Exact expressions of mean first-passage times and splitting probabilities for random walks in bounded rectangular domains

The exact expressions of the pseudo-Green functions involved in the mean first-passage times of a random walker in both cases of a rectangular domain with reflected boundaries and a rectangular domain with periodic boundary conditions were investigated. An equation for mean time taken by a random walker starting at a source of a particular position of a rectangular regular lattice with reflecting boundary conditions to reach a particular target of another position for the first time was described. It was found that in the case of a rectangular domain with reflected boundary the exact expression of the pseudo-Green function H involved in equations for the first-passage times may be computed explicitly with the help of Fourier analysis. In the case of a rectangular domain with periodic boundary conditions the equation for pseudo-Green function was presented and the results were found to be similar to those of the rectangular domain with reflected boundaries.