Title: First-passage properties of the jump process with a drift. The general case Show Chinese title

Title: 跳跃过程带漂移的第一 passage 性质 一般情况

Abstract: We study the first-passage properties of a jump process with constant drift where jump amplitudes and inter-arrival times follow arbitrary light-tailed distributions with smooth densities. Using a mapping to an effective discrete-time random walk, we identify three regimes determined by the drift strength: survival (weak drift), absorption (strong drift), and critical. We derive explicit expressions for exponential decay rates in the survival and absorption regimes, and characterize algebraic decay at the critical point. We also obtain asymptotic behavior of the mean first-passage time, number of jumps, and their variances for processes starting either close to the origin or far from it. Show Chinese abstract

Abstract: 我们研究了一个具有恒定漂移的跳跃过程的首次通过性质，其中跳跃幅度和到达时间间隔遵循任意轻尾分布且具有平滑密度。 通过映射到一个有效的离散时间随机游走，我们确定了由漂移强度决定的三个区域：存活（弱漂移）、吸收（强漂移）和临界。 我们推导了在存活和吸收区域中指数衰减率的显式表达式，并表征了临界点处的代数衰减。 我们还得到了从接近原点或远离原点开始的过程的平均首次通过时间、跳跃次数及其方差的渐近行为。