Interpolating between random walk and rotor walk

AbstractWe introduce a family of stochastic processes on the integers, depending on a parameter and interpolating between the deterministic rotor walk () and the simple random walk (). This p‐rotor walk is not a Markov chain but it has a local Markov property: for each the sequence of successive exits from 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 , where is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation urn:x-wiley:10429832:media:rsa20747:rsa20747-math-0008 for all . Here is a standard Brownian motion and are constants depending on the marginals of the initial rotors on and respectively. Chaumont and Doney have shown that Equation 1 has a pathwise unique solution , and that the solution is almost surely continuous and adapted to the natural filtration of the Brownian motion. Moreover, and . 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 .