Trinomial random walk with one or two imperfect absorbing barriers

Trinomial random walk, with one or two barriers, on the non-negative integers is discussed. At the barriers, the particle is either annihilated or reflects back to the system with respective probabilities 1 - rho, rho at the origin and 1 - omega, omega at L, 0 <= rho, omega <= 1. Theoretical formulae are given for the probability distribution, its generating function as well as the mean of the time taken before absorption. In the one-boundary case, very qualitatively different asymptotic forms for the result, depending on the relationship between transition probabilities and the annihilation probability, are obtained.