标题： 首通量性质的跳跃过程与漂移。 两个完全可解的情形 显示英文标题

标题： First-passage properties of the jump process with a drift. Two exactly solvable cases

摘要： 我们研究了一种具有恒定漂移的跳跃过程的第一通量性质，重点关注两个关键可观测量：首次穿越时间$\tau$和首次穿越事件之前跳跃次数$n$。 通过将问题映射到一个有效离散时间随机游走，我们利用广义 Pollaczek-Spitzer 公式推导出联合分布$\tau$和$n$的拉普拉斯变换的精确表达式。 这一结果随后被用于分析两种完全可解的情况：(i) 跳跃间隔和跳跃幅度都呈指数分布；(ii) 跳跃间隔呈指数分布而所有跳跃具有相同固定幅度。 我们展示了由漂移强度控制的两个不同区域的存在：(i) 生存区域，其中过程以有限概率始终为正；(ii) 吸收区域，其中最终会发生首次穿越；以及(iii) 这两个相之间的临界点。 我们刻画了每个区域生存概率的渐近行为：它们在生存区域内以指数方式衰减至常数，在吸收区域内以指数速度消失，并在临界点处表现出幂律衰减。 此外，在吸收区域内，我们推导了边际分布$\tau$和 n 的大偏差形式。通过广泛的数值模拟验证了这些解析预测。 显示英文摘要

摘要： We investigate the first-passage properties of a jump process with a constant drift, focusing on two key observables: the first-passage time $\tau$ and the number of jumps $n$ before the first-passage event. By mapping the problem onto an effective discrete-time random walk, we derive an exact expression for the Laplace transform of the joint distribution of $\tau$ and $n$ using the generalized Pollaczek-Spitzer formula. This result is then used to analyze the first-passage properties for two exactly solvable cases: (i) both the inter-jump intervals and jump amplitudes are exponentially distributed, and (ii) the inter-jump intervals are exponentially distributed while all jumps have the same fixed amplitude. We show the existence of two distinct regimes governed by the strength of the drift: (i) a survival regime, where the process remains positive indefinitely with finite probability; (ii) an absorption regime, where the first-passage eventually occurs; and (iii) a critical point at the boundary between these two phases. We characterize the asymptotic behavior of survival probabilities in each regime: they decay exponentially to a constant in the survival regime, vanish exponentially fast in the absorption regime, and exhibit power-law decay at the critical point. Furthermore, in the absorption regime, we derive large deviation forms for the marginal distributions of $\tau$ and n. The analytical predictions are validated through extensive numerical simulations.