First-passage times and solutions of the telegrapher equation with boundaries

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Abstract
The first-passage-time distribution function (FPTDF) is derived for a relativistic random walk (RW) in one-dimensional (1D) space limited by one boundary. Thereby two configurations are considered with the start of the RW directed towards or away from the boundary. For each configuration the distribution function (DF) of the RW path ends is evaluated subjected to either absorption or reflection occurring at the boundary. The DF’s are identical with the solutions of the telegrapher equation in 1D space under the same initial and boundary conditions. The calculus can be extended to two boundaries, where all combinations of reflecting and absorbing boundaries are possible, for which the Laplace transforms of the path-end DF’s and the FPTDF’s are presented. The derivation is demonstrated and cross-checked for the case of two absorbing boundaries.