Abstract

We consider the problem of the distribution of the sojourn time in a compact set \(\mathbb{Z}_{p}\) in the case of a \(p\)-adic random walk. We rely on the results of our previous studies of the distribution of the first return time for a \(p\)-adic random walk and the results of Takacs on the study of the sojourn time problem for a wide class of random processes. For a \(p\)-adic random walk we find the mean sojourn time of the trajectory in \(\mathbb{Z}_{p}\) and the asymptotics as \(t\rightarrow\infty\) of arbitrary moments of the distribution of the sojourn time in \(\mathbb{Z}_{p}\). We also discuss some possible applications of our results to the modeling of relaxation processes related to the conformational dynamics of protein.

中文翻译：

---

$$p$$-Adic 随机游走的逗留时间问题及其应用

摘要

在\(p\) -adic 随机游走的情况下，我们考虑了紧集\(\mathbb{Z}_{p}\)中逗留时间的分布问题。我们依赖于我们之前对\(p\) -adic 随机游走的首次返回时间分布的研究结果以及 Takacs 对大量随机过程停留时间问题的研究结果。对于\(p\) -adic 随机游走，我们发现轨迹在\(\mathbb{Z}_{p}\)中的平均逗留时间和任意时刻的渐近线\(t\rightarrow\infty\)逗留时间在\(\mathbb{Z}_{p}\)中的分布. 我们还讨论了我们的结果在与蛋白质构象动力学相关的松弛过程建模中的一些可能应用。