Asymptotic analysis of a random walk with a history-dependent step   length
  We study an unbiased, discrete time random walk on the nonnegative integers, with the origin absorbing. The process has a history-dependent step length: the walker takes steps of length v while in a region which has been visited before, and steps of length n when entering a region that has never been visited. The process provides a simplified model of spreading in systems with an infinite number of absorbing configurations. Asymptotic analysis of the probability generating function shows that, for large t, the survival probability decays as S(t) \sim t^{-delta}, with delta = v/2n. Our expression for the decay exponent is in agreement with results obtained via numerical iteration of the transition matrix. 
Ronald Dickman, Francisco Fontenele Araujo, Jr., and Daniel   ben-Avraham