Theory of Probability &Its Applications, Volume 67, Issue 1, Page 17-27, May 2022.
We consider special one-dimensional Markov processes, namely, asymmetric jump Lévy processes, which have values in a given interval and reflect from the boundary points. We show that in this case, in addition to the standard semigroup of operators generated by the Markov process, there also appears the family of “boundary” random operators that send functions defined on the boundary of the interval to elements of the space $L_2$ on the entire interval. This study is a continuation of our paper [Theory Probab. Appl., 64 (2019), 335--354], where a similar problem was solved for symmetric reflecting Lévy processes.

中文翻译：

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反映 Lévy 过程和相关的线性算子族。二

概率理论及其应用，第 67 卷，第 1 期，第 17-27 页，2022 年 5 月。
我们考虑特殊的一维马尔可夫过程，即不对称跳跃 Lévy 过程，其值在给定区间内并从边界点反射。我们表明，在这种情况下，除了马尔可夫过程生成的标准半群算子外，还出现了“边界”随机算子族，它们将定义在区间边界上的函数发送到空间 $L_2$ 的元素在整个区间。这项研究是我们论文 [Theory Probab. Appl., 64 (2019), 335--354]，其中解决了对称反射 Lévy 过程的类似问题。