Non-Markovian reaction sites and trapping

Most models of absorption sites for random walks or diffusion processes fall into one of two categories: (1) Perfect absorption, in which every encounter of a random walker with a trap produces a trapping event, and (2) imperfect absorption in which an encounter leads to a trapping event with probability α<1. We introduce the notion of a non-Markovian trap characterized by a set of probabilities {f j}, where f j is the probability that the jth encounter leads to a trapping event. Some consequences of this assumption are examined in the context of a one-dimensional trapping problem. It is shown that when the f j have an associated finite first moment the asymptotic survivial probability goes like n1/2 exp(−an1/3) where n is the step number and a is a constant. This is equivalent to the results one would obtain with a Markovian model. However, when f j is asymptotically proportional to 1/j1+α where 0<α<1 the survival probability falls off as 1/nα.