位于Integer顺序$ w^{m，p}的Sobolev空间中其中$ \ Mathcal {d} $是任意开放集。结果是根据对角线上协方差函数的Sobolev规则性形式来表达的。然后将其与与协方差函数及其跨衍生物相关的积分运算符的合适Mercer或其他核分解有关。在Hilbert案$ p = 2 $中，其他链接已成为W.R.T.上述积分运算符的Mercer分解，其跟踪和RKH的嵌入在$ w^{m，2}（\ Mathcal {d}）$中。我们提供了简单的例子，并部分恢复了与高斯过程的Sobolev规律性有关的最新结果。
In this article, we fully characterize the measurable Gaussian processes $(U(x))_{x\in\mathcal{D}}$ whose sample paths lie in the Sobolev space of integer order $W^{m,p}(\mathcal{D}),\ m\in\mathbb{N}_0,\ 1 <p<+\infty$, where $\mathcal{D}$ is an arbitrary open set. The result is phrased in terms of a form of Sobolev regularity of the covariance function on the diagonal. This is then linked to the existence of suitable Mercer or otherwise nuclear decompositions of the integral operators associated to the covariance function and its cross-derivatives. In the Hilbert case $p=2$, additional links are made w.r.t. the Mercer decompositions of the said integral operators, their trace and the imbedding of the RKHS in $W^{m,2}(\mathcal{D})$. We provide simple examples and partially recover recent results pertaining to the Sobolev regularity of Gaussian processes.
