Discrete random walk with geometric absorption

We consider a discrete random walk (RW) in n dimensions . The RW is adapted with a geometric absorption process: at any discrete time there is a constant probability that absorption occurs in the current state. To model the RW with geometric absorption we use the concept of a multiple function barrier (MFB). In a MFB there is a modification of the original RW: each transition probability in the original RW is multiplied by {\beta} and there is an additional probability (1-{\beta}) of absorption, where 0<{\beta}<1. We study three cases: one-dimensional simple asymmetric RW, n-dimensional simple symmetric RW (n>1) and a two level RW.