AI 中文总结
该研究延续前期工作，利用Eisenbaum同构定理，得到半直线上Feller布朗运动在空间区间$(0,1]$局部时的精确一致连续模，并推广至$[0,1]$及更一般扩散过程情形。
AI 中文摘要
我们研究半直线$[0,\infty)$上Feller布朗运动（FBM）局部时过程关于空间变量的连续模。简要来说，FBM是$[0,\infty)$上的强马尔可夫过程，在首次到达状态0之前，其运动类似$[0,\infty)$上的标准布朗运动；之后该过程返回$(0,\infty)$，返回方式要么是连续的（如反射布朗运动），要么是根据指定测度随机跳至某个正状态。本研究是我们与Michael Marcus早期工作的延续与应用，早期工作聚焦于通过拼接（“重生”）另一个有限寿命马尔可夫过程的路径构建的马尔可夫过程局部时的连续模。我们首先建立关于带特殊持有状态（FBM的状态0）的重生过程的预解式与局部时的一般结果，展示我们早期利用Eisenbaum同构定理在一族游程上的方法如何适用于该场景。随后，我们将该知识作为近似工具，得到关于空间区间$(0,1]$上FBM局部时的精确一致连续模这一主要结果；还在特定条件下将结果推广至更精细的空间区间$[0,1]$情形，并简要考虑$[0,\infty)$上更一般扩散过程的情况。
英文摘要
We examine the modulus of continuity, in the spatial variable, of the local time process of Feller Brownian motion (FBM) on the half-line $[0,\infty)$. Briefly, a FBM is a strong Markov process on $[0,\infty)$ that moves like standard Brownian motion on $[0,\infty)$ up until it first encounters the state $0$. The process returns to $(0,\infty)$, either continuously (like reflecting Brownian motion) or by jumping to a (random) positive state chosen according to a specified measure. The present work is a continuation and application of our earlier work with Michael Marcus on the moduli of continuity for the local times of a Markov process built by piecing together (``rebirthing") the paths of another Markov process with finite lifetime. We first establish a general result on the resolvent and local times for a rebirthed process with a special holding state (the state $0$ for FBM). We show how our earlier approach using the Eisenbaum Isomorphism Theorem on an assemblage of excursions works out in this context. This knowledge is then used as an approximation device to obtain our main result on the exact uniform moduli of continuity for the local time of FBM on a spatial interval of the form $(0,1]$. Extensions are made, under certain conditions, to the more delicate situation of the spatial interval $[0,1]$. We also consider briefly the case of more general diffusions on $[0,\infty)$.
