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| Local times of multi-parameter processes with stable components |
| XIONG Xian-zhu |
| College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China |
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Abstract Let $X_1,\cdots ,X_h$ be independent $ (N,d_1,\alpha _1 ),\cdots ,(N,d_h,\alpha _h)$-stable processes respectively, where $\alpha _1, \cdots , \alpha _h $ may be different real numbers in $(0,2]$. The corresponding multi-parameter process with stable components is defined as the following $N$-parameter random field on $\mathbf{R}^d\left(d= \sum\limits_{i=1}^{h}d_{i}\right)$, $X(t)= (X_1(t),X_2(t),\cdots,X_h(t)),$ $\forall t \in \mathbf{R}_+^N$. Under the condition that $N>d,$ it is proved that there is a (jointly continuous) local time for $X(t)$. The result is peculiar to the process $X(t)$, because there is no local time under any condition for the one-parameter process with stable components.
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Received: 13 January 2014
Published: 28 July 2018
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多指标稳定分量过程的局部时
设$X_1,\cdots ,X_h$分别是独立的$ (N,d_1,\alpha _1 ),\cdots ,(N,d_h,\alpha _h)$稳定过程, 其中$\alpha _1, \cdots , \alpha _h $可以是$(0,2]$中不同的数. 设$X(t)= (X_1(t),X_2(t),\cdots,X_h(t)),$ $\forall t \in \mathbf{R}_+^N$, 则称$X=\{X(t); t\in\mathbf{R}_+^N\}$为 多指标稳定分量过程. 在$N>\sum\limits_{i=1}^{h}d_{i}$的条件下, 证明了$X$存在(联合连续的)局部时, 该结果是多指标稳定分量过程所特有的, 因为单指标稳定分量过程在任何情况下都不存在局部时.
关键词：
多指标稳定分量过程,
局部时,
联合连续
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