Delayed random walks

The fluctuations about the stable point in a delayed dynamical system are modeled as a delayed random walk: i.e., a random walk in which the transition probability depends on the position of the walker at a time {τ} in the past and transitions in the direction of the stable point are more probable. It is shown that, depending on the magnitude of the delay, the root mean square displacement {}〈X²(t)〉 versus time interval approaches a limiting value in either an oscillatory or nonoscillatory fashion. This limiting value of {}〈X²(t)〉 is a linear function of {τ}.