This paper concerns a random walk that moves on the integer lattice and has zero mean and a finite variance. We obtain first an asymptotic estimate of the transition probability of the walk absorbed at the origin, and then, using the obtained estimate, that of the walk absorbed on a half line. The latter is used to evaluate the space-time distribution for the first entrance of the walk into the half line.
"One dimensional lattice random walks with absorption at a point/on a half line." J. Math. Soc. Japan 63 (2) 675 - 713, April, 2011. https://doi.org/10.2969/jmsj/06320675