2016/03/14 by Huss, Wilfried, Levine, Lionel, Sava-Huss, Ecaterina
#60F17 #60G42 #60J10 #60J65 #60K37 #82C41 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1603.04107
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 sups≤ t X(s) + b infs≤ 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 [CD99] have shown that the above equation 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, \limsup X(t) = +∞ and \liminf X(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