Conditional expectation of the duration of the classical gambler problem with defects
The effect of space inhomogeneities on a diffusing particle is studied in the framework of the 1D random walk. The typical time needed by a particle to cross a one-dimensional finite lane, the so-called residence time, is computed possibly in presence of a drift. A local inhomogeneity is introduced as a single defect site with jumping probabilities differing from those at all the other regular sites of the system. We find complex behaviors in the sense that the residence time is not monotonic as a function of some parameters of the model, such as the position of the defect site.