Interpolating between random walk and rotor walk

We introduce a family of stochastic processes on the integers, depending on a parameter p ∈ [0, 1] and interpolating between the deterministic rotor walk (p = 0) and the simple random walk (p = 1/2). This p-rotor walk is not a Markov chain but it has a local Markov property: for each x ∈ Z the sequence of successive exits from x is a Markov chain. The main result of this paper identifies the scaling limit of the p-rotor walk with two-sided i.i.d. initial rotors. The limiting process takes the form √︁ 1−p p X(t), where X is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation X(t) = B(t) + a sup s≤t X(s) + b inf s≤t X(s) (1) for all t ∈ [0,∞). Here B(t) is a standard Brownian motion and a, b < 1 are constants depending on the marginals of the initial rotors on N and−N respectively. Chaumont and Doney have shown that equation (1) has a pathwise unique solution X(t), and that the solution is almost surely continuous and adapted to the natural filtration of the Brownian motion [CD99]. Moreover, lim supX(t) = +∞ and lim infX(t) = −∞ [CDH00]. This last result, together with the main result of this paper, implies that the p-rotor walk is recurrent for any two-sided i.i.d. initial rotors and any 0 < p < 1. 2010 Mathematics Subject Classification. 60G42, 60F17, 60J10, 60J65, 60K37, 82C41.