标题： 鞅方法在马尔可夫过程的时间可加观测物首通问题中的应用 显示英文标题

标题： Martingale approach for first-passage problems of time-additive observables in Markov processes

摘要： 我们开发了一种基于鞅的方法来研究马尔可夫过程的时间可加可观测量首次离开有限宽度区间的第一通量问题。当区间宽度较大时，我们推导出分裂概率和首次通过时间累积量的一般表达式。这些表达式将首次通过量与时间可加可观测量的大偏差性质联系起来。我们发现，根据时间可加可观测量的大偏差速率函数的性质，存在三种定性上不同的情况。这些情况对应于在区间不可能边界处事件的指数、超指数或亚指数抑制。此外，我们证明了两个区间的首次通过时间统计量通常是不同的，即使对于对称阈值和在大区间宽度的极限下也是如此。虽然到达可能边界的统计量由原始过程中的时间可加可观测量的累积量决定，但到达不可能边界的统计量由一个对偶过程决定。我们从一个单参数族的正鞅（称为Perron鞅）得到这些结果，因为它们与定义马尔可夫过程的转移率矩阵的倾斜版本的Perron根有关。此外，我们证明了倾斜矩阵的每个特征对都有一个单参数族的鞅。为了求解有限阈值下的首次通过问题，我们通常需要包括非正鞅在内的所有单参数族的鞅。我们通过求解具有有限宽度区间的直奔-翻滚粒子的首次通过问题来说明这一点。 显示英文摘要

摘要： We develop a method based on martingales to study first-passage problems of time-additive observables exiting an interval of finite width in a Markov process. In the limit that the interval width is large, we derive generic expressions for the splitting probability and the cumulants of the first-passage time. These expressions relate first-passage quantities to the large deviation properties of the time-additive observable. We find that there are three qualitatively different regimes depending on the properties of the large deviation rate function of the time-additive observable. These regimes correspond to exponential, super-exponential, or sub-exponential suppression of events at the unlikely boundary of the interval. Furthermore, we show that the statistics of first-passage times at both interval boundaries are in general different, even for symmetric thresholds and in the limit of large interval widths. While the statistics of the times to reach the likely boundary are determined by the cumulants of the time-additive observables in the original process, those at the unlikely boundary are determined by a dual process. We obtain these results from a one-parameter family of positive martingales that we call Perron martingales, as these are related to the Perron root of a tilted version of the transition rate matrix defining the Markov process. Furthermore, we show that each eigenpair of the tilted matrix has a one-parameter family of martingales. To solve first-passage problems at finite thresholds, we generally require all one-parameter families of martingales, including the non-positive ones. We illustrate this by solving the first-passage problem for run-and-tumble particles exiting an interval of finite width.