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 form1 − p pX ( t ) , where X is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation 1 X ( t ) = B ( t ) + a sup ⁡ s ≤ t X ( s ) + b inf ⁡ s ≤ t X ( s ) 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. Moreover, lim ⁡ sup ⁡ X ( t ) = + ∞ and lim ⁡ inf ⁡ X ( t ) = − ∞ . 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 .