Method of images for one-dimensional discrete random walk under a reflecting barrier

The transition probability for a one-dimensional discrete symmetric random walk under a reflecting barrier was once given by the method of images [S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943)]. However, several inconsistencies have been reported when the method of images is applied in cases where a reflecting barrier is considered, even after the exact solution has been obtained. Here, we explicitly show that the method of images becomes applicable if the image position is shifted. We also discuss the implications of the fluctuation theorem in infinite discrete lattice models.