First passage time distribution in random walks with absorbing boundaries

We calculate the first passage time distribution in simple, unbiased random walks in presence of absorbing boundaries of various shapes. We obtain explicit solutions for the following geometries of the boundaries—a box in one dimension, circular, square and triangular boundaries in two dimensions and cubical box and spherical shell in three dimensions. The distribution in all cases shows scaling and the scaling function can be expressed in terms of the Jacobi Theta functions.