On multidimensional locally perturbed standard random walks

Abstract

Let d be a positive integer, and let A be a set in \({\mathbb{Z}}^{d}\) that contains finitely many points with integer coordinates. We consider a standard random walk X perturbed on the set A. This means that X is a Markov chain whose transition probabilities from the points outside A coincide with those of a standard random walk on \({\mathbb{Z}}^{d}\), whereas the transition probabilities from the points inside A are different. We investigate the impact of the perturbation on a scaling limit of X. It turns out that if d ⩾ 2, then in a typical situation the scaling limit of X coincides with that of the underlying standard random walk. This is unlike the case d = 1, in which the scaling limit of X is usually a skew Brownian motion, a skew stable Lévy process, or some other “skew” process. The distinction between the one-dimensional and multidimensional cases under comparable assumptions may simply be caused by transience of the underlying standard random walk in \({\mathbb{Z}}^{d}\) for d ⩾ 3. More interestingly, in the situation where the standard random walk in \({\mathbb{Z}}^{2}\) is recurrent, the preservation of its Donsker scaling limit is secured by the fact that the number of visits of X to the set A is much smaller than in the one-dimensional case. As a consequence, the influence of the perturbation vanishes upon the scaling. On the other edge of the spectrum, we have the situation in which the standard random walk admits a Donsker’s scaling limit, whereas its locally perturbed version does not because of huge jumps from the set A, which occur early enough.