Distribution of the span of one-dimensional confined random processes before hitting a target

We derive the distribution of the number of distinct sites visited by a random walker before hitting a target site of a finite one-dimensional (1D) domain. Our approach holds for the general class of Markovian processes with connected span-i.e., whose trajectories have no holes. We show that the distribution can be simply expressed in terms of splitting probabilities only. We provide explicit results for classical examples of random processes with relevance to target search problems, such as simple symmetric random walks, biased random walks, persistent random walks, and resetting random walks. As a by-product, explicit expressions for the splitting probabilities of all these processes are given. Extensions to reflecting boundary conditions, continuous processes, and an example of a random process with a nonconnected span are discussed.

Automated summary

This study examines the distribution of the number of distinct sites visited by a random walker before hitting a target site in a finite one-dimensional domain, expressed in terms of splitting probabilities. Explicit results are provided for classical random processes such as simple symmetric random walks and biased random walks. Additionally, extensions to reflecting boundary conditions and continuous processes are discussed.