论文标题
高斯随机场的命中概率和随机矩阵特征值的碰撞
Hitting probabilities of Gaussian random fields and collision of eigenvalues of random matrices
论文作者
论文摘要
Let $X= \{X(t), t \in \mathbb R^N\}$ be a centered Gaussian random field with values in $\mathbb R^d$ satisfying certain conditions and let $F \subset \mathbb R^d$ be a Borel set.在我们的主要定理中，我们为$ f $提供了足够的条件，即$ x $，即$ \ \ \ m athbb p \ big（x（t）\ in f \ hbox {in \ hbox {in \ mathbb r^n \ big）= 0 $，这在Dalang等人的主要结果中得到了显着改善。我们提供了多种高斯随机场的示例，我们的结果适用。此外，通过使用我们的主要定理，我们解决了一个问题，即与贾拉米洛（Jaramillo）和nualart [14]和Song et al [21]中留下的随机矩阵的特征值的碰撞存在。
Let $X= \{X(t), t \in \mathbb R^N\}$ be a centered Gaussian random field with values in $\mathbb R^d$ satisfying certain conditions and let $F \subset \mathbb R^d$ be a Borel set. In our main theorem, we provide a sufficient condition for $F$ to be polar for $X$, i.e. $\mathbb P \big( X(t) \in F \hbox{ for some } t \in \mathbb R^N \big) = 0$, which improves significantly the main result in Dalang et al [7], where the case of $F$ being a singleton was considered. We provide a variety of examples of Gaussian random field for which our result is applicable. Moreover, by using our main theorem, we solve a problem on the existence of collisions of the eigenvalues of random matrices with Gaussian random field entries that was left open in Jaramillo and Nualart [14] and Song et al [21].
