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First-passage time, the moment a process first satisfies a condition (e.g., reaching a specific spatial point for the first time), is a fundamental concept in stochastic dynamics. In this seminar, we explore recent tools for analyzing first-passage statistics in both continuous and discrete space-time. In the continuous space-time setting, we show how Martingales can be employed with remarkable efficiency in obtaining first-passage statistics. On the other hand, we use Montroll's defect technique to tackle the first-passage problems in discrete space-time. Using these approaches, we analyze first-passage problems in various Markovian and non-Markovian processes, including biased diffusion, its area functional, and run-and-tumble walks with applications in movement ecology. |