Limit theorems for persistent random walks in cookie environments

Abstract

Excited random walks (ERW) or random walks in a cookie environment is a modification of the nearest neighbor simple random walk such that in several first visits to each site of the integer lattice, the walk’s jump kernel gives a preference to a certain direction and assigns equal probabilities to the remaining directions. If the current location of the random walk has been already visited more than a certain number of times, then the walk moves to one of its nearest neighbors with equal probabilities. The model was introduced by Benjamini and Wilson and extended by Martin Zerner. In the cookies jargon, upon first several visits to every site of the lattice, the walker consumes a cookie providing them a boost toward a distinguished direction in the next step. The excited random walk is a popular mainstream model of theoretical probability. An interesting application of this model to the motion of DNA molecular motors has been discovered by Antal and Krapivsky (Phys. Review E, 2007), see also the article of Mark Buchanan Attack of the cyberspider in Nature Physics, 2009. Many basic asymptotic properties of excited random walk have their counterparts for random walk in random environment (RWRE). The major difference between two processes is that while the random (cookie) environment is dynamic and rapidly changes with time the environments considered in the RWRE process are stationary both in space and in time. The similarity between the asymptotic behaviors of these two classes of random walks can be explained using the fact that certain functionals (for instance, exit times and exit probabilities) of the local time (or occupation time, also referred to as the number of previous visits to a current location) process converge after a proper rescaling to diffusion processes with timeindependent coefficients. Thus phenomenon, discovered by Kosygina and Mpountford, can be exploited for a heuristic explanation of the analogy between the role of the local drift of ERW